Experimental analysis of heuristics for the bottleneck traveling salesman problem Abstract. In this paper we develop efficient heuristic algorithms to solve the bottleneck traveling salesman problem (BTSP). Results of extensive computational experiments are reported. Our heuristics produced optimal solutions for all the test problems considered from TSPLIB, JM-instances, National TSP instances, and VLSI TSP instances in very reasonable running time. We also conducted experiments with specially constructed 'hard' instances of the BTSP that produced optimal solutions for all but seven problems. Some fast construction heuristics are also discussed. Our algorithms could easily be modified to solve related problems such as the maximum scatter TSP and testing hamiltonicity of a graph. Let G = (V, E) be a complete undirected graph on n nodes. For each edge (i, j) ∈ E a non-negative cost c ij is prescribed. Then the bottleneck traveling salesman problem (BTSP) is to Minimize max{c ij : (i, j) ∈ H} Subject to H ∈ Π, where Π is the collection of all Hamiltonian cycles (tours) in G. The matrix C = (c ij ) n×n is called the cost matrix associated with the BTSP.. To the best of our knowledge, BTSP was introduced by Gilmore and Gomory [11] in 1964, where the edge costs were assumed to be specially structured. Related works on special cases of the BTSP include various extensions of the Gilmore and Gomory class of problems [17, 18, 20, 23, 40, 41] and BTSP on a Halin graph [27] . Many of these special cases can be solved in polynomial time. For a state-of-the-art discussion on polynomially solvable cases of the BTSP, we refer to book chapter by Kabadi and Punnen [18] . The BTSP with a general cost matrix was first studied by Gabovich et al. [9] in 1971. The BTSP is well known to be NP-hard. In fact, unless P = NP, This work was supported by NSERC discovery grants awarded to Abraham P. Punnen and Eric Aubanel.. there does not exist a polynomial time -approximation algorithm for the BTSP for any > 0 [6, 26, 34] . However, polynomial algorithms with guaranteed performance ratios exist for various special cases. For example, when the edge costs satisfy the τ -triangle inequality (i.e. c ij ≤ τ (c ik + c kj ) for all i, j, k ∈ V and τ ≥ 1 2 ), a 2τ -approximate solution can be obtained in polynomial time [18] and this performance bound seems to be the best possible for problems in this class. That is, unless P = NP, there exists no polynomial 2τ − approximation algorithm for the BTSP for any > 0 even if the edge costs satisfy the τ -triangle inequality for τ > 1/2 [18] . The special case of the above result where τ = 1 (i.e. the edge costs satisfy the triangle inequality) has been obtained independently by Parker and Rardin [26] and Doroshko and Sarvanov [6] . For similar results on other bottleneck problems, including the BTSP, we refer to Hochbaum and Shmoys [13] . It is well known that the Euclidean TSP can be solved by a polynomial approximation scheme [3] . However, for any > 0 it is NP-hard to obtain a (2 − )-approximate solution for the BTSP even with a Euclidean cost matrix [34] . In this sense the Euclidean version of BTSP is harder than its TSP counterpart.. Despite these negative results, our main heuristic algorithms produced optimal solutions (in many cases with a proof of optimality) for almost all of the test problems considered, even for problems with over 30,000 nodes. Furthermore, these excellent computational results were achieved in a very reasonable computational time.. All of the known exact algorithms for the BTSP are of an enumerative type. Garfinkel and Gilbert [8] proposed a branch and bound algorithm to solve the BTSP and its directed version. They reported computational results on random problems with the number of nodes ranging from 10 to 100. Carpeneto, Martello and Toth [5] reported computational results with branch and bound algorithms on random problems of size between 40 to 200 nodes. Sergeev [35, 36] proposed dynamic programming based algorithms to solve the BTSP. One of our heuristic algorithms can easily be modified to get an exact optimal solution for the BTSP.. Although heuristic algorithms for the traveling salesman problem (TSP) have been thoroughly investigated in literature with well designed experimental analysis [16] , systematic experimental analysis of heuristic algorithms for the BTSP reported in literature are limited to small-size problems. Garfinkel and Gilbert reported experimental results using a construction heuristic on random problems of sizes up to 100 nodes. The algorithm produced optimal solutions in many cases. In one set of problems the solution quality appeared to be decreasing as the problem size varied from 10 to 100. Ramakrishnan, Punnen and Sharma [30] considered a simple binary search based threshold algorithm using the 2-max bound [23] as a starting lower bound and the objective function value of the nearest neighbor heuristic as the upper bound. The 2 − max bound is obtained as follows: For each node i find the second smallest cost of an edge incident on i, counting multiplicities. Let ζ(i) be this value. Note that if the smallest cost of an arc incident on node i is not unique, then ζ(i) takes the value of this smallest cost. The 2-max bound is defined as max{ζ(i) | i ∈ V }. Computational results on 72 TSPLIB problems of sizes up to 783 nodes are reported in [30] with optimality guaranteed for 37 problems. The optimality of these solutions were established by showing that the objective function value of the solution produced was equal to a lower bound to the optimal objective function value. The optimality of the remaining 35 problems was not proven because of the low quality of the 2-max lower bound. Our experiments show that the solutions produced by the algorithm of Ramakrishnan et al [30] for the remaining 35 problems were also optimal. They also discussed a generalized threshold algorithm without computational results.. In this paper we develop several heuristics for the BTSP. One of our algorithms could be viewed as a generalization of the threshold heuristic of Ramakrishnan et al. [30] . Results of extensive computational experiments conducted using these algorithms are reported. The algorithms produced provably optimal solutions for all problems from TSPLIB with sizes varying from 14 to 18512 nodes with the exception of linhp318 1 . We could not attempt to solve pla33810 and pla85900 due to memory limitations. Our algorithms also produced provably optimal solutions on random problems generated by the TSP instance generator of Johnson and McGeoch [14, 16, 32] of sizes 1000 to 30,000 from the categories of Euclidean problems with random uniform points, Euclidean problems with random clustered points, and random distance matrices. Further, we obtained optimal solutions to the National TSP instances and VLSI TSP instances of [37] . All these excellent experimental results were achieved within a very reasonable computational time. Our implementation used an explicit cost matrix and hence we could not solve problems of size more than 32000 cities due to memory limitations.. The optimality of many of the solutions is established by showing that the objective function values of the solutions obtained are equal to a lower bound. For some solutions without such proof of optimality, we used an exact solver obtained by modifying our heuristic algorithm to verify optimality. Since for most of the standard TSP test problems the lower bound value was optimal, we have tested the algorithms on specially constructed hard instances that are generated in such a way that our lower bound for the optimal objective function value is not equal to the optimal objective function value of the corresponding BTSP. Out of 81 such hard instances all except seven problems produced optimal solutions. In addition to these powerful (heuristic) algorithms, we also have results on some simple and 1 We did not test linhp318 because its format in TSPLIB was different from other problems and required separate processing to be consistent with our input format fast heuristics which are adaptations of the corresponding TSP heuristics. As expected, the performance of these heuristics was not as good as that of our primary algorithms in terms of solution quality, but these algorithms computed reasonable solutions quickly.. Another problem closely related to the BTSP is the maximum scatter traveling salesman problem (MSTSP) [2] which is defined as follows:. Maximize min{c ij : (i, j) ∈ H} Subject to H ∈ Π Any instance of the MSTSP can be reduced to an equivalent instance of the BTSP using cost matrixC = (c ij ) n×n wherec ij = max rs∈E c rs − c ij . Although this transformation preserves optimality, it does not preserve theoretical approximation ratios. Nevertheless, our BTSP heuristics can be used to solve MSTSP using these transformations.. Applications of the BTSP model are not widely discussed in the literature as that of the TSP. However, many applications of the TSP model have an interpretation in the BTSP context. Specific applications of the BTSP model discussed in the literature include sequencing problems on single state variable machines [11] , minimizing makespan in a two machine flow-shop [31] , printed circuit board assembly [4] , reconstructing sequential order from inaccurate adjacency information [19] , medical image processing [2] , and obtaining riveting sequences in joining metals in the aircraft industry [2] .. The paper is organized as follows: In Section 2 we discuss our heuristic algorithms for BTSP. Computational results with these algorithms are reported in Section 3 and concluding remarks are given in Section 4.

B I O O O O O B I I O O O O O O O B I O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O B I I O O O O O O O O O O B I O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O


Lateness minimization with Tabu search for job shop scheduling problem with sequence dependent setup times Abstract We tackle the job shop scheduling problem with sequence dependent setup times and maximum lateness minimization by means of a tabu search algorithm. We start by defining a disjunctive model for this problem, which allows us to study some properties of the problem. Using these properties we define a new local search neighborhood structure, which is then incorporated into the proposed tabu search algorithm. To assess the performance of this algorithm, we present the results of an extensive experimental study, including an analysis of the tabu search algorithm under different running conditions and a comparison with the state-of-the-art algorithms. The experiments are performed across two sets of conventional benchmarks with 960 and 17 instances respectively. The results demonstrate that the proposed tabu search algorithm is superior to the state-of-the-art methods both in quality and stability. In particular, our algorithm establishes new best solutions for 817 of the 960 instances of the first set and reaches the best known solutions in 16 of the 17 instances of the second set. In this paper we confront the Job Shop Scheduling Problem with Sequence Dependent Setup Times (SDST-JSP) and maximum lateness minimization. This is a variant of the classical Job Shop Scheduling Problem (JSP) of great interest that has been considered in a number of surveys and research works, (Ovacik and Uzsoy, 1994; Balas et al, 2008; Artigues and Feillet, 2008; Oddi et al, 2009 ). This kind of problems arise, for instance, in automobile, printing, semiconductor, chemical or pharmaceutical industries (Allahverdi et al, 2008) . For instance, (Ovacik and Uzsoy, 1994) presented an extensive study of a digital testing facility of large semiconductor manufacturing, where numerous integrated circuits are tested and the test workcenters have sequence dependent setup times. A benchmark inspired on these problems was proposed therein which has since then been used by a number of researches. Another example is described in (Pinedo, 2008) regarding a paper bag factory producing bags for different uses. The production process consists of three stages (printing, gluing and sewing) and setup times are required when a machine switches from one bag to another, their values depending on the similarity between bags (color, size, etc.). Setup considerations are a relevant characteristic that changes the nature of scheduling problems, so well-known results and techniques for the JSP are not directly applicable to the SDST-JSP. Some extensions are however possible, as done for the case of makespan minimization in (Brucker and Thiele, 1996; Vela et al, 2010) .. In the sequel, we propose to exploit neighbourhood structures to deal with lateness minimization for the SDST-JSP. We start by defining a new disjunctive graph representation for the SDST-JSP with lateness minimization which is similar to other representations proposed for the classical JSP, for example for lateness minimization (Mehta and Uzsoy, 1998) or considering deadlines and makespan minimization (Balas and Lancia, 1998) . Given this new graph, we extend a neighborhood structure proposed for makespan minimization in (González et al, 2008) . The resulting structure, denoted N S L , is then incorporated to a Tabu Search (TS) algorithm, T S − N S L . For the experimental study we consider the benchmarks proposed in (Ovacik and Uzsoy, 1994) and in (Brucker and Thiele, 1996) , and compare T S − N S L with state-of-the-art approaches from (Balas et al, 2008) and (Oddi et al, 2009 ), illustrating the good performance of our proposal and obtaining new best-known values in some cases.. Setup considerations in scheduling started almost four decades ago in (Wilbrecht and Prescott, 1969) , where the authors discovered through a simulation study that sequencedependent setup times play a critical role in the performance of a job shop operating near full capacity. Since then, a number of approaches to scheduling problems have been proposed that take into account setup times. These have been reviewed in (Armentano and Filho, 2007) ; more recent work on a variety of scheduling problems involving setup times can be found, for example, in (Ying et al, 2011; Fleszar et al, 2011; Huang et al, 2011) . The SDST-JSP with makespan minimization was first considered in (Brucker and Thiele, 1996) , where the authors proposed a branch and bound algorithm and experimented on the BT set. More recently, in González et al, 2008 ) two hybrid approaches have been proposed that combine a genetic algorithm with local search procedures, obtaining the best-known solutions for the largest instances of the BT set.. In (Uzsoy and Velásquez, 2008 ) the authors make it clear the relevance of lateness as optimization criteria, motivating the formulation of the shop-floor scheduling problem as that of minimizing maximum lateness (L max ), "since this will avoid making some jobs early at the expense of others being extremely delayed". In this paper, the authors establish a relationship between both optimization criteria, L max and C max . In fact, their local search algorithm, termed MEDD, attempts to reduce the makespan at an intermediate step, and this reduction is expected to lead a further reduction in L max as well.. The SDST-JSP with maximum lateness minimization has been considered in a number of research works. For instance, (Ovacik and Uzsoy, 1994) proposed a family of heuristics that use global information to make local decisions at the machine level and experimented with the above mentioned instances, inspired by semiconductor industry. Their results were later improved in (Balas et al, 2008) with the algorithm SB-GLS. More recently, in (Oddi et al, 2009 ) the authors have proposed an Iterative-Sampling Search (ISS) method, which consists of a constraint-based search embedded within an iterative sampling framework.. Many solving methods for combinatorial optimization use heuristic techniques. One of the most popular heuristics is Tabu Search (TS) (Glover, 1989) , with a solid record of good empirical performance in problem solving. In particular, in the field of scheduling, the first TS algorithm for the JSP with makespan minimization was proposed in (Taillard, 1993) , outperforming earlier methods based on simulated annealing or shifting bottleneck heuristics. In (Dell' Amico and Trubian, 1993) new neighborhood structures were introduced and used used in a TS algorithm, using new elements such as makespan estimations, a dynamic tabu list which only stores inverted arcs instead of complete solutions or a simple mechanism to avoid cycles. In (Nowicki and Smutnicki, 1996) , the authors proposed the TSAB algorithm, introducing the elite solution stack. This algorithm was improved and combined with path-relinking in (Nowicki and Smutnicki, 2005) , resulting in the method termed i-TSAB. This has recently been combined with a constructive search procedure in (Beck et al, 2010) . In (Grabowski and Wodecki, 2005) , some new properties of critical blocks were analyzed and exploited in a TS algorithm. In (Zhang et al, 2008) , TS was combined with simulated annealing to introduce new solutions in the elite solution stack; this algorithm and i-TSAB are among the best algorithms for the JSSP with makespan minimization. In (Watson et al, 2006) , a thorough study of the i-TSAB algorithm was carried out showing that it is the combination of all the elements of i-TSAB that yields so good results.. TS has also been applied to formal scheduling problems other than JSP with makespan minimization: the parallel machine scheduling problem with setup times and tardiness minimization (Bilge et al, 2004) , the JSP with release dates and time lags (DeBontridder, 2005) or the one machine sequencing problem with setup times (Stecco and Cordeau, 2009) , to name just a few. Moreover, real scheduling problems have also been solved by means of TS algorithms. For example, in (Yan et al, 2003) , the authors propose a solution to an integrated planning and scheduling problem in automobile assembling lines that optimizes several objective functions. In (Meeran and Morshed, 2011) , TS is combined with a genetic algorithm and this hybrid approach is able to solve a number of real-life job shop problems.. The rest of the paper is organized as follows. In Section 2 we formulate the SDST-JSP and introduce the notation used across the paper. Then, in Sections 3 and 4, the TS algorithm and the neighborhood structure are described. Section 5 reports results from the experimental study. Finally, in Section 6 we summarize the main conclusions and propose ideas for future work. We have proposed a method, T S − N S L , to solve the SDST-JSP with maximum lateness minimization. We have presented a series of experiments which show that T S − N S L outperforms SB-GLS -one of the best methods in the literature, under equivalent running times. It also improves the results of ISS, although in this case we have to be aware of the difference in the running conditions. Additionally, we have shown the robustness of our method to changes in the parameters used to define problem instances from (Ovacik and Uzsoy, 1994) . Finally, we have seen how the results can be further improved when T S − N S L is given longer running time. Indeed, in our experiments, T S − N S L has established new best-known solutions for 817 of the 960 instances in this difficult benchmark, which should serve as reference for future research. We may conclude that T S − N S L has proved to be a highly competitive method for the SDST-JSP with maximum lateness minimization. We believe the strongest points of T S − N S L to be the powerful tabu search algorithm, already successful in other similar problems (Dell' Amico and Trubian, 1993; Nowicki and Smutnicki, 1996) as well as the neighborhood structure N S L . This structure has been specifically designed for this problem and is therefore very well adapted to its characteristics, in particular, to setup times (even those not fulfilling the triangular inequality) and the objective function (maximum lateness). In principle, N S L requires exploring a large number of moves. However, the theoretical results in Theorems 1 and 2 allow to discard a great number of candidate neighbors which either are not feasible or cannot improve the current schedule, so the number of actual neighbors in N S L can be managed efficiently. Additionally, the algorithm given in Section 4.1 is very efficient and accurate in estimating the maximum lateness of a new neighbor, giving T S − N S L the chance of selecting one of the best neighbors. Given the specialization of the defined neighborhood, the algorithm would require a series of modifications in order to solve other versions of this problem with different characteristics or objective functions. In particular, a new disjunctive model would be required that accurately represents these characteristics and new formal results should be established to devise a new efficient neighborhood structure.. A first task for future work is to analyze the advantages and disadvantages of incorporating to our TS algorithm different strategies for handling long term memory, such as those used by two of the most competitive algorithms for classical JSP: path relinking of i-TSAB from (Nowicki and Smutnicki, 2005) or simulated annealing from (Zhang et al, 2008) . In particular, it will be necessary to study its effectiveness when faced with the increase in complexity caused by setup times. We also intend to combine T S − N S L with a genetic algorithm, expecting that the combined approach will further improve the solutions. Additionally, we plan to extend our approach to other variants or extensions of classical JSP which are closer to real environments and usually result harder to solve. For example the JSP with uncertain durations (González Rodríguez et al, 2010) or the JSP with other objective functions such as total flow time (Sierra and Varela, 2010) or weighted tardiness, as well as multiobjective problems.

B I O B I O O O B O O O O B I O O O O O O B O O O O B I O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B I O O O B I O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B I O O O O O O O O O O O O O O O O O B O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Maximal f-vectors of Minkowski sums of large numbers of polytopes It is known that in the Minkowski sum of r polytopes in dimension d, with r < d, the number of vertices of the sum can potentially be as high as the product of the number of vertices in each summand [2] . However, the number of vertices for sums of more polytopes was unknown so far.In this paper, we study sums of polytopes in general orientations, and show a linear relation between the number of faces of a sum of r polytopes in dimension d, with r ≥ d, and the number of faces in the sums of less than d of the summand polytopes. We deduce from this exact formula a tight bound on the maximum possible number of vertices of the Minkowski sum of any number of polytopes in any dimension. In particular, the linear relation implies that a sum of r polytopes in dimension d has a number of vertices in O(n d−1 ) of the total number of vertices in the summands, even when r ≥ d. This bound is tight, in the sense that some sums do have that many vertices. The Minkowski sum of two polytopes is defined as P 1 + P 2 = {x 1 + x 2 : x 1 ∈ P 1 , x 2 ∈ P 2 }. Minkowski sums are of interest in various fields of theoretical and applied mathematics. While some applications only require sums of two polytopes in low dimensions (e.g. motion planning [7] [8]), others use iterative sums of many polytopes in higher dimensions [9] [12] . It is therefore desirable to study the complexity of such sums.. A trivial bound on the number of vertices of a sum is found as follows. Every vertex of a Minkowski sum decomposes into a sum of vertices of the summands. Therefore, there cannot be more vertices in the sum than there are possible decompositions. Thus, a trivial bound on the number of vertices in a Minkowski sum is the product of the number of vertices in the summands. That is, if P 1 , . . . , P r are polytopes, and f 0 (P ) is the number of vertices of a polytope, then f 0 (P 1 + · · · + P r ) ≤ f 0 (P 1 ) · · · f 0 (P r ).. If we sum r polytopes in dimension d, with r < d, then the trivial bound is tight, that is, it is possible to choose summands with any number of vertices so that their sum has as many vertices as the trivial bound [2] .. However, if we sum r polytopes in dimension d with r ≥ d, the trivial bound cannot be reached, except when summing d segments [10] [11] . We assume here and in the rest of the article that all polytopes have at least two vertices, since a summand of only one vertex can be ignored without changing the properties of the sum.. The f-vector of a polytope encodes its number of faces of different dimensions. Maximal f-vectors are obtained for a particular class of Minkowski sums, called sums of polytopes in general orientations. We will therefore restrict our study to such sums.. We recently presented in [1] a result on sums of 3-dimensional polytopes in general orientations. We showed that the number of vertices in a sum of r summands can be deduced from the number of vertices in the summands and the number of vertices in sums of each of the r 2 pairs of summands. Using this result, we found a tight upper bound on the number of vertices and facets in sums of 3-dimensional polytopes.. The basic reasoning of this previous result is to define a unique witness, called western-most corner, for all but two vertices of a polytope. These witnesses have the property that a western-most corner for a Minkowski sum of any number of summands is also necessarily a western-most corner for the sum of some pair of the summands. So the number of western-most corners in the total sum, and thus its number of vertices, can be found by examining sums of one or two summands only.. This prompted us to examine the possibility of extending the reasoning to higher dimensions and other faces, which resulted in this article. Our main result is presented in Theorem 1. It is a linear relation between the number of faces of a sum of r polytopes and the number of faces in the sums of less than d of the summand polytopes:. where C r j is the family of subsets of {1, . . . , r} of cardinality j, P S is the sum of polytopes i∈S P i ; α = 2 if k = 0 and d is odd, α = 0 otherwise.. A slightly more general result also applies when summands are not full-dimensional. The intuitive explanation of the theorem is that for any face of the whole sum, we can find a witness of its existence by examining the faces of the same dimension in sums of d − 1 summands. However, if that witness exists in some sum of d − 2 summands, we will find it in many different sums of d − 1 summands. So we need to offset this by removing an appropriate number of times the witnesses in sums of d − 2 summands. But that in turn removes too many times witnesses that exist in some sum of d − 3 summands, so we need to add them back, etc. This implies that the total sum is smaller than the term for j = d − 1:. From this result, we deduce bounds for the maximum possible number of vertices in a Minkowski sum of polytopes, for fixed number of vertices in the summands.. We find in particular that a sum of r polytopes in dimension d, r ≥ d, where summands have n vertices in total, has less than. In the case where each summand has at most n vertices, then the number of vertices of the sum is less than. . This is better than the previous known bound which was in O(r d−1 n 2(d−1) ) [5] .. In the rest of the article, we shortly present the theory in Section 2. We first give an introduction to the concepts of west and western-most corner in three dimensions in Section 3, then extend them formally to higher dimensions in Section 4. We examine in Section 5 what are the maximum possible number of faces of a Minkowski sum and conclude in Section 6. We have extended the intuitive concept of west from three dimensions to higher dimensions. Thanks to the properties of the concept, we were able to prove a relation on the number of vertices in sums of many polytopes, and show that this number has a comparatively low order of complexity. For faces of higher dimensions, the result also shows that the complexity of Minkowski sums of many polytopes is not much more complex than that of d − 1 summands.. The only remaining case is that for any p and p ′ distinct in K, θ 1 (p) = θ 1 (p ′ ); for any p and p ′ in K such that θ 1 (p) + π = θ 1 (p ′ ), p and p ′ must be antipodal; and K ∩ U is empty.. Note that if a great circle of S d−1 does not intersect U , then θ 1 is different in any two points of the great circle. Since the parametrization is smooth on S d−1 \U , θ 1 augments monotonically and continuously in one direction around the great circle, except in one point when it drops from 2π to 0.. Let us recall Lemma 5 before proving it: Proof. Let T 1 (C) be the set of values of θ 1 over C in our parametrization. Since C does not intersect U , T 1 (C) is connected. Suppose T 1 (C) is [0; 2π), then there are two points p, p ′ in C with θ 1 (p) + π = θ 1 (p ′ ). Because C is spherically convex, any shortest arc of great circle between p and p ′ is contained in C. If p and p ′ are antipodal, then C is the whole sphere and intersects U . Otherwise, by Lemma 12, the arc of great circle again contains a point in U . This is a contradiction. Otherwise, suppose without loss of generality that the supremum of T 1 (C) is 3π/2. Then either C is in the hemisphere defined by θ 1 ∈ [π/2, 3π/2], or there is a δ > 0 such that there are two points p, p ′ in C with θ 1 (p) + π = θ 1 (p ′ ) = 3/2 − δ. As above, this implies that C contains a point in U , which is a contradiction.

O B O B I O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B I O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Extending twin support vector machine classifier for multi-category classification problems Abstract. Twin support vector machine classifier (TWSVM) was proposed by Jayadeva et al., which was used for binary classification problems. TWSVM not only overcomes the difficulties in handling the problem of exemplar unbalance in binary classification problems, but also it is four times faster in training a classifier than classical support vector machines. This paper proposes one-versus-all twin support vector machine classifiers (OVA-TWSVM) for multi-category classification problems by utilizing the strengths of TWSVM. OVA-TWSVM extends TWSVM to solve k-category classification problems by developing k TWSVM where in the ith TWSVM, we only solve the Quadratic Programming Problems (QPPs) for the ith class, and get the ith nonparallel hyperplane corresponding to the ith class data. OVA-TWSVM uses the well known one-versus-all (OVA) approach to construct a corresponding twin support vector machine classifier. We analyze the efficiency of the OVA-TWSVM theoretically, and perform experiments to test its efficiency on both synthetic data sets and several benchmark data sets from the UCI machine learning repository. Both the theoretical analysis and experimental results demonstrate that OVA-TWSVM can outperform the traditional OVA-SVMs classifier. Further experimental comparisons with other multiclass classifiers demonstrated that comparable performance could be achieved. Standard support vector machines (SVMs) [1] [2] [3] [4] , introduced by Vapnik et al., are an excellent tool for binary classification problems and have been successfully and widely applied in many fields [5] [6] [7] [8] [9] [10] .. One key research topic about SVMs has been to develop the efficient learning algorithms and models. Over the past few decades, many improvements to SVMs have emerged, such as lagrangian support vector machines (LSVM) [11] , a smooth support vector machine for classification (SSVM) [12] , reduced SVMs (RSVM) [13] , least squares support vector machine classifier (LS-SVM) [14] , proximal support vector machine classifiers (PSVM) [15] , and the generalized eigenvalue proximal SVMs (GEPSVM) for multiclass classification problems [16] .. Recently, Jayadeva et al., motivated by GEPSVM, proposed a twin support vector machine (TWSVM) classifier for binary classification problem [17] . TWSVM determines two nonparallel hyperplanes by solving two smaller and related Quadratic Programming Problems (QPPs), in which each hyperplane is closer to one of the two classes and is as far away as possible from the other class. The strategy of solving two smaller QPPs rather than solving a single large QPP in traditional SVMs makes the learning speed of TWSVM approximately four times faster than that of a classical SVMs, whilst overcoming the potential problem of the exemplar unbalance in binary classification problems by introducing two penalty variables for two classes. Some extensions of TWSVM have been made, including a smooth TWSVM [18] , least squares twin support vector machines for pattern classification (LS-TSVM) [19] , nonparallel plane proximal classifier (NPPC) [20, 21] , and twin support vector machine for regression (TSVR) [22] . In addition, Cong et al. applied TWSVM to text independent speaker recognition, and obtained better results than that of traditional SVMs obtained [23] .. SVMs were originally developed for binary classification problems. How to effectively extend it to multiclass classification problems is still an ongoing research issue. Currently there are two kinds of approaches for multiclass SVMs. One is by constructing and combining several binary classifiers, while the other is by directly considering all data in one optimization formulation. Over the past few decades, several algorithms have been proposed based on these two types of approaches. In particular the following models are widely discussed: One-Versus-All support vector machines (OVA-SVMs) [2, 24] where one class is separated from the remaining classes; One-Versus-One SVMs (OVO-SVMs) [24] where any one class is separated from any other class; Error-correcting-output code SVMs (ECOC SVMs) [25] where error correcting codes are used for improving the generation ability; directed acyclic graph SVMs (DAGSVMs) proposed in [26, 27] , in which the training phase is the same as One-Versus-One support vector machines by solving k(k − 1)/2 binary SVMs, but its testing phase is different from the One-Versus-One SVMs; all-at-once SVMs [2] where all the decision functions are determined at once; multicategory proximal support vector machine classifiers (MPSVM) [28] which extend PSVM [15] to multiclass classification; and multiclass least squares support vector machines [29] which is the extension of LS-SVM [14] for multicategory.. However, the speed in learning a model and the method for dealing with the potential unbalance of exemplars in different classes are still two main problems for multiclass classification problems in SVMs. TWSVM overcomes the exemplar unbalance problem in two classes by choosing two different penalty variables for different classes, and is four times faster in learning a model by solving two smaller QPPs. In this paper, we keep the strengths of TWSVM, and extend it to solve the multiclass classification problems. We combine TWSVM with the well known and simple one-versus-all methodology which has been very popular in solving multiclass classification problems to propose one-versus-all twin support vector machine classifiers (OVA-TWSVM) for multiclass classification problems. In our algorithm we combine the advantages of TWSVM and the well-known one-versus-all approach to learn a classifier for the multiclass classification problem. Our OVA-TWSVM consists of solving k QPPs, one for each class, so that we obtain k nonparallel hyperplanes for k classes, respectively. In OVA-TWSVM we use oneversus-all approach to construct a TWSVM classifier, where in ith TWSVM classifier, we only solve one QPP corresponding to the ith class to determine the hyperplane for the ith class. We overcome the unbalance problem of exemplars existing in ith TWSVM by choosing the proper penalty variable C i for the ith class which TWSVM supported. Solving one QPPs in one TWSVM classifier guarantees the speed of learning the model for the k-category classification problems. Extensive experimental comparisons of OVA-SVMs, OVO-SVMs, DAGSVMs and our OVA-TWSVM classifier have been made on six UCI benchmark datasets. Experimental results show that our OVA-TWSVM classifier achieved better performances than OVA-SVMs, while still giving comparable performances to OVO-SVMs and DAGSVMs.. The paper is organized as follows: Section 2 introduces the basic notations we use in this paper. Section 3 briefly describes TWSVMs and its properties. Section 4 introduces our proposed multiclass TWSVM classifier and gives the detailed analysis for it. At the same time, linear and nonlinear OVA-TWSVM classifier algorithms are described in Subsections 4.1 and 4.2, respectively. In Section 5 we demonstrate the experimental results of our OVA-TWSVM and other three SVMs for multiclass classification. Finally, the paper is concluded in Section 6. In this paper, we extend TWSVM to solve multi-category data classification problems, and propose one-versus-all twin support vector machine (OVA-TWSVM) classifiers for multiclass classification problems. In OVA-TWSVM, we solve quadratic programming problems and obtain nonparallel hyperplanes, one for each class. In each TWSVM, we only solve the quadratic programming problem for the corresponding class, and get the nonparallel hyperplane for it, so that we save time. The theoretical analysis of our OVA-TWSVM uncovers its efficiency. Experimental results on synthetic datasets and on benchmark datasets from UCI machine learning repository show that our OVA-TWSVM classifier achieves consistently better performances than traditional OVA-SVMs. The picture becomes more mixed when OVA-TWSVM is compared to OVO-SVMs and DAGSVMs, but the results are promising as there are benchmark datasets where OVA-TWSVM outperforms these methods. Future work would include improving the computational efficiency of obtaining the optimal parameters for the kernel function of SVMs.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Should telecoms liberalisation stop at call termination? In recent years concerns about the degree of competition in the provision of telecommunications call termination services have emerged. While the general consensus is that call origination is becoming more and more competitive, regulatory attention to call termination has widened in scope beyond the incumbent public telephone operators, leading to direct regulation of mobile termination charges in some countries and a debate on whether regulation should also be extended to`non-dominanta networks. This paper assesses whether these concerns are justi"ed, extending the analysis to review the economic literature on reciprocal setting of termination charges between network operators. We conclude that while ex ante regulation of call termination simpli"es the work of regulators it does not appear justi"able in all circumstances. Access to the incumbent's telecommunications network is crucial for the development of competition. However, the dependency of service provision on access regulation stimulated the search for ways to reduce its importance. The so-called network competition (sometimes called facilities-based competition) appeared to provide the solution. Recent regulatory decisions, however, suggest that, at least for a special type of interconnection known as call termination, regulation will not only be required even under fully-#edged network competition, but it will also extend in scope.. This article examines whether the current regulatory concerns over call termination are justi"ed. Section 2 provides some background information on the access issue and its regulatory framework. Section 3 examines the two main regulatory arguments behind cost regulation of call termination.. The exception is the New Zealand model where there is no industry regulator and the only control is left to the application of competition law. This resulted in Telecom Corporation of New Zealand attempting to delay any interconnection agreement with the new entrant Clear Communications or requiring high access charges. For an account see Mueller (1998). First, the claim that call termination is a bottleneck monopoly, giving rise to an`inherent market failurea, is reviewed in the light of the UK Competition Commission's (CC) enquiry on "xed-tomobile call charges. Second, a concern often mentioned by the UK O$ce of Telecommunications (OFTEL), the Australian Competition & Consumers Commission (ACCC) and the European Commission relates to the potential collusive use of interconnection negotiations. A critical review of the literature on reciprocal network interconnection in Section 4 shows that only under particular circumstances is there an economic justi"cation for extensively regulating call termination.. The article concludes that, although access and interconnection still require regulatory vigilance, consideration of the competitive aspects of the interconnecting telecommunications network operators does not call for ubiquitous regulation of call termination. This is especially true for call termination charges that are set between network operators. These conclusions are particularly relevant in the light of the EU 1999 Communications Review, which will shape the telecommunications regulatory framework in the years to come. This article explored the increasing concern of regulators over call termination. Two types of concerns are put forward to support increasing regulation of call termination. The extreme argument is that call termination is a bottleneck monopoly and that, therefore, network operators charge a monopoly or excessive price to terminate calls to their networks. The characterisation of call termination as a bottleneck monopoly was relied upon by OFTEL for examining "xed-tomobile calls and although rejected by the CC still features in recent OFTEL's documents. The CC con"rmed that call termination cannot be described as a bottleneck monopoly, but that the market power of the terminating network operators varies depending on the circumstances of the case. Competitive constraints over call termination might be weak in some cases, but call termination cannot be always described as a bottleneck monopoly. Therefore, a more #exible approach is necessary in order to assess the competitive conditions that exist in each case.. The risks of using reciprocal access negotiations in order to collude in the retail market have been emphasised in OFTEL, the ACCC and the European Commission's consultative documents and legislation. These risks are often based on a super"cial interpretation of the economic literature. This brief literature review indicates that a collusive use of reciprocal interconnection agreements cannot be ruled out. However, the conditions that make this likely are rather restrictive. Therefore, it cannot be assumed that the risks from an improper use of interconnection negotiations are su$ciently high to justify pre-emptive regulation of call termination.. The EU 1999 Communications Review is a process of revision of the existing EU regulatory framework. It is important that the issues raised by call termination be carefully examined. While call termination is likely to continue to require regulation, especially in the case of service providers seeking access to the PTOs' networks, extending regulation to call termination services exchanged between network operators does not appear justi"able. Terming call termination a bottleneck monopoly simpli"es the work of regulators at the risk of introducing unnecessary regulation in a period in which the e!ects of the telecommunications liberalisation start to unfold.

O O B O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O B I O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B O O O


Pattern matching in pseudo real-time It has recently been shown how to construct online, non-amortised approximate pattern matching algorithms for a class of problems whose distance functions can be classified as being local. Informally, a distance function is said to be local if for a pattern P of length m and any substring T [i, i+m−1] of a text T , the distance between P and, where ∆ is any distance function between individual characters. We show in this work how to tackle online approximate matching when the distance function is nonlocal. We give new solutions which are applicable to a wide variety of matching problems including function and parameterised matching, swap matching, swapmismatch, k-difference, k-difference with transpositions, overlap matching, edit distance/LCS and L 1 and L 2 rearrangement distances. The resulting online algorithms bound the worst case running time per input character to within a log factor of their comparable offline counterpart. A great deal of progress has been made in finding fast algorithms for a variety of important forms of approximate matching in the last few decades. The most common computational model in which these algorithms have been analysed assumes that the text and pattern are to be held in fast primary storage and that each query to the data has constant cost. However, increasingly it has become apparent that new applications such as those found in telecommunications or monitoring Internet traffic require a fresh approach. It may no longer be possible to store the entirety of the text and the worst case time per input character is often more important than the overall running time of any algorithm.. The model that we consider is a deterministic variant of data streaming where we assume we are given a pattern in advance and the text to which it is to be matched arrives one character at a time. The overall task is to report matches between the pattern and text as soon as they occur and to bound the worst case time per input character. It is an important feature of this model that the asymptotic time complexities must not be amortised. We refer to this as the pseudo real-time (PsR) model by analogy to the real-time model where the time per input must be constant. The algorithms we present are in the RAM model with a word size of Ω(log n) (where n is the total text length).. Previous work in this model showed how to convert offline algorithms for approximate pattern matching problems with simple distance functions into efficient online ones using a black box approach [7] . The main restriction for this black box solution was that the distance function defined by the approximate matching problem had to have the property of being local. Informally, we think of a local distance function as one where the distance between a pattern P and a substring of the text T can be written as Σ j ∆(P [j], T [i + j]), where ∆ is any distance function between individual characters. In other words, the distance was simply measured as the sum of the distances between individual symbols. Formally we define a local distance function as being expressible in the form where • is any associative operator with arity two. Throughout we will assume that both ∆ and • are computable in constant time. When • is defined to be addition, the definition simplifies to Σ j ∆(P [j], T [i + j]) as above and includes matching under the Hamming norm and numerical measures such as L p p norms for constant p ≥ 1. When • is defined to be the logical and operation, the definition simplifies to j ∆(P [j], T [i + j]) and includes exact matching and exact matching with wildcards. The previous work of [7] also considered the k-mismatch problem which can also be shown to be local under the given definition. In this case, we define a • b to be min(a + b, k + 1).. To appreciate the challenges that arise in online pattern matching when the distance function is non-local, consider for example the problem of function matching [2] . There is a function match between pattern P and text substring T [i, i + m − 1] if there exists a function f (possibly different for each i) from the input alphabet Σ to itself such that T [i + j] = f (P [j]) for all 0 ≤ j < m. For example P = aba has a function match with T = xyx but not T = xyy as a cannot be mapped to two different letters. In the previous black box approach that we briefly describe in Section 2, distances are found independently for different substrings of the pattern and the results combined. However, in this case whether P [2] = a matches T [2] = y depends on the function chosen to map the characters in P [0, 1] and vice versa. Therefore, any matchings for the substrings of P have to depend on the results for all other substrings. In general for non-local distance functions, we must find a way to handle efficiently the dependencies between different parts of the pattern.. Our contribution is to present three general methods which can be applied successfully to convert a wide variety of non-local approximate matching problem into efficient non-amortised online ones. The techniques are necessarily no longer black box, depending in detail on the specific offline algorithm being considered. The running time per input character of the new online pattern matching algorithms is guaranteed in each case to be within a log factor of its offline counterpart.. Our results are summarised in Table 1 which gives the time complexity of the best known offline algorithm divided by the text length and the multiplicative penalty incurred by our onlinization process. All algorithms listed use only O(m) space. In some cases, the methods we present allow us to create online pattern matching algorithms whose running time is the same as in the offline case, that is without any asymptotic overhead at all. For completeness, we also give examples of offline algorithms which translate immediately to the PsR setting with little or no modification.

O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Exploiting Service Usage Information for Optimizing Server Resource Management It is often difficult to tune the performance of modern component-based Internet services because: (1) component middleware are complex software systems that expose several independently tuned server resource management mechanisms; (2) session-oriented client behavior with complex data access patterns makes it hard to predict what impact tuning these mechanisms has on application behavior; and (3) component-based Internet services themselves exhibit complex structural organization with requests of different types having widely ranging execution complexity. In this article we show that exposing and using detailed information about how clients use Internet services enables mechanisms that achieve two interconnected goals: (1) providing improved QoS to the service clients, and (2) optimizing server resource utilization. To differentiate among levels of service usage (service access) information, we introduce the notion of the service access attribute and identify four related groups of service access attributes, encompassing different aspects of service usage information, ranging from the high-level structure of client web sessions to low-level fine-grained information about utilization of server resources by different requests. To show how the identified service usage information can be collected, we implement a request profiling infrastructure in the JBoss Java application server. In the context of four representative service management problems, we show how collected service usage information is used to improve service performance, optimize server resource utilization, or to achieve other problem-specific service management goals. In this article we looked at several performance and service management issues encountered in modern component-based Internet services. We showed that exposing and using detailed information about how clients use Internet services enables mechanisms that achieve the two interconnected goals: (1) providing improved QoS to the service clients, and (2) optimizing server resource utilization. To differentiate among different levels of service usage (service access) information, we introduced the notion of the service access attribute and identified four related groups of service access In the last decade, the role of the Internet has undergone a transition from simply being a data repository to one providing access to a variety of network-accessible services Authors' addresses: A. Totok, Google Inc., 76 9th Ave, 6th Floor, New York, NY 10011; email: totok@google.com; V. Karamcheti, Courant Institute of Mathematical Sciences, New York University, 715 Broadway, 7th Floor, New York, NY 10003; email: vijayk@cs.nyu.edu Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from1:2 A. Totok and V. Karamcheti such as e-mail, social networking, banking, shopping, and entertainment. The emergence of these web portals marked a shift from monolithically structured web sites with static read-only content, which were typical for the early Internet, to complex services that provide richer functionality, and dominate the modern Internet. These services share several commonalities in the way they are structured and the way they are used by their clients.. -Session-oriented usage by clients. The typical interaction of users with such services is organized into sessions, a sequence of related requests, which together achieve a higher-level user goal. An example of such an interaction is an online shopping scenario for an e-commerce web site, which involves multiple requests that search for particular products, retrieve information about a specific item (e.g., quantity and price), add it to the shopping cart, initiate the check-out process, and finally commit the order. The success of the whole session now becomes the ultimate client goal [Cherkasova and Phaal 2002] , which contrasts with per-request success performance metrics of the early Internet. -Complex data access patterns. Application data no longer has read-only access, as was typical for the older content-providing web sites. In scenarios such as the one just mentioned, certain service requests not only read but also write application data. Moreover, concurrent requests coming from different clients can access and modify shared application data. Data access patterns become even more complicated when the application data is replicated (e.g., for failover purposes) or partitioned (e.g., due to business requirements). The outcome of a request execution depends on the datasources it accesses, and may be influenced by concurrent user requests. For some services, it becomes crucial to preserve the correctness of a session's execution, w.r.t. the data it accesses. -Use of component middleware. A growing number of services utilize component middleware such as the Java Platform Enterprise Edition (Java EE) framework [Java EE 2011] as their building platform. Such services are structured as aggregations of multiple application components communicating with each other and with the back-end databases, while the middleware provides commonly required support for communication, security, persistence, clustering, and transactions. Consequently, a web application's behavior depends not only on the way it is programmed, but also on the way it is assembled, deployed, and managed at runtime. As an example, changing middleware policies such as transaction demarcation may significantly impact not only application performance, but also certain aspects of application logic and the correctness of data presented to the users.. These characteristics have implications for how service providers ensure reasonable service quality for client requests. Providing good performance quality of service (QoS) has been a classic problem in the context of Internet services, and, the characteristics outlined above have made this problem even harder. Not only is providing performance guarantees more difficult for modern Internet services, but service providers now also need to take care of ensuring correctness of service (application) logic and providing expected data quality to the service clients.. 1.1.1 Performance Quality. GVU WWW User Surveys [2001] showed that around 19% of the people surveyed attribute the bad experiences they had with Internet services to bad performance. The primary performance concern for service users is request response time [Barnes and Mookerjee 2009; Selvridge et al. 2001] . One study [Moskalyuk 2006] showed that 33% of shoppers on a slow-loading e-commerce web site abandoned the site entirely, and 75% of visitors would never shop on that site again.. The key contributing factor to long response delays is service overload, when the user load nears or exceeds server capacity, which causes request rejections and increases request response times, even in the absence of network disruptions. To improve service performance in the situation of service overload, service providers have traditionally used server-side resource management mechanisms to improve utilization of server resources. But this is harder to do for modern Internet services for the following reasons. First, middleware usually exposes several mechanisms that can be independently tuned in an attempt to improve application performance and optimize server resource utilization. However, these mechanisms do not provide a unique server configuration, which would be optimal for all request loads. Second, complex session-oriented client behavior makes it hard to predict what impact tuning a server resource management mechanism would have on server performance, which may vary for different incoming request mixes. Finally, component-based applications exhibit complex structural organization, where different sets of application components and middleware services are used to execute requests of different types. Some requests, for example, may need to access a back-end database, some may need CPU-intensive processing, while others may need exclusive access to a component or a critical resource.. The inability of service providers to predict the exact effects of using the server resource management mechanisms on service performance for a given client load accounts for the fact that these mechanisms are often used in an ad-hoc or "best-guess" manner. This results in suboptimal usage of server resources, not tailored for the specific incoming request load, and, as a consequence, the service clients do not get the best performance quality they could potentially get.

O B I I O O O O B O O O O O O O B O O O O O O O B I O O O O O O O O O O O B O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O B O O O O O B I I O O O O O B O O O O O O O O O O B O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O


Language as a Cognitive Tool Abstract The standard view of classical cognitive science stated that cognition consists in the manipulation of language-like structures according to formal rules. Since cognition is 'linguistic' in itself, according to this view language is just a complex communication system and does not influence cognitive processes in any substantial way. This view has been criticized from several perspectives and a new framework (Embodied Cognition) has emerged that considers cognitive processes as non-symbolic and heavily dependent on the dynamical interactions between the cognitive system and its environment. But notwithstanding the successes of the embodied cognitive science in explaining low-level cognitive behaviors, it is still not clear whether and how it can scale up for explaining high-level cognition. In this paper we argue that this can be done by considering the role of language as a cognitive tool: i.e. how language transforms basic cognitive functions in the highlevel functions that are characteristic of human cognition. In order to do that, we review some computational models that substantiate this view with respect to categorization and memory. Since these models are based on a very rudimentary form of non-syntactic 'language' we argue that the use of language as a cognitive tool might have been an early discovery in hominid evolution, and might have played a substantial role in the evolution of language itself. A crucial, but often neglected, characteristic of human language is that language is used not only for communicating with others but also for communicating with Language as a Cognitive Tool 525 oneself, whereas we seem not to have evidence for this type of use of animal communication systems. Talking to oneself, in the form of both private and inner speech, has tremendous consequences for the development of the human mind. Indeed, we have argued that considering the cognitive role of language can provide the missing link for addressing the high-level cognitive capacities which characterize humans within the new, emerging framework which considers cognition as ''environmentally embedded, corporeally embodied, and neurally embrained.' ' (van Gelder 1999, p. 244) . In the present paper we have described some simple computer simulations that show that language can improve one's categories and can be an useful aid to memory, both if it mediates social communication and if it is used to talk to oneself as private or inner speech. But we argue that the use of language for oneself does not improve only categorization and memory, but almost any human cognitive function (for a discussion of other possible roles of language in enhancing cognitive processes, see Mirolli and Parisi 2009 ). Therefore, much more work needs to be done in order to understand the relationships between language and cognition. And we think that computer simulations will play an important role in our understanding of this fundamental topic.

B O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B O O B I O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O B I O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O B O O O O O O O O O O O O O B O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O


Exploiting persistent mappings in cross-domain analogical learning of physical domains Cross-domain analogies are a powerful method for learning new domains. This paper extends the Domain Transfer via Analogy (DTA) method with persistent mappings, correspondences between domains that are incrementally built up as a cognitive system gains experience with a new domain. DTA uses analogies between pairs of textbook example problems, or worked solutions, to create a domain mapping between a familiar and a new domain. This mapping enables the initialization of a new domain theory. Another analogy is then made between the domain theories themselves, providing additional conjectures about the new domain. After these conjectures are verified, the successful mappings are stored as persistent mappings to constrain future analogies between the domains. We show that DTA plus persistent mappings enables a Companion, the first structure mapping cognitive architecture, to learn the equation schemas and control knowledge necessary to solve problems in three domains (rotational mechanics, electricity, and heat) by analogy with linear mechanics. We provide a detailed analysis categorizing transfer failures. As with people, the most difficult step in cross-domain analogy is identifying an appropriate example. Once an analogous example has been found, DTA successfully transfers the domain knowledge necessary to solve the problem in the new domain 78% of the time. Furthermore, we illustrate how persistent mappings assist in retrieval of analogous examples and overcoming two types of mapping failures. Cognitive science research has shown how cross-domain analogies are useful to scientists in producing paradigm shifts [19, 25, 5] and how students use cross-domain analogies to learn new domains [17] . Textbook authors routinely exploit this aspect of human adaptability when introducing new concepts by providing a useful base domain explicitly and giving hints about correspondences. For example, a typical physics textbook states "the dynamics of rotation is analogous to the dynamics of linear motion" [20] . Students are expected to use this analogy to understand the phenomena and make modeling decisions in rotational mechanics.. Domain Transfer via Analogy (DTA) is a method of learning via cross-domain analogies [28] . DTA consists of four steps: (1) learning the correspondences between the domains, or domain mapping, (2) initializing the target domain, (3) extending the target domain, and (4) verifying transferred knowledge. Operating with domain theories, defined in predicate calculus, DTA transfers knowledge from an understood domain to a new domain. In this work, DTA transfers schemas representing equations and control knowledge about their use to new domains. It uses analogies between textbook example problems, or worked solutions, as evidence about what non-identical predicate matches will be productive, re-using those correspondences to transfer equation schemas and control knowledge from the base to the target, creating a new target domain theory.. In previous work [28] we showed that DTA was able to transfer knowledge between kinematic domains. This paper applies DTA to larger and more distant domains. Our previous analyses suggested that learning complex domains by crossdomain analogy does not happen all at once. Instead, the learner's understanding of the analogy grows incrementally. Therefore, we extended DTA with persistent mappings [24] to enable the incremental accumulation of knowledge about correspondences between the domains. Learned from successful analogies, persistent mappings constrain subsequent crossdomain analogies, ensuring that a structurally consistent set of knowledge is imported to the target domain. The experiment described here shows that adding persistent mappings to DTA improves performance. By using multiple conditions, we identify several types of failures, and show how persistent mappings address some of them.. We implemented Domain Transfer via Analogy in the Companion Cognitive Architecture [13] . A major hypothesis of the Companion architecture, differentiating it from others, is that analogical processing is central to human reasoning and learning [16] . Given the gap between current AI systems and human abilities, we believe that building human-like systems will likely lead to more flexible AI systems. The Companions architecture includes cognitive models of analogy, similaritybased retrieval and generalization which have been used to account for psychological data. These models have been used in many AI tasks from learning strategy games [24] to everyday physical reasoning [30] . For these reasons, the Companion Cognitive Architecture is an excellent platform for Domain Transfer via Analogy, and in the rest of this article, we refer to the entire system as "the Companion". The Cognitive Systems paradigm [32] investigates intelligence building off of five assumptions: multi-step inference, heuristics, structured knowledge, links to human cognition, and systems-level research. All five appear in this work, placing our efforts squarely within this community. A consequence of doing systems-level research with cognitive models is that Domain Transfer via Analogy could be implemented with different cognitive models of analogical matching and retrieval, providing that the interfaces were the same.. We begin by discussing the DTA algorithm with a running example. In the course of this example, we describe the cognitive models for analogical retrieval and matching used by DTA as well as the representations for physics problems, worked solutions and domain theories. Next, we discuss how persistent mappings enable incremental cross-domain analogy and how they are incorporated into DTA. Then, we describe the experiment and present a detailed analysis of the results. We conclude with a discussion of related and future work. First, these results indicate DTA is able to transfer domain knowledge from linear mechanical systems to solve problems from related domains. In addition to successful transfer to rotational mechanics, DTA was able to transfer linear mechanical knowledge to the superficially dissimilar domains of electrical and thermal systems when the analogous worked solution was retrieved. In this experiment, DTA transfers not only equation schemas but also control knowledge. While the analogous portion of these domains contains 7 equations relating 11 quantities, there are 33 total equations and 56 total quantities used in the 4 domain theories and 22 worked solutions. In the next sections, we analyze the results with respect to the retrieval, transfer and persistent mapping aspects of the algorithm in detail. The major threads of related work concern computational models of analogy from cognitive science and transfer learning methods from artificial intelligence. The base domain for this experiment is linear mechanics, because it is the dynamical analogy domain most students learn first. The seven linear mechanics problems and worked solutions were added to the case library. The base domain theory consists of the eleven equation schemas and control knowledge necessary to solve these seven problems. The target domains for this experiment are rotational, electrical, and thermal systems. It would be impossible to learn these domains entirely by analogy. Therefore, we provided the companion initial target domain theories including the non-analogous equation schemas necessary to solve the problems (e.g., equations for computing moment of inertia). This includes 4 rotational, 0 electrical, and 2 thermal equation schemas. None of the target problems are solvable with only these equation schemas.. This experiment evaluates DTA on a per problem basis in four independent conditions. In the first condition, the entire DTA algorithm was run on each problem. To isolate the effects of the retrieval mechanism, the Companion was only allowed one attempt to retrieve a worked solution from the base domain. In the second condition, each problem was presented with the correct retrieval from linear mechanics. The third and fourth conditions address the effects of persistent mappings. In the third condition, each problem was presented along with the persistent mappings resulting from a successful transfer using DTA on the most similar problem. For example, for problem Gia-21-37-P, the persistent mappings were provided from a successful transfer between Gia-6-3 and Gia-21-46-P. In the fourth condition, the persistent mappings and the correct retrieval were provided.. We measured two aspects of performance, the Companion's ability to solve the problem in the target domain after DTA and if retrieval selected the analogous base worked solution. For each problem, the problem was scored correct if the Companion was able to solve the problem after transfer and the problem's retrieval was scored correct using the rows in Table 2 . Therefore, each problem has only one correct retrieval out of 7 worked solutions in the linear mechanics case library.. The goal of this experiment was to assess DTA's ability to learn rotational mechanical, electrical, and thermal systems through cross-domain analogy with linear mechanics. Using the four conditions, the retrieval, mapping and persistent mapping components were evaluated independently. n/a a Problem sources are indicated by their prefixes. The first number indicates the chapter and the second problem indicates the example within that chapter. The 'P' designation is used for problems from the end of the chapter. "Gia" are from [20] , "Oga" are from [38] , "She" are from [45] , "Rea" are from [7] and "Phy" are from [26]. Table 3 shows the results of the four trials for each of the three transfer domains. In condition 1, DTA transferred the necessary equation schemas and control knowledge to solve 3 out of 15 problems from the target domains (20%). While the Companion solved problems from rotational mechanical and electrical domains, Table 3 DTA results using linear mechanics as the base domain.

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