algorithmic complexity of weakly semiregular partitioning and the representation number abstract a graph g is weakly semiregular if there are two numbers a b such that the degree of every vertex is a or b the weakly semiregular number of a graph g denoted by w r g is the minimum number of subsets into which the edge set of g can be partitioned so that the subgraph induced by each subset is a weakly semiregular graph we present a polynomial time algorithm to determine whether the weakly semiregular number of a given tree is two on the other hand we show that determining whether w r g 2 for a given bipartite graph g with at most three numbers in its degree set is np complete among other results for every tree t we show that w r t 2 log 2 t o 1 where t denotes the maximum degree of t a graph g is a d d s graph if the degree of every vertex of g lies in the interval d d s a d d 1 graph is said to be semiregular the semiregular number of a graph g denoted by s r g is the minimum number of subsets into which the edge set of g can be partitioned so that the subgraph induced by each subset is a semiregular graph we prove that the semiregular number of a tree t is t 2 on the other hand we show that determining whether s r g 2 for a given bipartite graph g with g 6 is np complete in the second part of the work we consider the representation number a graph g has a representation modulo r if there exists an injective map l v g z r such that vertices v and u are adjacent if and only if l u l v is relatively prime to r the representation number denoted by r e p g is the smallest r such that g has a representation modulo r narayan and urick conjectured that the determination of r e p g for an arbitrary graph g is a difficult problem 38 in this work we confirm this conjecture and show that if n p p then for any 0 there is no polynomial time 1 n 2 approximation algorithm for the computation of representation number of regular graphs with n vertices

solving the vehicle routing problem with time windows and multiple routes exactly using a pseudo polynomial model in this paper we address a variant of the vehicle routing problem called the vehicle routing problem with time windows and multiple routes it considers that a given vehicle can be assigned to more than one route per planning period we propose a new exact algorithm for this problem our algorithm is iterative and it relies on a pseudo polynomial network flow model whose nodes represent time instants and whose arcs represent feasible vehicle routes this algorithm was tested on a set of benchmark instances from the literature the computational results show that our method is able to solve more instances than the only other exact method described so far in the literature and it clearly outperforms this method in terms of computing time

performance and design of cycles in consensus networks abstract this work explores the role of cycles in consensus seeking networks for analysis and synthesis purposes cycles are critical for many reasons including improving the convergence rate of the system resilience to link failures and the overall performance of the system the focus of this work examines how cycles impact the h 2 performance of consensus networks a first contribution shows that the addition of cycles always improves the performance of the system we provide an analytic characterization of how the addition of edges improves the performance and show that it is related to the inverse of the cycle lengths and the number of shared edges between independent cycles these results are then used to consider the design of consensus networks in this direction we present an l 1 relaxation method that leads to a convex program for adding a fixed number of edges to a consensus networks we also demonstrate how this relaxation can be used to embed additional performance criteria such as maximization of the algebraic connectivity of the graph

the np completeness of some edge partition problems we show that for each fixed n geqq 3 it is np complete to determine whether an arbitrary graph can be edge partitioned into subgraphs isomorphic to the complete graph k n the np completeness of a number of other edge partition problems follows immediately

an extension of kleene s and ochman ski s theorems to infinite traces abstract as was noted by mazurkiewicz traces constitute a convenient tool for describing finite behaviour of concurrent systems extending in a natural way mazurkiewicz s original definition infinite traces have recently been introduced enabling one to deal with infinite behaviour of nonterminating concurrent systems in this paper we examine the basic families of recognizable sets and of rational sets of infinite traces the seminal kleene characterization of recognizable subsets of the free monoid and its subsequent extensions to infinite words due to buchi and to finite traces due to ochmanski are the cornerstones of the corresponding theories the main result of our paper is an extension of these characterizations to the domain of infinite traces using recognizing and weakly recognizing morphisms as well as a generalization of the schutzenberger product of monoids we prove various closure properties of recognizable trace languages moreover we establish normal form representations for recognizable and rational sets of infinite traces

recognizable closures and submonoids of free partially commutative monoids abstract the main result of this paper is the presentation of two sufficient conditions for the closure x of x and the closure x of x respectively in any free partially commutative monoid for any regular language x to be recognizable several previously obtained sufficient conditions for x to be recognizable can be seen to imply one of those two conditions

recognizable subsets of some partially abelian monoids we show that in a free partially abelian monoid generated by a finite alphabet a the subset x of a containing all the words equivalent to a product of words of x is rational if x is a finite set of words each word containing at least one occurrence of any letter of a we suppose that the graph the vertices of which are letters of a and the edges of which correspond to noncommuting pairs of letters is connected

edge disjoint packing of stars and cycles abstract we study the parameterized complexity of two graph packing problems edge disjoint k packing of s stars and edge disjoint k packing of s cycles with respect to the choice of parameters we show that although the two problems are fpt with both k and s as parameters they are unlikely to be fixed parameter tractable when parameterized by only k or only s in terms of kernelization complexity we show that edge disjoint k packing of s stars has a kernel with size polynomial in both k and s but in contrast unless np conp poly edge disjoint k packing of s cycles does not have a kernel with size polynomial in both k and s and moreover does not have a kernel with size polynomial in s for any fixed k we also show that edge disjoint k packing of 4 cycles admits a 96 k 2 kernel in general graphs and a 96 k kernel in planar graphs

an exact algorithm for a single vehicle routing problem with time windows and multiple routes this paper describes an exact algorithm for solving a problem where the same vehicle performs several routes to serve a set of customers with time windows the motivation comes from the home delivery of perishable goods where vehicle routes are short and must be combined to form a working day a method based on an elementary shortest path algorithm with resource constraints is proposed to solve this problem the method is divided into two phases in the first phase all non dominated feasible routes are generated in the second phase some routes are selected and sequenced to form the vehicle workday computational results are reported on euclidean problems derived from benchmark instances of the classical vehicle routing problem with time windows

parameterized complexity of k chinese postman problem we consider the following problem called the k chinese postman problem k cpp given a connected edge weighted graph g and integers p and k decide whether there are at least k closed walks such that every edge of g is contained in at least one of them and the total weight of the edges in the walks is at most p the problem k cpp is np complete and van bevern et al 4 and sorge 14 asked whether the k cpp is fixed parameter tractable when parameterized by k we prove that the k cpp is indeed fixed parameter tractable in fact we prove a stronger result the problem admits a kernel with o k 2logk vertices we prove that the directed version of k cpp is np complete and ask whether the directed version is fixed parameter tractable when parameterized by k

hardness and approximation of traffic grooming traffic grooming is a central problem in optical networks it refers to packing low rate signals into higher speed streams in order to improve bandwidth utilization and reduce network cost in wdm networks the most accepted criterion is to minimize the number of electronic terminations namely the number of sonet add drop multiplexers adms in this article we focus on ring and path topologies on the one hand we provide an inapproximability result for traffic grooming for fixed values of the grooming factor g answering affirmatively the conjecture of chow and lin t chow p lin the ring grooming problem networks 44 2004 194 202 more precisely we prove that ring traffic grooming for fixed g 1 and path traffic grooming for fixed g 2 are apx complete that is they do not accept a ptas unless p np both results rely on the fact that finding the maximum number of edge disjoint triangles in a tripartite graph and more generally cycles of length 2g 1 in a 2g 1 partite graph of girth 2g 1 is apx complete on the other hand we provide a polynomial time approximation algorithm for ring and path traffic grooming based on a greedy cover algorithm with an approximation ratio independent of g namely the approximation guarantee is o n 1 3log 2n for any g 1 n being the size of the network this is useful in practical applications since in backbone networks the grooming factor is usually greater than the network size finally we improve this approximation ratio under some extra assumptions about the request graph

on recognizable subsets of free partially commutative monoids on montre que dans un mono de partiellement commutatif libre engendre par un alphabet fini a le sous ensemble x de a forme des mots equivalents a un des mots de x est reconnaissable si x est reconnaissable et si tout facteur iterant de x a un graphe de non commutation connexe

the effect of number of hamiltonian paths on the complexity of a vertex coloring problem a generalization of dobkin and lipton s element uniqueness problem is introduced for any fixed undirected graph g on vertex set v 1 v 2 cdots v n the problem is to determine given n real numbers x 1 x 2 cdots x n whether x i ne x j for every edge upsilon i upsilon j in g this problem is shown to have upper and lower bounds of theta n log n linear comparisons if g is any dense graph the proof of the lower bound involves showing that any dense graph must contain a subgraph with many hamiltonian paths and demonstrating the relevance of these hamiltonian paths to a geometric argument in addition we exhibit relatively sparse graphs for which the same lower bound holds and relatively dense graphs for which a linear upper bound holds

branch and price algorithms for the solution of the multi trip vehicle routing problem with time windows we investigate the exact solution of the vehicle routing problem with time windows where multiple trips are allowed for the vehicles in contrast to previous works in the literature we specifically consider the case in which it is mandatory to visit all customers and there is no limitation on duration we develop two branch and price frameworks based on two set covering formulations a traditional one where columns variables represent routes that is a sequence of consecutive trips and a second one in which columns are single trips one important difficulty related to the latter is the way mutual temporal exclusion of trips can be handled it raises the issue of time discretization when solving the pricing problem our dynamic programming algorithm is based on concept of groups of labels and representative labels we provide computational results on modified small sized instances 25 customers from solomon s benchmarks in order to evaluate and compare the two methods results show that some difficult instances are out of reach for the first branch and price implementation while they are consistently solved with the second

traffic grooming on the path in a wdm network routing a request consists in assigning it a route in the physical network and a wavelength if each request uses at most 1 c of the bandwidth of the wavelength we will say that the grooming factor is c that means that on a given edge of the network we can groom group at most c requests on the same wavelength with this constraint the objective can be either to minimize the number of wavelengths related to the transmission cost or minimize the number of add drop multiplexers shortly adm used in the network related to the cost of the nodes we consider here the case where the network is a path on n nodes pn thus the routing is unique for a given grooming factor c minimizing the number of wavelengths is an easy problem well known and related to the load problem but minimizing the number of adms is np complete for a general set of requests and no results are known here we show how to model the problem as a graph partition problem and using tools of design theory we completely solve the case where c 2 and where we have a static uniform all to all traffic one request for each pair of vertices

covering the edges of bipartite graphs using k2 2 graphs an optimization problem arising in the design of optical networks is shown here to be abstracted by the following model of covering the edges of a bipartite graph with a minimum number of 4 cycles or k 2 2 given a bipartite graph g l r e over the node set l r with e u v u l v r and the implicit collection of all four node cycles in the complete bipartite graph over l r the goal is to cover all the edges in e with a sub collection of graphs g 1 g 2 g p of minimum size where each g i is a subgraph of a cycle over four nodes a 4 cycle since a subgraph of a 4 cycle can cover up to 4 edges this covering problem is a special case of the unweighted 4 set cover problem this specialization allows us to obtain an improved approximation guarantee whereas the currently best known approximation algorithm for the general unweighted 4 set cover problem has an approximation ratio of h 4 98195 1 58077 where h p lnp denotes the p th harmonic number we show that for every e 0 there is a polynomial time 1310 e approximation algorithm for our problem our analysis of the greedy algorithm shows that when applied to covering a bipartite graph using copies of k q q bicliques it returns a feasible solution whose cost is at most h q 2 h q 1 opt 1 where opt denotes the optimal cost thus improving the approximation bound for unweighted q 2 set cover by a factor of almost 2

approximation algorithms for grooming in optical network design we study traffic grooming in optical network design where the goal is to aggregate low bandwidth traffic streams to utilize efficiently high bandwidth media such as wavelength channels more precisely given traffic demands to be routed in a network the design problem is to define a collection of light paths such that each demand can follow a sequence of consecutive light paths each light path has a unit wavelength bandwidth and multiple sub wavelength demands may share a common light path traffic must enter and depart from a light path at its two endpoints only most previous work on grooming focused on the ring topology and typically involved only uniform bandwidth demands whereas we deal with more general settings two objectives are considered one is to minimize the total cost of equipment necessary to support the light paths the other is simply to minimize the number of light paths even for the extremely restricted special case of a line topology and traffic demands that request half wavelength bandwidth we show that both objectives are apx hard to optimize which means we cannot approximate the optimum arbitrarily closely on the other extreme of generality for arbitrary network topologies and traffic demands that request arbitrary amounts of bandwidth we show a logarithmic approximation for cost minimization and a 2 approximation for minimizing light path counts furthermore we discover that the special case of half wavelength demands has rich combinatorial properties closely related to graph problems such as cycle packing and pattern matching problems such as interchange distance in strings we show how to approximate both objectives up to small constant factors in this case while similarly improving the approximation and hardness of the interchange distance problem as well

bounding the maximum size of a packet radio network the author addresses the problem arising in the topology design of packet radio networks prns which use time division multiplexing and have a diameter constraint of what is the maximum number n sub c f k of users which can be contained in a diameter k prn with f time slots per frame the author assumed that users cannot transmit and receive simultaneously and cannot transmit receive more than one packet at a time this assumption implies that no two channels accessed by the same user may be assigned the same time slot it is shown that the problem of determining n sub c f k is identical to the problem of determining the largest number of vertices which can be contained in an f edge colorable directed graph with diameter k lower bounds on n sub c f k for f 2 k 3 4 5 are obtained by generating large graphs of this type the graphs are constructed and colored by simple and fast procedures which are similar for different values of f and k an extensive bibliography on the edge coloring problem is included

the np completeness of edge coloring we show that it is np complete to determine the chromatic index of an arbitrary graph the problem remains np complete even for cubic graphs

the complexity of computing the sign of the tutte polynomial we study the complexity of computing the sign of the tutte polynomial of a graph as there are only three possible outcomes positive negative and zero this seems at first sight more like a decision problem than a counting problem surprisingly however there are large regions of the parameter space for which computing the sign of the tutte polynomial is actually p hard as a trivial consequence approximating the polynomial is also p hard in this case thus approximately evaluating the tutte polynomial in these regions is as hard as exactly counting the satisfying assignments to a cnf boolean formula for most other points in the parameter space we show that computing the sign of the polynomial is in fp whereas approximating the polynomial can be done in polynomial time with an np oracle as a special case we completely resolve the complexity of computing the sign of the chromatic polynomial this is easily computable at q 2 and when q leq 32 27 and is np hard to compute for all other va

new results on the star problem in trace monoids we prove that if it is decidable whether x is recognizable for a recognizable subset x of a free partially commutative monoid then it is decidable whether a recognizable subset of a free partially commutative monoid possesses the finite power property we prove that if any trace of a set x is connected we can decide whether x possesses the finite power property finally we also show that if x is a finite set containing at most four traces or at most two connected traces it is decidable whether x is recognizable

iterative aggregation and disaggregation algorithm for pseudo polynomial network flow models with side constraints this paper develops a general solution framework based on aggregation techniques to solve np hard problems that can be formulated as a circulation model with specific side constraints the size of the extended mixed integer linear programming formulation is generally pseudo polynomial to efficiently solve exactly these large scale models we propose a new iterative aggregation and disaggregation algorithm at each iteration it projects the original model onto an aggregated one producing an approximate model the process iterates to refine the current aggregated model until the opti mality is proved the computational experiments on two hard optimization problems a variant of the vehicle routing problem and the cutting stock problem show that a generic implementation of the proposed framework allows us to out perform previous known methods

on the decidability of some problems about rational subsets of free partially commutative monoids let i a b be a partially commutative alphabet such that two letters commute if one of them belongs to a and the other one belongs to b let m a b denote the free partially commutative monoid generated by i we consider the following six problems for rational given by regular expressions subsets x y of m q1 x y o q2 x y di x y q4 x m l25 m x finite x is recognizable it was proved by choffrut see 2 that all these problems are undecidable if card a 1 and card b 1 and they are decidable if card a card b 1 card u denotes the cardinality of u it was conjectured see 2 p 79 that these problems are decidable in the remaining cases where card a 1 and card b 1 in this paper we show that if card a 1 and card b 1 then the problem 01 is decidable and problems q2 06 are undecidable our paper is an application of results concerning reversal bounded nondeterministic multicounter machines and nondeterministic general sequential machines

algorithms for the vehicle routing and scheduling problems with time window constraints this paper considers the design and analysis of algorithms for vehicle routing and scheduling problems with time window constraints given the intrinsic difficulty of this problem class approximation methods seem to offer the most promise for practical size problems after describing a variety of heuristics we conduct an extensive computational study of their performance the problem set includes routing and scheduling environments that differ in terms of the type of data used to generate the problems the percentage of customers with time windows their tightness and positioning and the scheduling horizon we found that several heuristics performed well in different problem environments in particular an insertion type heuristic consistently gave very good results

an approximation algorithm for diagnostic test scheduling in multicomputer systems the problem of diagnostic test scheduling dts is to assign to each edge e of a diagnostic graph g a time interval of length l e so that intervals corresponding to edges at any given vertex do not overlap and the overall finishing time is minimum in this correspondence we show that the dts problem is np complete then we present a longest first sequential scheduling algorithm which runs in worst case time o dm log n and uses o m space to produce a solution of length less than four times optimal then we show that the general performance bound can be strengthened to 3 opt g for low degree graphs and to 2 opt g in some special cases of binomial diagnostic graphs

an exact algorithm for a vehicle routing problem with time windows and multiple use of vehicles the vehicle routing problem with multiple use of vehicles is a variant of the classical vehicle routing problem it arises when each vehicle performs several routes during the workday due to strict time limits on route duration e g when perishable goods are transported the routes are defined over customers with a revenue a demand and a time window given a fixed size fleet of vehicles it might not be possible to serve all customers thus the customers must be chosen based on their associated revenue minus the traveling cost to reach them we introduce a branch and price approach to address this problem where lower bounds are computed by solving the linear programming relaxation of a set packing formulation using column generation the pricing subproblems are elementary shortest path problems with resource constraints computational results are reported on euclidean problems derived from well known benchmark instances for the vehicle routing problem with time windows

on the minimum monochromatic or multicolored subgraph partition problems let g v e be an edge colored graph a subgraph h is said to be monochromatic if all the edges of h have the same color and multicolored if no two edges of h have the same color we investigate the complexity of the problems for finding the minimum number of monochromatic or multicolored subgraphs such as cliques cycles trees and paths partitioning v g depending on the number of colors used and the maximal number of times a color appears in a coloring we also present a greedy scheme that yields a lnm 1 approximation for the problem of finding the minimum number of monochromatic cliques partitioning v g for a k 4 free graph g where m is the size of the largest monochromatic clique in g by a slightly modification of the approximation algorithm it can be used for the multicolored case we show that unless np dtime n o l o g l o g n for any e 0 there is no approximation algorithm for finding the minimum number of multicolored trees partitioning v g with performance 50 521 1 e ln v

on some equations in free partially commutative monoids abstract we characterize the solutions of all the equations in two unknowns in the free partially commutative monoids and we show that the connected solutions are cyclic next we define the transposition relation m and n are transposed iff m xy and n yx and the conjugacy relation m and n are conjugate iff m n we show that the conjugacy is the transitive closure of the transposition and that the set of conjugacy factors is a recognizable subset of the free partially commutative monoid

algorithmic problems in right angled artin groups complexity and applications abstract in this paper we consider several classical and novel algorithmic problems for right angled artin groups some of which are closely related to graph theoretic problems and study their computational complexity we study these problems with a view towards applications to cryptography

string matching problems over free partially commutative monoids abstract this paper studies two string matching problems over free partially commutative monoids we analyze these two problems in detail and present two efficient polynomial time algorithms for solving them

graph decomposition is np complete a complete proof of holyer s conjecture an h decomposition of a graph g v e is a partition of e into subgraphs isomorphic to h given a fixed graph h the h decomposition problem is to determine whether an input graph g admits an h decomposition in 1980 holyer conjectured that h decomposition is np complete whenever h is connected and has three edges or more some partial results have been obtained since then a complete proof of holyer s conjecture is the content of this paper the characterization problem of all graphs h for which h decomposition is np complete is hence reduced to graphs where every connected component contains at most two edges