random k sat two moments suffice to cross a sharp threshold many np complete constraint satisfaction problems appear to undergo a phase transition from solubility to insolubility when the constraint density passes through a critical threshold in all such cases it is easy to derive upper bounds on the location of the threshold by showing that above a certain density the first moment expectation of the number of solutions tends to zero we show that in the case of certain symmetric constraints considering the second moment of the number of solutions yields nearly matching lower bounds for the location of the threshold specifically we prove that the threshold for both random hypergraph 2 colorability property b and random not all equal k sat is 2 k 1 ln 2 o 1 as a corollary we establish that the threshold for random k sat is of order theta 2 k resolving a long standing open problem

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

interactive proofs and the hardness of approximating cliques the contribution of this paper is two fold first a connection is established between approximating the size of the largest clique in a graph and multi prover interactive proofs second an efficient multi prover interactive proof for np languages is constructed where the verifier uses very few random bits and communication bits last the connection between cliques and efficient multi prover interaction proofs is shown to yield hardness results on the complexity of approximating the size of the largest clique in a graph of independent interest is our proof of correctness for the multilinearity test of functions

lower bounds for the happy coloring problems in this paper we study the maximum happy vertices and the maximum happy edges problems mhv and mhe for short very recently the problems attracted a lot of attention and were studied in agrawal 17 aravind et al 16 choudhari and reddy 18 misra and reddy 17 main focus of our work is lower bounds on the computational complexity of these problems established lower bounds can be divided into the following groups np hardness of the above guarantee parameterization kernelization lower bounds answering questions of misra and reddy 17 exponential lower bounds under the set cover conjecture and the exponential time hypothesis and inapproximability results moreover we present an mathcal o ell k randomized algorithm for mhv and an mathcal o 2 k algorithm for mhe where ell is the number of colors used and k is the number of required happy vertices or edges these algorithms cannot be improved to subexponential taking proved lower bounds into account

approximation algorithms for node deletion problems on bipartite graphs with finite forbidden subgraph characterization in this paper we develop approximation algorithms for a few node deletion problems when the input is restricted to be a bipartite graph we look at node deletion problems for non trivial properties which can be characterized by forbidden structure which has a bounded intersection with both the bipartitions the approximation factors obtained directly depend upon the size of the largest such intersection special instances of this general problem include problems such as the minimum chain vertex deletion minimum dissociation vertex deletion minimum bipartite claw vertex deletion minimum bi complement vertex deletion and minimum bipartite threshold vertex deletion problems the algorithms are based upon the techniques of linear programming and iterative rounding we also use the node deletion algorithms to marginally improve the trivial approximation factor for complementary problem of determining the size of the maximum sized vertex induced subgraph lying in the given graph class and prove the apx completeness of all of these problems

property testing and its connection to learning and approximation in this paper we consider the question of determining whether a function f has property p or is e far from any function with property p a property testing algorithm is given a sample of the value of f on instances drawn according to some distribution in some cases it is also allowed to query f on instances of its choice we study this question for different properties and establish some connections to problems in learning theory and approximation in particular we focus our attention on testing graph properties given access to a graph g in the form of being able to query whether an edge exists or not between a pair of vertices we devise algorithms to test whether the underlying graph has properties such as being bipartite k colorable or having a p clique clique of density p with respect to the vertex set our graph property testing algorithms are probabilistic and make assertions that are correct with high probability while making a number of queries that is independent of the size of the graph moreover the property testing algorithms can be used to efficiently i e in time linear in the number of vertices construct partitions of the graph that correspond to the property being tested if it holds for the input graph

satisfiability allows no nontrivial sparsification unless the polynomial time hierarchy collapses consider the following two player communication process to decide a language l the first player holds the entire input x but is polynomially bounded the second player is computationally unbounded but does not know any part of x their goal is to decide cooperatively whether x belongs to l at small cost where the cost measure is the number of bits of communication from the first player to the second player for any integer d 3 and positive real e we show that if satisfiability for n variable d cnf formulas has a protocol of cost o nd e then conp is in np poly which implies that the polynomial time hierarchy collapses to its third level the result even holds when the first player is conondeterministic and is tight as there exists a trivial protocol for e 0 under the hypothesis that conp is not in np poly our result implies tight lower bounds for parameters of interest in several areas namely sparsification kernelization in parameterized complexity lossy compression and probabilistically checkable proofs by reduction similar results hold for other np complete problems for the vertex cover problem on n vertex d uniform hypergraphs this statement holds for any integer d 2 the case d 2 implies that no np hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of o k2 e edges unless conp is in np poly where k denotes the size of the deletion set kernels consisting of o k2 edges are known for several problems in the class including vertex cover feedback vertex set and bounded degree deletion

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

quadratic vertex kernel for split vertex deletion abstract a graph is called a split graph if its vertex set can be partitioned into a clique and an independent set split graphs have rich mathematical structure and interesting algorithmic properties making it one of the most well studied special graph classes in the split vertex deletion problem given a graph and a positive integer k the objective is to test whether there exists a subset of at most k vertices whose deletion results in a split graph in this paper we design a kernel for this problem with o k 2 vertices improving upon the previous cubic bound known also by giving a simple reduction from the vertex cover problem we establish that split vertex deletion does not admit a kernel with o k 2 edges for any 0 unless np conp poly

kernel bounds for disjoint cycles and disjoint paths in this paper we show that the problems disjoint cycles and disjoint paths do not have polynomial kernels unless np conp poly thus these problems do not allow polynomial time preprocessing that results in instances whose size is bounded by a polynomial in the parameter at hand we build upon recent results by bodlaender et al 6 and fortnow and santhanam 20 that show that np complete problems that are or compositional do not have polynomial kernels unless np conp poly to this machinery we add a notion of transformation and obtain that disjoint cycles and disjoint paths do not have polynomial kernels unless np conp poly for the proof we introduce a problem on strings called disjoint factors and first show that this problem has no polynomial kernel unless np conp poly we also show that the related disjoint cycles packing problem has a kernel of size o klogk

the strong perfect graph theorem a graph g is perfect if for every induced subgraph h the chromatic number of h equals the size of the largest complete subgraph of h and g is berge if no induced subgraph of g is an odd cycle of length at least five or the complement of one the strong perfect graph conjecture berge 1961 asserts that a graph is perfect if and only if it is berge a stronger conjecture was made recently by conforti cornu ejols and vuiskovi c that every berge graph either falls into one of a few basic classes or admits one of a few kinds of separation designed so that a minimum counterexample to berge s conjecture cannot have either of these properties in this paper we prove both of these conjectures

linear time algorithms for linear programming in r 3 and related problems linear time algorithms for linear programming in r 2 and r 3 are presented the methods used are applicable for other graphic and geometric problems as well as quadratic programming for exam

improved upper bounds for vertex cover this paper presents an o 1 2738 k kn time polynomial space algorithm for vertex cover improving the previous o 1 286 k kn time polynomial space upper bound by chen kanj and jia most of the previous algorithms rely on exhaustive case by case branching rules and an underlying conservative worst case scenario assumption the contribution of the paper lies in the simplicity uniformity and obliviousness of the algorithm presented several new techniques as well as generalizations of previous techniques are introduced including general folding struction tuples and local amortized analysis the algorithm also improves the o 1 2745 kk 4 kn time exponential space upper bound for the problem by chandran and grandoni

worst case analysis of greedy heuristics for integer programming with nonnegative data we give a worst case analysis for two greedy heuristics for the integer programming problem minimize cx ax b 0 x u x integer where the entries in a b and c are all nonnegative the first heuristic is for the case where the entries in a and b are integral the second only assumes the rows are scaled so that the smallest nonzero entry is at least 1 in both cases we compare the ratio of the value of the greedy solution to that of the integer optimal the error bound grows logarithmically in the maximum column sum of a for both heuristics

polynomial kernels for hitting forbidden minors under structural parameterizations abstract we investigate polynomial time preprocessing for the problem of hitting forbidden minors in a graph using the framework of kernelization for a fixed finite set of connected graphs f the f deletion problem is the following given a graph g and integer k is it possible to delete k vertices from g to ensure the resulting graph does not contain any graph from f as a minor earlier work by fomin lokshtanov misra and saurabh focs 12 showed that when f contains a planar graph an instance g k can be reduced in polynomial time to an equivalent one of size k o 1 in this work we focus on structural measures of the complexity of an instance with the aim of giving nontrivial preprocessing guarantees for instances whose solutions are large motivated by several impossibility results we parameterize the f deletion problem by the size of a vertex modulator whose removal results in a graph of constant treedepth we prove that for each set f of connected graphs and constant the f deletion problem parameterized by the size of a treedepth modulator has a polynomial kernel our kernelization is fully explicit and does not depend on protrusion reduction or well quasi ordering which are sources of algorithmic non constructivity in earlier works on f deletion our main technical contribution is to analyze how models of a forbidden minor in a graph g with modulator x interact with the various connected components of g x using the language of labeled minors we analyze the fragments of potential forbidden minor models that can remain after removing an optimal f deletion solution from a single connected component of g x by bounding the number of different types of behavior that can occur by a polynomial in x we obtain a polynomial kernel using a recursive preprocessing strategy our results extend earlier work for specific instances of f deletion such as vertex cover and feedback vertex set it also generalizes earlier preprocessing results for f deletion parameterized by a vertex cover which is a treedepth one modulator

improved bound for complexity of matrix multiplication in 1968 strassen 16 described a method of multiplying two 2 2 matrices using only seven multiplications by applying it recursively it was shown that two n n matrices could be multiplied using o n operations in all where log2 7 2 81 this discovery led to a period of activity devoted to the improvement of this bound one can define an exponent of complexity which is in effect the smallest number such that two n n matrices can be multiplied using o n operations using an method of the type described by strassen a precise definition is given in 2 then strassen s result says that log2 7 so it is natural to wish to determine the true value of in the decade or so after 1978 several successive reductions in the upper bound for were achieved 2 3 6 8 10 12 14 17 leading to the bound 2 375477 obtained by coppersmith and winograd 7 it is an open question whether 2 we present a small improvement of the coppersmith winograd bound namely 2 3736897 which was first obtained in stothers 15 the proof uses an extension of the ideas of 7 and relies heavily on a combinatorial construction from 7 which depends on probabilistic arguments and a theorem of salem and spencer on sets of integers containing no three terms in arithmetical progression we have tried to organize the proof so that it is reasonably concise but also self contained to make it more accessible in the belief that at least parts of the argument may be of wider interest outside the algebraic complexity community to this end we derive the results we need from algebraic complexity theory in 2 which also contains general background material in 3 we present the essence of the combinatorial construction as a lemma which we subsequently apply several times we first apply it in 4 to recover the coppersmith winograd bound although our presentation is rather different the proof is essentially the same as that of 7 except for the proof of lemma 4 1 for which we give a somewhat simpler argument using 4 as preparation and motivation we obtain our new bound in 5 the argument is similar but more complicated and technical

nondeterminism within p classes of machines using very limited amounts of nondeterminism are studied the p np question is related to questions about classes lying within p complete sets for these classes are given

parameterized complexity of vertex deletion into perfect graph classes vertex deletion problems are at the heart of parameterized complexity for a graph class f the f deletion problem takes as input a graph g and an integer k the question is whether it is possible to delete at most k vertices from g such that the resulting graph belongs to f whether perfect deletion is fixed parameter tractable and whether chordal deletion admits a polynomial kernel when parameterized by k have been stated as open questions in previous work we show that perfect deletion and weakly chordal deletion are w 2 hard when parameterized by k in search of positive results we study a restricted variant of the f deletion problem in this restricted variant the deleted vertices must be taken from a specified set x and we parameterize by x we show that for perfect deletion and weakly chordal deletion although this restriction immediately ensures fixed parameter tractability it is not enough to yield polynomial kernels unless np conp poly on the positive side for chordal deletion the restriction enables us to obtain a kernel with o x 4 vertices

subexponential parameterized algorithm for minimum fill in the minimum fill in problem is used to decide if a graph can be triangulated by adding at most k edges in 1994 kaplan shamir and tarjan showed that the problem is solvable in time mathcal o 2 mathcal o k k 2 nm on graphs with n vertices and m edges and thus is fixed parameter tractable here we give the first subexponential parameterized algorithm solving minimum fill in in time mathcal o 2 mathcal o sqrt k log k k 2 nm this substantially lowers the complexity of the problem techniques developed for minimum fill in can be used to obtain subexponential parameterized algorithms for several related problems including minimum chain completion chordal graph sandwich and triangulating colored graph

kernels for packing and covering problems abstract we show how the notion of combinatorial duality related to the well known notion of duality from linear programming may be used for translating kernel results obtained for packing problems into kernel results for covering problems we exemplify this approach by having a closer look at the problems of packing a graph with vertex disjoint trees or vertex disjoint stars with r edges the case r 2 has been studied in several other papers by establishing a general notion of a crown we show how linear size vertex kernels can be efficiently achieved for the mentioned problems

fuzzy facility location problem with preference of candidate sites models so far considered as facility location problems treat either min max or max min criterion that is the facility is either desirable or undesirable we propose the following model considering the satisfaction degree with respect to the distance from the facility for each customer residents and preference of the site in an urban area the objective is to find the site of the facility which maximizes the minimal satisfaction degree among all demand points and maximizes the preference of the site since generally speaking there exists no site that maximizes both criteria we seek non dominated sites after defining non domination

sharp thresholds of graph properties and the sat problem consider g n p to be the probability space of random graphs on n vertices with edge probability p we will be considering subsets of this space defined by monotone graph properties a monotone graph property p is a property of graphs such that a p is invariant under graph automorphisims b if graph h has property p then so does any graph g having h as a subgraph a monotone symmetric family of graphs is a family defined by such a property one of the first observations made about random graphs by erdos and renyi in their seminal work on random graph theory 12 was the existence of threshold phenomena the fact that for many interesting properties p the probability of p appearing in g n p exhibits a sharp increase at a certain critical value of the parameter p bollobas and thomason proved the existence of threshold functions for all monotone set properties 6 and in 14 it is shown that this behavior is quite general and that all monotone graph properties exhibit threshold behavior i e the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval more precise analysis of the size of the threshold interval is done in 7 this threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics one of the main questions that arises in studying phase transitions is how sharp is the transition for example one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random kcnf boolean formula nati linial who introduced me to this problem suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition answering the question posed in 14 as to the relation between the length of the threshold interval and the value of the critical probability in this paper we indeed introduce a simple condition and prove it is sufficient stated roughly in the setting of random graphs the main theorem states that if a property has a coarse threshold then it can be approximated by the property of having certain given graphs as a subgraph this condition can be applied in a more

representative sets and irrelevant vertices new tools for kernelization we continue the development of matroid based techniques for kernelization initiated by the present authors 47 we significantly extend the usefulness of matroid theory in kernelization by showing applications of a result on representative sets due to lovasz 51 and marx 53 as a first result we show how representative sets can be used to derive a polynomial kernel for the elusive almost 2 sat problem where the task is to remove at most k clauses to make a 2 cnf formula satisfiable solving a major open problem in kernelization this result also yields a new o log opt approximation for the problem improving on the o log n approximation of agarwal et al 3 and an implicit o log opt approximation due to even et al 24 we further apply the representative sets tool to the problem of finding irrelevant vertices in graph cut problems that is vertices that can be made undeletable without affecting the answer to the problem this gives the first significant progress towards a polynomial kernel for the multiway cut problem in particular we get a kernel of o ks 1 vertices for multiway cut instances with at most s terminals both these kernelization results have significant spin off effects producing the first polynomial kernels for a range of related problems more generally the irrelevant vertex results have implications for covering min cuts in graphs for a directed graph g v e and sets s t v let r be the size of a minimum s t vertex cut which may intersect s and t we can find a set z v of size o s t r that contains a minimum a b vertex cut for every a s b t similarly for an undirected graph g v e a set of terminals x v and a constant s we can find a set z v of size o x s 1 that contains a minimum multiway cut for every partition of x into at most s pairwise disjoint subsets both results are polynomial time we expect this to have further applications in particular we get direct reduction rule based kernelizations for all problems above in contrast to the indirect compression based kernel previously given for odd cycle transversal 47 all our results are randomized with failure probabilities that can be made exponentially small in n due to needing a representation of a matroid to apply the representative sets tool

a 13k kernel for planar feedback vertex set via region decomposition we show a kernel of at most 13k vertices for the feedback vertex set problem restricted to planar graphs i e a polynomial time algorithm that transforms an input instance g k to an equivalent instance with at most 13k vertices to this end we introduce a few new reduction rules however our main contribution is an application of the region decomposition technique in the analysis of the kernel size we show that our analysis is tight up to a constant additive term

some consequences of non uniform conditions on uniform classes we obtain some results of the form if certain complexity classes satisfy a non uniform condition then some unlikely consequences follow more precisely 1 1 if the non uniform polynomial time hierarchy collapses at level i 0 i e ipoly ipoly then the meyer stockmeyer hierarchy collapses at level i 2 i e i 2 i 2 this strengthens a generalization of a result of karp and lipton 1980 2 2 if co np is conjunctively reducible to a sparse set then p np this generalizes a theorem of fortune 1979 3 3 if np is conjunctively and disjunctively reducible to a sparse np complete set then p np this is a partial generalization of a result of mahaney 1980 conjuctive and disjunctive reducibility were introduced by ladner lynch and selman 1975 4 4 if co np is reducible to a sparse set then np co np reducibility was introduced 5 by adleman and manders 1977

a hybrid modified pso approach to var based facility location problems with variable capacity in fuzzy random uncertainty this paper studies a facility location model with fuzzy random parameters and its swarm intelligence approach a value at risk var based fuzzy random facility location model var frflm is built in which both the costs and demands are assumed to be fuzzy random variables and the capacity of each facility is unfixed but a decision variable assuming continuous values under this setting the var frflm is inherently a two stage mixed 0 1 integer fuzzy random programming problem to which analytical nonlinear programming methods are not applicable a hybrid modified particle swarm optimization mpso approach is proposed to solve the var frflm in this hybrid mechanism an approximation algorithm is utilized to compute the fuzzy random var objective a continuous nbest gbest based pso and a genotype phenotype based binary pso vehicles are designed to deal with the continuous capacity decisions and the binary location decisions respectively and two mutation operators are incorporated into the pso to further decrease the possibility of becoming trapped in the local optima a numerical experiment illustrates the application of the proposed hybrid mpso algorithm and lays out its robustness to the system parameter settings the comparison shows that the hybrid mpso exhibits better performance than that when hybrid regular continuous binary pso and genetic algorithm ga are used to solve the var frflm

energy of sections of the deligne hitchin twistor space we study a natural functional on the space of holomorphic sections of the deligne hitchin moduli space of a compact riemann surface generalizing the energy of equivariant harmonic maps corresponding to twistor lines we show that the energy is the residue of the pull back along the section of a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by hitchin as a byproduct we show the existence of a hyper kahler potentials for new components of real holomorphic sections of twistor spaces of hyper kahler manifolds with rotating s 1 action additionally we prove that for a certain class of real holomorphic sections of the deligne hitchin moduli space the energy functional is basically given by the willmore energy of corresponding equivariant conformal map to the 3 sphere as an application we use the functional to distinguish new components of real holomorphic sections of the deligne hitchin moduli space from the space of twistor lines

improving the performance guarantee for approximate graph coloring the performance guarantee of a graph coloring algorithm is the worst case ratio between the number of colors it uses on the input graph and the chromauc number of this graph the previous best known polynomial time algorithm had a performance guarantee o n logn for graphs on n vertices this result stood unchallenged for eight years this paper presents an efficient algorithm with performance guarantee of o n loglog n 2 logn 2

small approximate pareto sets for biobjective shortest paths and other problems we investigate the problem of computing a minimum set of solutions that approximates within a specified accuracy epsilon the pareto curve of a multiobjective optimization problem we show that for a broad class of biobjective problems containing many important widely studied problems such as shortest paths spanning tree matching and many others we can compute in polynomial time an epsilon pareto set that contains at most twice as many solutions as the minimum set furthermore we show that the factor of 2 is tight for these problems i e it is np hard to do better we present upper and lower bounds for three or more objectives as well as for the dual problem of computing a specified number k of solutions which provide a good approximation to the pareto curve

the pcp theorem by gap amplification the pcp theorem arora and safra 1998 arora et al 1998 says that every language in np has a witness format that can be checked probabilistically by reading only a constant number of bits from the proof the celebrated equivalence of this theorem and inapproximability of certain optimization problems due to feige et al 1996 has placed the pcp theorem at the heart of the area of inapproximability in this work we present a new proof of the pcp theorem that draws on this equivalence we give a combinatorial proof for the np hardness of approximating a certain constraint satisfaction problem which can then be reinterpreted to yield the pcp theorem our approach is to consider the unsat value of a constraint system which is the smallest fraction of unsatisfied constraints ranging over all possible assignments for the underlying variables we describe a new combinatorial amplification transformation that doubles the unsat value of a constraint system with only a linear blowup in the size of the system the amplification step causes an increase in alphabet size that is corrected by a standard pcp composition step iterative application of these two steps yields a proof for the pcp theorem the amplification lemma relies on a new notion of graph powering that can be applied to systems of binary constraints this powering amplifies the unsat value of a constraint system provided that the underlying graph structure is an expander we also extend our amplification lemma towards construction of assignment testers alternatively pcps of proximity which are slightly stronger objects than pcps we then construct pcps and locally testable codes whose length is linear up to a polylog factor and whose correctness can be probabilistically verified by making a constant number of queries namely we prove sat pcp 1 2 1 log2 n poly log n o 1

on the compressibility of mathcal np instances and cryptographic applications we study compression that preserves the solution to an instance of a problem rather than preserving the instance itself our focus is on the compressibility of mathcal np decision problems we consider mathcal np problems that have long instances but relatively short witnesses the question is whether one can efficiently compress an instance and store a shorter representation that maintains the information of whether the original input is in the language or not we want the length of the compressed instance to be polynomial in the length of the witness and polylog in the length of original input such compression enables succinctly storing instances until a future setting will allow solving them either via a technological or algorithmic breakthrough or simply until enough time has elapsed in this paper we first develop the basic complexity theory of compression including reducibility completeness and a stratification of mathcal np with respect to compression we then show that compressibility say of sat would have vast implications for cryptography including constructions of one way functions and collision resistant hash functions from any hard on average problem in mathcal np and cryptanalysis of key agreement protocols in the bounded storage model when mixed with time complexity based cryptography

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