non deterministic polynomial optimization problems and their approximations abstract a unified and general framework for the study of nondeterministic polynomial optimization problems npop is presented and some properties of npop s are investigated a characterization of npop s with regard to their approximability properties is given by proving necessary and sufficient conditions for two approximability schemes known approximability results are shown to fit within the general frame developed in the paper finally npop s are classified and studied with regard to the possibility or impossibility of reducing certain types of npop s to other types in a sense specified in the text

the traveling salesman problem with distances one and two we present a polynomial time approximation algorithm with worst case ratio 7 6 for the special case of the traveling salesman problem in which all distances are either one or two we also show that this special case of the traveling salesman problem is max snp hard and therefore it is unlikely that it has a polynomial time approximation scheme

approximate solution of np optimization problems abstract this paper presents the main results obtained in the field of approximation algorithms in a unified framework most of these results have been revisited in order to emphasize two basic tools useful for characterizing approximation classes that is combinatorial properties of problems and approximation preserving reducibilities in particular after reviewing the most important combinatorial characterizations of the classes ptas and fptas we concentrate on the class apx and as a concluding result we show that this class coincides with the class of optimization problems which are reducible to the maximum satisfiability problem with respect to a polynomial time approximation preserving reducibility

2 matchings the traveling salesman problem and the subtour lp a proof of the boyd carr conjecture determining the precise integrality gap for the subtour linear programming lp relaxation of the traveling salesman problem is a significant open question with little progress made in thirty years in the general case of symmetric costs that obey triangle inequality boyd and carr boyd s carr r 2011 finding low cost tsp and 2 matching solutions using certain half integer subtour vertices discrete optim 8 525 539 prior version accessed june 27 2011 http www site uottawa ca sylvia recentpapers halftri pdf observe that we do not even know the worst case upper bound on the ratio of the optimal 2 matching to the subtour lp they conjecture the ratio is at most 10 9 in this paper we prove the boyd carr conjecture in the case that the support of a fractional 2 matching has no cut edge we can further prove that an optimal 2 matching has cost at most 10 9 times the cost of the fractional 2 matching

robust characterizations of polynomials withapplications to program testing the study of self testing and self correcting programs leads to the search for robust characterizations of functions here the authors make this notion precise and show such a characterization for polynomials from this characterization the authors get the following applications simple and efficient self testers for polynomial functions are constructed the characterizations provide results in the area of coding theory by giving extremely fast and efficient error detecting schemes for some well known codes this error detection scheme plays a crucial role in subsequent results on the hardness of approximating some np optimization problems

viscosity solutions of a degenerate parabolic elliptic system arising in the mean field theory of superconductivity in a type ii superconductor the magnetic field penetrates the superconducting body through the formation of vortices in an extreme type ii superconductor these vortices reduce to line singularities because the number of vortices is large it seems feasible to model their evolution by an averaged problem known as the mean field model of superconductivity we assume that the evolution law of an individual vortex which underlies the averaging process involves the current of the generated magnetic field as well as the curvature vector in the present paper we study a two dimensional reduction assuming all vortices to be perpendicular to a given direction since both the magnetic field h and the averaged vorticity are curl free we may represent them via a scalar magnetic potential q and a scalar stream function respectively we study existence uniqueness and asymptotic behaviour of solutions q of the resulting degenerate elliptic parabolic system with curvature taken into account or not by means of viscosity and weak solutions in addition we relate q to solutions h of the mean field equations without curvature finally we construct special solutions of the corresponding stationary equations with two or more superconducting phases

proof verification and the hardness of approximation problems we show that every language in np has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof if a string is in the language then there exists a proof such that the verifier accepts with probability 1 i e for every choice of its random string for strings not in the language the verifier rejects every provided proof with probability at least 1 2 our result builds upon and improves a recent result of arora and safra 1998 whose verifiers examine a nonconstant number of bits in the proof though this number is a very slowly growing function of the input length as a consequence we prove that no max snp hard problem has a polynomial time approximation scheme unless np p the class max snp was defined by papadimitriou and yannakakis 1991 and hard problems for this class include vertex cover maximum satisfiability maximum cut metric tsp steiner trees and shortest superstring we also improve upon the clique hardness results of feige et al 1996 and arora and safra 1998 and show that there exists a positive e such that approximating the maximum clique size in an n vertex graph to within a factor of n e is np hard

the steiner traveling salesman problem with online edge blockages abstract we consider the online steiner traveling salesman problem in this problem we are given an edge weighted graph g v e and a subset d v of destination vertices with the optimization goal to find a minimum weight closed tour that traverses every destination vertex of d at least once during the traversal the salesman could encounter at most k non recoverable blocked edges the edge blockages are real time meaning that the salesman knows about a blocked edge whenever it occurs we first show a lower bound on the competitive ratio and present an online optimal algorithm for the problem while this optimal algorithm has non polynomial running time we present another online polynomial time near optimal algorithm for the problem experimental results show that our online polynomial time algorithm produces solutions very close to the offline optimal solutions

approximating the pareto curve with local search for the bicriteria tsp 1 2 problem local search has been widely used in combinatorial optimization local search in combinatorial optimization wiley new york 1997 however in the case of multicriteria optimization almost no results are known concerning the ability of local search algorithms to generate good solutions with performance guarantee in this paper we introduce such an approach for the classical traveling salesman problem tsp problem proc stoc 00 2000 pp 126 133 we show that it is possible to get in linear time a 3 2 approximate pareto curve using an original local search procedure based on the 2 opt neighborhood for the bicriteria tsp 1 2 problem where every edge is associated to a couple of distances which are either 1 or 2 math oper res 18 1 1993 1

tests of spatial randomness adjusted for an inhomogeneity a general framework in many applications it is of interest to test whether a spatial point pattern is randomly generated after adjusting for an underlying spatial inhomogeneity a great variety of different test statistics have been proposed for this purpose by scientists in different fields these are reviewed in this article despite apparent dissimilarities in terms of their original formulations most of these statistics can be placed into one general framework of which they are special cases this makes it easier to see exactly how they relate to one another and also to determine which test to use for a particular application the general framework can also be used for proposing new tests by combining properties of existing tests for developing theoretical foundations for these types of test statistics for doing structured comparative evaluations and for software development

polynomial approximation algorithms with performance guarantees an introduction by example we present a short overview on polynomial approximation of np hard problems we present the main approximability classes together with examples of problems belonging to them we also describe the general concept of approximability preserving reductions together with a discussion about their capacities and their limits finally we present a quick description of what it is commonly called inapproximability results such results provide limits on the approximability of the problems tackled

the stationary mean field model of superconductivity partial regularity of the free boundary abstract we prove a partial regularity result for a free boundary problem arising in the mean field theory of superconductivity to this end we extend a regularity result of caffarelli and friedman for the zero set of functions satisfying a bound v c v to functions only satisfying a bound of the form v c v where v x supp 0 t 1 v tx is the radially maximal function in addition we extend the unique continuation theorem of aronszajn and cordes to cover this case

improved integrality gap upper bounds for traveling salesperson problems with distances one and two abstract we prove new integrality gap upper bounds for the traveling salesperson problem with distances one and two 1 2 tsp we obtain these bounds by investigating the structure of the support graph of optimal solutions to the subtour elimination linear programming relaxation with one additional cutting plane inequality for undirected 1 2 tsp our main results are as follows all instances have an integrality gap of at most 5 4 instances admitting half integral solutions have integrality gap at most 7 6 instances admitting subcubic solutions of cost at most the order of the instance have integrality gap at most 10 9 even without the cutting plane this bound is tight and holds in particular for basic solutions in the fractional 2 matching polytope with cost at most the order of the instance for directed 1 2 tsp instances we show an integrality gap upper bound of 3 2 for general instances and of 5 4 for instances admitting half integral solutions we prove these bounds by providing local search algorithms that in polynomial time find 2 matchings with few components in the support of the solution we show that the run times of our algorithms cannot be considerably improved under standard complexity theoretic assumptions we show that finding improved tsp solutions via local search is intractable for edge change parameterized by the size of the neighborhoods even for instances with distances one and two this strengthens a result of daniel marx

probabilistic checking of proofs a new characterization of np we give a new characterization of np the class np contains exactly those languages l for which membership proofs a proof that an input x is in l can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof we discuss implications of this characterization specifically we show that approximating clique and independent set even in a very weak sense is np hard

a linear time approximation scheme for tsp in undirected planar graphs with edge weights we give an algorithm requiring o c 1 epsilon 2 n time to find an epsilon optimal traveling salesman tour in the shortest path metric defined by an undirected planar graph with nonnegative edge lengths for the case of all lengths equal to 1 the time required is o c 1 epsilon n

the traveling salesman problem on cubic and subcubic graphs we study the traveling salesman problem tsp on the metric completion of cubic and subcubic graphs which is known to be np hard the problem is of interest because of its relation to the famous 4 3 conjecture for metric tsp which says that the integrality gap i e the worst case ratio between the optimal value of a tsp instance and that of its linear programming relaxation the subtour elimination relaxation is 4 3 we present the first algorithm for cubic graphs with approximation ratio 4 3 the proof uses polyhedral techniques in a surprising way which is of independent interest in fact we prove constructively that for any cubic graph on n vertices a tour of length 4n 3 2 exists which also implies the 4 3 conjecture as an upper bound for this class of graph tsp recently momke and svensson presented an algorithm that gives a 1 461 approximation for graph tsp on general graphs and as a side result a 4 3 approximation algorithm for this problem on subcubic graphs also settling the 4 3 conjecture for this class of graph tsp the algorithm by momke and svensson is initially randomized but the authors remark that derandomization is trivial we will present a different way to derandomize their algorithm which leads to a faster running time all of the latter also works for multigraphs

on the hardness of approximating minimization problems we prove results indicating that it is hard to compute efficiently good approximate solutions to the graph coloring set covering and other related minimization problems specifically there is an e 0 such that graph coloring cannot be approximated with ratio n e unless p np set covering cannot be approximated with ratio c log n for any c 1 4 unless np is contained in dtime n poly log n similar results follow for related problems such as clique cover fractional chromatic number dominating set and others

second order analysis of inhomogeneous spatial point processes using case control data methods for the statistical analysis of stationary spatial point process data are now well established methods for nonstationary processes less so one of many sources of nonstationary point process data is a case control study in environmental epidemiology in that context the data consist of a realization of each of two spatial point processes representing the locations within a specified geographical region of individual cases of a disease and of controls drawn at random from the population at risk in this article we extend work by baddeley moller and waagepetersen 2000 statistica neerlandica54 329 350 concerning estimation of the second order properties of a nonstationary spatial point process first we show how case control data can be used to overcome the problems encountered when using the same data to estimate both a spatially varying intensity and second order properties second we propose a semiparametric method for adjusting the estimate of intensity so as to take account of explanatory variables attached to the cases and controls our primary focus is estimation but we also propose a new test for spatial clustering that we show to be competitive with existing tests we describe an application to an ecological study in which juvenile and surviving adult trees assume the roles of controls and cases

detection of spatial disease clusters with lisa functions detection of disease clusters is an important tool in epidemiology that can help to identify risk factors associated with the disease and in understanding its etiology in this article we propose a method for the detection of spatial clusters where the locations of a set of cases and a set of controls are available the method is based on local indicators of spatial association functions lisa functions particularly on the development of a local version of the product density which is a second order characteristic of spatial point processes the behavior of the method is evaluated and compared with kulldorff s spatial scan statistic by means of a simulation study it is shown that the lisa method yields high sensitivity and specificity when it is used to detect simulated clusters of different sizes and shapes it also performs better than the spatial scan statistic when they are used to detect clusters of irregular shape however it presents relatively high type i error in situations where the number of cases is high both methods are applied for detecting spatial clusters of kidney disease in the city of valencia spain in the year 2008

approximation algorithms for the scaffolding problem and its generalizations scaffolding is one of the main stages in genome assembly during this stage we want to merge contigs assembled from the paired end reads into bigger chains called scaffolds for this purpose the following graph theoretical problem has been proposed given an edge weighted complete graph g and a perfect matching d of g we wish to find a hamiltonian path p in g such that all edges of d appear in p and the total weight of edges in p but not in d is maximized this problem is np hard and the previously best polynomial time approximation algorithm for it achieves a ratio of 12 in this paper we design a new polynomial time approximation algorithm achieving a ratio of 5 5 9 8 for any constant 0 1 several generalizations of the problem have also been introduced in the literature and we present polynomial time approximation algorithms for them that achieve better approximation ratios than the previous bests in particular one of the algorithms answers an open question

traveling salesman problems in temporal graphs abstract in this work we introduce the notion of time to some well known combinatorial optimization problems in particular we study problems defined on temporal graphs a temporal graph d v a may be viewed as a time sequence g 1 g 2 g l of static graphs over the same static set of nodes v each g t d t v a t is called the instance of d at time t and l is called the lifetime of d our main focus is on analogues of traveling salesman problems in temporal graphs a sequence of time labeled edges e g a tour is called temporal if its labels are strictly increasing we begin by considering the problem of exploring the nodes of a temporal graph as soon as possible in contrast to the positive results known for the static case we prove that it cannot be approximated within cn for some constant c 0 in a special case of temporal graphs and within 2 e for every constant e 0 in another special case in which d t is strongly connected for all 1 t l both unless p np we then study the temporal analogue of tsp 1 2 abbreviated ttsp 1 2 where for all 1 t l d t is a complete weighted graph with edge costs from 1 2 and the cost of an edge may vary from instance to instance the goal is to find a minimum cost temporal tsp tour we give several polynomial time approximation algorithms for ttsp 1 2 our best approximation is 1 7 e for the generic ttsp 1 2 and 13 8 e for its interesting special case in which the lifetime of the temporal graph is restricted to n in the way we also introduce temporal versions of other fundamental combinatorial optimization problems for which we obtain polynomial time approximation algorithms and hardness results

reductions completeness and the hardness of approximability in computability and in complexity theory reductions are widely used for mapping sets into sets in order to prove undecidability or hardness results in the study of the approximate solvability of hard discrete optimization problems suitable kinds of reductions called approximation preserving reductions can also be used to transfer from one problem to another either positive results solution techniques or negative results non approximability results in this paper various kinds of approximation preserving reductions are surveyed and their properties discussed the role of completeness under approximation preserving reductions is also analyzed and its relationship with hardness of approximability is explained

maximizing the number of spanning trees in a connected graph we study the problem of maximizing the number of spanning trees in a connected graph with n vertices and m edges by adding at most k edges from a given set of q candidate edges a problem that has applications in many domains we give both algorithmic and hardness results for this problem 1 we give a greedy algorithm that obtains an approximation ratio of 1 1 e epsilon in the exponent of the number of spanning trees for any epsilon 0 in time widetilde o m epsilon 1 n q epsilon 3 where widetilde o cdot hides mathrm poly log n factors our running time is optimal with respect to the input size up to logarithmic factors and improves on the o n 3 running time of the previous proposed greedy algorithm with an approximation ratio 1 1 e in the exponent notably the independence of our running time of k is novel compared to conventional top k selections on graphs that usually run in omega mk time 2 we show the exponential inapproximability of this problem by proving that there exists a constant c 0 such that it is np hard to approximate the optimum number of spanning trees in the exponent within 1 c

approximation algorithms for the maximum hamiltonian path problem with specified endpoint s abstract this paper deals with the problem of constructing hamiltonian paths of optimal weight called hpps t if the two endpoints are specified hpps if only one endpoint is specified we show that hpps t is 1 2 differential approximable and hpps is 2 3 differential approximable moreover we observe that these problems cannot be differential approximable better than 741 742 based upon these results we obtain new bounds for standard ratio a 1 2 standard approximation for m ax hpps t and a 2 3 for m ax hpps which can be improved to 2 3 for m ax hpps t a 2a all the edge weights are within an interval a 2a to 5 6 for m ax hpps a 2a and to 2 3 for m in hpps t a 2a to 3 4 for m in hpps a 2a

on approximating restricted cycle covers a cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle an l cycle cover is a cycle cover in which the length of every cycle is in the set l the weight of a cycle cover of an edge weighted graph is the sum of the weights of its edges we come close to settling the complexity and approximability of computing l cycle covers on the one hand we show that for almost all l computing l cycle covers of maximum weight in directed and undirected graphs is mathsf apx hard most of our hardness results hold even if the edge weights are restricted to zero and one on the other hand we show that the problem of computing l cycle covers of maximum weight can be approximated within a factor of 2 for undirected graphs and within a factor of 8 3 in the case of directed graphs this holds for arbitrary sets l

on the complexity of approximating the independent set problem we show that for some positive constant c it is not feasible to approximate independent set for graphs of n nodes within a factor of n c provided maximum 2 satisfiability does not have a randomized polynomial time approximation scheme we also study reductions preserving the quality of approximations and exhibit complete problems

the checkpoint problem in this paper we consider the checkpoint problem the input consists of an undirected graph g a set of source destination pairs s 1 t 1 s 2 t 2 s k t k and a collection p of paths connecting the s i t i pairs a feasible solution is a multicut e namely a set of edges whose removal disconnects every source destination pair for each email protected p we define cp e p email protected e in the sum checkpoint scp problem the goal is to minimize p pcp e p while in the maximum checkpoint mcp problem the goal is to minimize max p pcp e p these problems have several natural applications e g in urban transportation and network security in a sense they combine the multicut problem and the minimum membership set cover problem for the sum objective we show that weighted scp is equivalent with respect to approximability to undirected multicut thus there exists an o logn approximation for scp in general graphs our current approximability results for the max objective have a wide gap we provide an approximation factor of o nlogn opt for mcp and a hardness of 2 under the assumption p np the hardness holds for trees this solves an open problem of nelson 2009 25 we complement the lower bound by an almost matching upper bound with an asymptotic approximation factor of 2 on trees with all s i t i having an ancestor descendant relation we give a combinatorial exact algorithm besides the algorithm being combinatorial its running time improves by many orders of magnitude the lp algorithm that follows from total unimodularity finally we show strong hardness for the well known problem of finding a path with minimum forbidden pairs which in a sense can be considered the dual to the checkpoint problem despite various works on this problem hardness of approximation was not known prior to this work we show that the problem cannot be approximated within cn for some constant c 0 unless p np this is the strongest type of hardness possible it carries over to directed acyclic graphs and is a huge improvement over the plain np hardness of gabow h n gabow finding paths and cycles of superpolylogarithmic length siam j comput 36 6 2007 1648 1671

linearity testing in characteristic two let dist f g pr sub u f u spl ne g u denote the relative distance between functions f g mapping from a group g to a group h and let dist f denote the minimum over all linear functions homomorphisms g of dist f g given a function f g spl rarr h we let err f pr sub u spl upsi f u f spl upsi spl ne f u spl upsi denote the rejection probability of the blum luby rubinfeld 1993 linearity test linearity testing is the study of the relationship between err f and dist f and in particular lower bounds on err f in terms of dist f we discuss when the underlying groups are g gf 2 sup n and h gf 2 in this case the collection of linear functions describe a hadamard code of block length 2 sup n and for an arbitrary function f mapping gf 2 sup n to gf 2 the distance dist l measures its distance to a hadamard code err f is a parameter that is easy to measure and linearity testing studies the relationship of this parameter to the distance of f the code and corresponding test are used in the construction of efficient probabilistically checkable proofs and thence in the derivation of hardness of approximation improved analyses translate into better nonapproximability results we present a description of the relationship between err f and dist f which is nearly complete in all its aspects and entirely complete in some we present functions l u 0 1 spl rarr 0 1 such that for all x spl isin 0 1 we have l x spl les err f spl les u x whenever dist f x with the upper bound being tight on the whole range and the lower bound tight on a large part of the range and close on the rest part of our strengthening is obtained by showing a new connection between linearity testing and fourier analysis

differential approximation results for the traveling salesman problem with distances 1 and 2 abstract we prove that both minimum and maximum traveling salesman problems on complete graphs with edge distances 1 and 2 denoted by min tsp12 and max tsp12 respectively are approximable within 3 4 based upon this result we improve the standard approximation ratio known for maximum traveling salesman with distances 1 and 2 from 3 4 to 7 8 finally we prove that for any 0 it is np hard to approximate both problems better than within 741 742 the same results hold when dealing with a generalization of min and max tsp12 where instead of 1 and 2 edges are valued by a and b

hardness of approximation for vertex connectivity network design problems in the survivable network design problem sndp the goal is to find a minimum cost spanning subgraph satisfying certain connectivity requirements we study the vertex connectivity variant of sndp in which the input specifies for each pair of vertices a required number of vertex disjoint paths connecting them we give the first strong lower bound on the approximability of sndp showing that the problem admits no efficient 2 log 1 epsilon n ratio approximation for any fixed epsilon 0 unless np subseteq dtime n polylog n we show hardness of approximation results for some important special cases of sndp and we exhibit the first lower bound on the approximability of the related classical np hard problem of augmenting the connectivity of a graph using edges from a given set

the approximability of the weighted hamiltonian path completion problem on a tree given a graph the hamiltonian path completion problem is to find an augmenting edge set such that the augmented graph has a hamiltonian path in this paper we show that the hamiltonian path completion problem will unlikely have any constant ratio approximation algorithm unless np p this problem remains hard to approximate even when the given subgraph is a tree moreover if the edge weights are restricted to be either 1 or 2 the hamiltonian path completion problem on a tree is still np hard then it is observed that this problem is strongly np hard so it does not have any fully polynomial time approximation scheme fptas unless np p when the given tree is a k tree we give an approximation algorithm with performance ratio 1 5

partial regularity of the zero set of solutions of linear and superlinear elliptic equations on considere l ensemble zero u 0 des solutions de u f x u u dans ou est un ensemble ouvert dans r m et f x u u a u b u ou a 0 b 0 et 1 1

linear approximation of shortest superstrings we consider the following problem given a collection of strings s 1 s m find the shortest string s such that each s i appears as a substring a consecutive block of s although this problem is known to be np hard a simple greedy procedure appears to do quite well and is routinely used in dna sequencing and data compression practice namely repeatedly merge the pair of distinct strings with maximum overlap until only one string remains let n denote the length of the optimal superstring a common conjecture states that the above greedy procedure produces a superstring of length o n in fact 2 n yet the only previous nontrivial bound known for any polynomial time algorithm is a recent o n log n result we show that the greedy algorithm does in fact achieve a constant factor approximation proving an upper bound of 4 n furthermore we present a simple modified version of the greedy algorithm that we show produces a superstring of length at most 3 n we also show the superstring problem to be maxsnp hard which implies that a polynomial time approximation scheme for this problem is unlikely

a better differential approximation ratio for symmetric tsp in this paper we study the approximability properties of symmetric tsp under an approximation measure called the differential ratio more precisely we improve up to 3 email protected for any e 0 the best differential ratio of 2 3 known so far given in hassin and khuller r hassin s khuller z approximations j algorithms 41 2 2001 429 442

completeness in standard and differential approximation classes poly d apx and d ptas completeness several problems are known to be apx dapx ptas or poly apx pb complete under suitably defined approximation preserving reductions but to our knowledge no natural problem is known to be ptas complete and no problem at all is known to be poly apx complete on the other hand dptas and poly dapx completeness have not been studied until now we first prove in this paper the existence of natural poly apx and poly dapx complete problems under the well known ptas reduction and under the dptas reduction defined in g ausiello c bazgan m demange and v th paschos completeness in differential approximation classes mfcs 03 respectively next we deal with ptas and dptas completeness we introduce approximation preserving reductions called ft and dft respectively and prove that under these new reductions natural problems are ptas complete or dptas complete then we deal with the existence of intermediate problems under our reductions and we partially answer this question showing that the existence of npo intermediate problems under turing reductions is a sufficient condition for the existence of intermediate problems under both ft and dft reductions finally we show that min coloring is dapx complete under dptas reductions this is the first dapx complete problem that is not simultaneously apx complete

fast detection of arbitrarily shaped disease clusters disease cluster detection and evaluation have commonly used spatial statistics methods that scan the map with a fixed circular window to locate candidate clusters recently there has been interest in searching for clusters with arbitrary shape the circular scan test retains high power of detecting a cluster but does not necessarily identify the exact regions contained in a non circular cluster particularly well we propose implement and evaluate a new procedure that is fast and produces clusters estimates of arbitrary shape in a rich class of possible cluster candidates we showed that our methods contain the so called upper level set method as a particular case we present a power study of our method and among other results the main conclusion is that the likelihood based arbitrarily shaped scan method is not appropriate to find a cluster estimate when the parameter space includes the set of all possible spatial clusters in a map a large and discrete parameter space maximum likely cluster estimates tend to overestimate the true cluster by a large extent this calls for a new approach different from the maximum likelihood method for this important public health problem

on unapproximable versions of np complete problems we prove that all of karp s 21 original np complete problems have a version that is hard to approximate these versions are obtained from the original problems by adding essentially the same simple constraint we further show that these problems are absurdly hard to approximate in fact no polynomial time algorithm can even approximate log k of the magnitude of these problems to within any constant factor where logk denotes the logarithm iterated k times unless np is recognized by slightly superpolynomial randomized machines we use the same technique to improve the constant epsilon such that max clique is hard to approximate to within a factor of n epsilon finally we show that it is even harder to approximate two counting problems counting the number of satisfying assignments to a monotone 2sat formula and computing the permanent of 1 0 1 matrices

on the held karp relaxation for the asymmetric and symmetric traveling salesman problems a long standing conjecture in combinatorial optimization says that the integrality gap of the famous held karp relaxation of the metric stsp symmetric traveling salesman problem is precisely 4 3 in this paper we show that a slight strengthening of this conjecture implies a tight 4 3 integrality gap for a linear programming relaxation of the metric atsp asymmetric traveling salesman problem our main tools are a new characterization of the integrality gap for linear objective functions over polyhedra and the isolation of hard to round solutions of the relaxations

two algorithmic results for the traveling salesman problem for any norm in a euclidean space and for any number 0 we present a polynomial time algorithm which computes a hamiltonian circuit with the given vertices in the space whose length approximates with relative error less than the largest length of a hamiltonian circuit with these vertices we also present an algorithm which for a given graph with n vertices computes the number of hamiltonian circuits in this graph with 2n olog n time complexity and a polynomial in n space complexity as a by product of our approach we prove that the permanent of a matrix can be computed in polynomial time provided the rank of the matrix is fixed

on the integrality gap of the subtour lp for the 1 2 tsp in this paper we study the integrality gap of the subtour lp relaxation for the traveling salesman problem in the special case when all edge costs are either 1 or 2 for the general case of symmetric costs that obey triangle inequality a famous conjecture is that the integrality gap is 4 3 little progress towards resolving this conjecture has been made in 30 years we conjecture that when all edge costs c ij in 1 2 cij 1 2 the integrality gap is 10 9 we show that this conjecture is true when the optimal subtour lp solution has a certain structure under a weaker assumption which is an analog of a recent conjecture by schalekamp et al we show that the integrality gap is at most 7 6 when we do not make any assumptions on the structure of the optimal subtour lp solution we can show that integrality gap is at most 5 4 this is the first bound on the integrality gap of the subtour lp strictly less than 4 3 known for an interesting special case of the tsp we show computationally that the integrality gap is at most 10 9 for all instances with at most 12 cities

a faster better approximation algorithm for the minimum latency problem we give a 7 18 approximation algorithm for the minimum latency problem that uses only o n log n calls to the prize collecting steiner tree pcst subroutine of goemans and williamson this improves the previous best algorithms in both performance guarantee and running time a previous algorithm of goemans and kleinberg for the minimum latency problem requires an approximation algorithm for the k minimum spanning tree k mst problem which is called as a black box for each value of k their algorithm can achieve an approximation factor of 10 77 while making o n n log c log n pcst calls a factor of 8 98 using o n 3 n log c log n pcst calls or a factor of 7 18 epsilon using n o 1 epsilon log c pcst calls via the k mst algorithms of garg arya and ramesh and arora and karakostas respectively here n denotes the number of nodes in the instance and c is the largest edge cost in the input in all cases the running time is dominated by the pcst calls since the pcst subroutine can be implemented to run in o n 2 time the overall running time of our algorithm is o n 3 log n we also give a faster randomized version of our algorithm that achieves the same approximation guarantee in expectation but uses only o log 2 n pcst calls and derandomize it to obtain a deterministic algorithm with factor 7 18 epsilon using o frac 1 epsilon log 2 n pcst calls the basic idea for our improvement is that we do not treat the k mst algorithm as a black box this allows us to take advantage of some special situations in which the pcst subroutine delivers a 2 approximate k mst we are able to obtain the same approximation ratio that would be given by goemans and kleinberg if we had access to 2 approximate k msts for all values of k even though we have them only for some values of k that we are not able to specify in advance we also extend our algorithm to a weighted version of the minimum latency problem

guillotine subdivisions approximate polygonal subdivisions a simple polynomial time approximation scheme for geometric tsp k mst and related problems we show that any polygonal subdivision in the plane can be converted into an m guillotine subdivision whose length is at most 1 c over m times that of the original subdivision for a small constant c m guillotine subdivisions have a simple recursive structure that allows one to search for the shortest of such subdivisions in polynomial time using dynamic programming in particular a consequence of our main theorem is a simple polynomial time approximation scheme for geometric instances of several network optimization problems including the steiner minimum spanning tree the traveling salesperson problem tsp and the k mst problem

approximation algorithms for asymmetric tsp by decomposing directed regular multigraphs a directed multigraph is said to be d regular if the indegree and outdegree of every vertex is exactly d by hall s theorem one can represent such a multigraph as a combination of at most n2 cycle covers each taken with an appropriate multiplicity we prove that if the d regular multigraph does not contain more than d 2 copies of any 2 cycle then we can find a similar decomposition into n2 pairs of cycle covers where each 2 cycle occurs in at most one component of each pair our proof is constructive and gives a polynomial algorithm to find such a decomposition since our applications only need one such a pair of cycle covers whose weight is at least the average weight of all pairs we also give an alternative simpler algorithm to extract a single such pair this combinatorial theorem then comes handy in rounding a fractional solution of an lp relaxation of the maximum traveling salesman problem tsp problem the first stage of the rounding procedure obtains two cycle covers that do not share a 2 cycle with weight at least twice the weight of the optimal solution then we show how to extract a tour from the 2 cycle covers whose weight is at least 2 3 of the weight of the longest tour this improves upon the previous 5 8 approximation with a simpler algorithm utilizing a reduction from maximum tsp to the shortest superstring problem we obtain a 2 5 approximation algorithm for the latter problem which is again much simpler than the previous one for minimum asymmetric tsp the same technique gives two cycle covers not sharing a 2 cycle with weight at most twice the weight of the optimum assuming triangle inequality we then show how to obtain from this pair of cycle covers a tour whose weight is at most 0 842 log 2 n larger than optimal this improves upon a previous approximation algorithm with approximation guarantee of 0 999 log 2 n other applications of the rounding procedure are approximation algorithms for maximum 3 cycle cover factor 2 3 previously 3 5 and maximum asymmetric tsp with triangle inequality factor 10 13 previously 3 4

on the power of multi prover interactive protocols abstract we look at complexity issues of interactive proof systems with multiple provers separated from each other this model developed by ben or et al 1988 allows the verifier to play the provers off each other we show this model equivalent to an alternative interactive proof system model using oracles as provers we also show that every language accepted by these models lies in nondeterministic exponential time we exhibit a relativized world where a co np language does not have multiple prover interactive proofs finally we show a simple example that one cannot parallelize multiple prover protocols as easily as the single prover model

the geometric maximum traveling salesman problem we consider the traveling salesman problem when the cities are points in n x211d d for some fixed d and distances are computed according to geometric distances determined by some norm we show that for any polyhedral norm the problem of finding a tour of maximum length can be solved in polynomial time if arithmetic operations are assumed to take unit time our algorithms run in time o nf 2 log n where f is the number of facets of the polyhedron determining the polyhedral norm thus for example we have o n2 log n algorithms for the cases of points in the plane under the rectilinear and sup norms this is in contrast to the fact that finding a minimum length tour in each case is np hard our approach can be extended to the more general case of quasi norms with a not necessarily symmetric unit ball where we get a complexity of o n2f 2 log n for the special case of two dimensional metrics with f 4 which includes the rectilinear and sup norms we present a simple algorithm with o n running time the algorithm does not use any indirect addressing so its running time remains valid even in comparison based models in which sorting requires n log n time the basic mechanism of the algorithm provides some intuition on why polyhedral norms allow fast algorithms complementing the results on simplicity for polyhedral norms we prove that for the case of euclidean distances in n x211d d for d 3 the maximum tsp is np hard this sheds new light on the well studied difficulties of euclidean distances

the complexity of approximating a nonlinear program we consider the problem of finding the maximum of a multivariate polynomial inside a convex polytope we show that there is no polynomial time approximation algorithm for this problem even one with a very poor guarantee unless p np we show that even when the polynomial is quadratic i e quadratic programming there is no polynomial time approximation unless np is contained in quasi polynomial time