knapsack problems a parameterized point of view abstract the knapsack problem kp is a very famous np hard problem in combinatorial optimization also its generalization to multiple dimensions named d dimensional knapsack problem d kp and to multiple knapsacks named multiple knapsack problem mkp are well known problems in this paper we give a first study on the fixed parameter tractability of these three problems the idea behind fixed parameter tractability is to split the complexity into two parts one part that depends purely on the size of the input and one part that depends on some parameter of the problem that tends to be small in practice further we consider the closely related question whether the sizes and the values can be reduced such that their bit length is bounded polynomially or even constantly in a given parameter i e the existence of kernelizations is studied we give several upper and some lower bounds on the parameterized complexity and kernel sizes for the following parameters the number of items the threshold value for the profit the sizes the profits the number d of dimensions and the number m of knapsacks we also consider the connection of parameterized knapsack problems to linear programming approximation and pseudo polynomial algorithms

on a singular perturbation problem involving a circular well potential we study the asymptotic behavior as a small parameter goes to 0 of the minimizers for a variational problem which involves a circular well potential i e a potential vanishing on a closed smooth curve in r2 we thus generalize previous results obtained for the special case of the ginzburg landau potential

describing parameterized complexity classes we describe parameterized complexity classes by means of classical complexity theory and descriptive complexity theory for every classical complexity class we introduce a parameterized analogue in a natural way in particular the analogue of polynomial time is the class of all fixed parameter tractable problems we develop a basic complexity theory for the parameterized analogues of classical complexity classes and give among other things complete problems and logical descriptions we then show that most of the well known intractable parameterized complexity classes are not analogues of classical classes nevertheless for all these classes we can provide natural logical descriptions

on the characterizing properties of the permanental polynomials of graphs abstract let g be a graph and a g the adjacency matrix of g the permanental polynomial of g is defined as g x per x i a g if two graphs g and h have the same permanental polynomial then g is called a per cospectral mate of h a graph g is said to be characterized by its permanental polynomial if all the per cospectral mates of g are isomorphic to g it is shown that complete graphs stars regular complete bipartite graphs and odd cycles are characterized by their permanental polynomials we prove that in general the permanental polynomial cannot characterize the paths and even cycles in particular for each l 1 and m 2 we can find non isomorphic per cospectral mates of p 4 l 3 and c 4 m respectively when we restrict our consideration to connected graphs both the paths and even cycles c 4 l 2 are characterized by their permanental polynomials

algorithms for vertex partitioning problems on graphs with fixed clique width many vertex partitioning problems can be expressed within a general framework introduced by telle and proskurowski they showed that optimization problems in this framework can be solved in polynomial time on classes of graphs with bounded tree width in this paper we consider a very similar framework in relationship with more general classes of graphs we propose a polynomial time algorithm on classes of graphs with bounded clique width for all the optimization problems in our framework these classes of graphs are more general than the classes of graphs with bounded tree width in the sense that classes of graphs with bounded tree width have also bounded clique width but not necessarily the inverse our framework includes problems such as independent dominating set p dominating set induced bounded degree subgraph induced p regular subgraph perfect matching cut graph k coloring and graph list k coloring with cardinality constraints fixed k this paper thus provides a second distinct framework within which the optimization problems can be solved in polynomial time on classes of graphs with bounded clique width after a first framework called ms1 due to the work of courcelle makowsky and rotics for which they obtained a linear time algorithm

a characterization of graphs with rank 5 abstract the rank of a graph g is defined to be the rank of its adjacency matrix in this paper we consider the following problem what is the structure of a connected graph g with rank 5 or equivalently what is the structure of a connected n vertex graph g whose adjacency matrix has nullity n 5 in this paper we completely characterize connected graphs g whose adjacency matrix has rank 5

parameterized learnability of juntas we study the parameterized complexity of learning k juntas and some variations of juntas we show the hardness of learning k juntas and subclasses of k juntas in the pac model by reductions from a w 2 complete problem on the other hand as a consequence of a more general result we show that k juntas are exactly learnable with improper equivalence queries and access to a w p oracle

permanental polynomials of graphs abstract the first section surveys recent results on the permanental polynomial of a square matrix a i e per xi a the second section concerns the permanental polynomial of the adjacency matrix of a graph the final section is an introduction to the permanental polynomial of the laplacian matrix of a graph an appendix lists some of these latter polynomials

approximation algorithms for knapsack problems with cardinality constraints abstract we address a variant of the classical knapsack problem in which an upper bound is imposed on the number of items that can be selected this problem arises in the solution of real life cutting stock problems by column generation and may be used to separate cover inequalities with small support within cutting plane approaches to integer linear programs we focus our attention on approximation algorithms for the problem describing a linear storage polynomial time approximation scheme ptas and a dynamic programming based fully polynomial time approximation scheme fptas the main ideas contained in our ptas are used to derive ptas s for the knapsack problem and its multi dimensional generalization which improve on the previously proposed ptas s we finally illustrate better ptas s and fptas s for the subset sum case of the problem in which profits and weights coincide

fixed parameter complexity in ai and nonmonotonic reasoning many relevant intractable problems become tractable if some problem parameter is fixed however various problems exhibit very different computational properties depending on how the runtime required for solving them is related to the fixed parameter chosen the theory of parameterized complexity deals with such issues and provides general techniques for identifying fixed parameter tractable and fixed parameter intractable problems we study the parameterized complexity of various problems in ai and nonmonotonic reasoning we show that a number of relevant parameterized problems in these areas are fixed parameter tractable among these problems are constraint satisfaction problems with bounded treewidth and fixed domain restricted forms of conjunctive database queries restricted satisfiability problems propositional logic programming under the stable model semantics where the parameter is the dimension of a feedback vertex set of the program s dependency graph and circumscriptive inference from a positive k cnf restricted to models of bounded size we also show that circumscriptive inference from a general propositional theory when the attention is restricted to models of bounded size is fixed parameter intractable and is actually complete for a novel fixed parameter complexity class

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

fixed parameter tractability and completeness i basic results for many fixed parameter problems that are trivially soluable in polynomial time such as k dominating set essentially no better algorithm is presently known than the one which tries all possible solutions other problems such as k feedback vertex set exhibit fixed parameter tractability for each fixed k the problem is soluable in time bounded by a polynomial of degree c where c is a constant independent of k we establish the main results of a completeness program which addresses the apparent fixed parameter intractability of many parameterized problems in particular we define a hierarchy of classes of parameterized problems fpt subseteq w 1 subseteq w 2 subseteq cdots subseteq w sat subseteq w p and identify natural complete problems for w t for t geq 2 in other papers we have shown many problems complete for w 1 dominating set is shown to be complete for w 2 and thus is not fixed parameter tractable unless independent set clique irredundant set and many other natural problems in w 2 are also fixed parameter tractable we also give a compendium of currently known hardness results as an appendix

on the parameterized complexity of multiple interval graph problems multiple interval graphs are a natural generalization of interval graphs where each vertex may have more than one interval associated with it many applications of interval graphs also generalize to multiple interval graphs often allowing for more robustness in the modeling of the specific application with this motivation in mind a recent systematic study of optimization problems in multiple interval graphs was initiated in this sequel we study multiple interval graph problems from the perspective of parameterized complexity the problems under consideration are k independent set k dominating set and k clique which are all known to be w 1 hard for general graphs and np complete for multiple interval graphs we prove that k clique is in fpt while k independent set and k dominating set are both w 1 hard we also prove that k independent dominating set a hybrid of the two above problems is also w 1 hard our hardness results hold even when each vertex is associated with at most two intervals and all intervals have unit length furthermore as an interesting byproduct of our hardness results we develop a useful technique for showing w 1 hardness via a reduction from the k multicolored clique problem a variant of k clique we believe this technique has interest in its own right as it should help in simplifying w 1 hardness results which are notoriously hard to construct and technically tedious

on the complexity of covering vertices by faces in a planar graph the pair g d consisting of a planar graph g v e with n vertices together with a subset of d special vertices d subseteq v is called k planar if there is an embedding of g in the plane so that at most k faces of g are required to cover all of the vertices in d checking 1 planarity can be done in linear time since it reduces to a problem of checking planarity of a related graph we present an algorithm which given a graph g and a value k either determines that g is not k planar or generates an appropriate embedding and associated minimum cover in o c k n time where c is a constant hence the algorithm runs in linear time for any fixed k the fact that the time required by the algorithm grows exponentially in k is to be expected since we also show that for arbitrary k the associated decision problem is strongly np complete even when the planar graph has essentially a unique planar embedding d theta n and all facial cycles have bounded length these results provide a polynomial t

a framework for exponential time hypothesis tight algorithms and lower bounds in geometric intersection graphs we give an algorithmic and lower bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds it can be applied to intersection graphs

threshold dominating sets and an improved characterization of w 2 the threshold dominating set problem is that of determining for a graph g v e whether there is a subset v v of size k such that for each vertex v v there are at least r elements of the closed neighborhood n v that belong to v we consider the complexity of the problem parameterized by the pair k r it is trivial to observe that this is hard for w 2 it can also be easily shown to belong to a natural extension w 2 of w 2 defined in terms of circuit families of depth bounded by a function of the parameter we prove membership in w 2 and thus w 2 completeness using this as a starting point we prove that w 2 w 2

on the hardness of labeled correlation clustering problem motivated by practical applications the labeled correlation clustering problem a variant of correlation clustering problem is formally defined and studied in this paper since the problem is np complete we consider the parameterized complexities three different parameterizations are considered and the corresponding parameterized complexities are studied for the two parameterized problems which are fixed parameter tractable the lower bounds of them are analyzed under seth strong exponential time hypothesis

cylindrical algebraic decomposition i the basic algorithm given a set of r variate integral polynomials a cylindrical algebraic decomposition cad of euclidean r space e r partitions e r into connected subsets compatible with the zeros of the polynomials by compatible with the zeros of the polynomials we mean that on each subset of e r each of the polynomials either vanishes everywhere or nowhere for example consider the bivariate polynomial y 4 2 y 3 y 2 3 x 2 y 2 x 4

transducing markov sequences a markov sequence is a basic statistical model representing uncertain sequential data and it is used within a plethora of applications including speech recognition image processing computational biology radio frequency identification rfid and information extraction the problem of querying a markov sequence is studied under the conventional semantics of querying a probabilistic database where queries are formulated as finite state transducers specifically the complexity of two main problems is analyzed the first problem is that of computing the confidence probability of an answer the second is the enumeration of the answers in the order of decreasing confidence with the generation of the top k answers as a special case or in an approximate order thereof in particular it is shown that enumeration in any subexponential approximate order is generally intractable even for some fixed transducers and a matching upper bound is obtained through a proposed heuristic due to this hardness a special consideration is given to restricted yet common classes of transducers that extract matches of a regular expression subject to prefix and suffix constraints and it is shown that these classes are indeed significantly more tractable

the hardness of perfect phylogeny feasible register assignment and other problems on thin colored graphs in this paper we consider the complexity of a number of combinatorial problems namely intervalizing colored graphs dna physical mapping triangulating colored graphs perfect phylogeny directed modified colored cutwidth feasible register assignment and module allocation for graphs of bounded pathwidth each of these problems has as a characteristic a uniform upper bound on the tree or path width of the graphs in yes instances for all of these problems with the exceptions of feasible register assignment and module allocation a vertex or edge coloring is given as part of the input our main results are that the parameterized variant of each of the considered problems is hard for the complexity classes w t for all t n we also show that intervalizing colored graphs triangulating colored graphs and colored cutwidth are np complete

on the parametric complexity of schedules to minimize tardy tasks given a set t of tasks each of unit length and having an individual deadline d t z a set of precedence constraints on t and a positive integer k t we can ask is there a one processor schedule for t that obeys the precedence constraints and contains no more than k late tasks this is a well known np complete problem we might also inquire is there a one processor schedule for t that obeys the precedence constraints and contains at least k tasks that are on time i e no more than t k late tasks within the framework of classical complexity theory these two questions are merely different instances of the same problem within the recently developed framework of parameterized complexity theory however they give rise to two separate problems that may be studied independently of one another we investigate these problems from the parameterized point of view we show that in the general case both these problems are hard for the parameterized complexity class w 1 in contrast in the case where the set of precedence constraints can be modelled by a partial order of bounded width we show that both these problems are fixed parameter tractable

the effect of a singular perturbation on nonconvex variational problems we study the effect of a singular perturbation on certain nonconvex variational problems the goal is to characterize the limit of minimizers as some perturbation parameter 0 the technique utilizes the notion of convergence of variational problems developed by de giorgi the essential idea is to identify the first nontrivial term in an asymptotic expansion for the energy of the perturbed problem in so doing one characterizes the limit of minimizers as the solution of a new variational problem for the cases considered here these new problems have a simple geometric nature involving minimal surfaces and geodesics

a characterization of graphs with rank 4 the rank of a graph g is defined to be the rank of its adjacency matrix in this paper we consider the following problem what is the structure of a connected graph with rank 4 this question has not yet been fully answered in the literature and only some partial results are known in this paper we resolve this question by completely characterizing graphs g whose adjacency matrix has rank 4

efficient algorithms for the domination problems on interval and circular arc graphs this paper first presents a unified approach to design efficient algorithms for the weighted domination problem and its three variants i e the weighted independent connected and total domination problems on interval graphs given an interval model with endpoints sorted these algorithms run in time o n or o n log log n where n is the number of vertices the results are then extended to solve the same problems on circular arc graphs in o n m time where m is the number of edges of the input graph

a fixed parameter algorithm for the directed feedback vertex set problem the parameterized feedback vertex set problem on directed graphs i e the dfvs problem is defined as follows given a directed graph g and a parameter k either construct a feedback vertex set of at most k vertices in g or report that no such a set exists it has been a well known open problem in parameterized computation and complexity whether the dfvs problem is fixed parameter tractable that is whether the problem can be solved in time f k no 1 for some function f in this article we develop new algorithmic techniques that result in an algorithm with running time 4k ke no 1 for the dfvs problem therefore we resolve this open problem

the complexity of homomorphism and constraint satisfaction problems seen from the other side we give a complexity theoretic classification of homomorphism problems for graphs and more generally relational structures obtained by restricting the left hand side structure in a homomorphism for every class c of structures let hom c be the problem of deciding whether a given structure a c has a homomorphism to a given arbitrary structure s we prove that under some complexity theoretic assumption from parameterized complexity theory hom c is in polynomial time if and only if c has bounded tree width modulo homomorphic equivalence translated into the language of constraint satisfaction problems our result yields a characterization of the tractable structural restrictions of constraint satisfaction problems translated into the language of database theory it implies a characterization of the tractable instances of the evaluation problem for conjunctive queries over relational databases

per spectral characterizations of graphs with extremal per nullity abstract a graph g is said to be determined by its permanental spectrum if any graph having the same permanental spectrum as g is isomorphic to g in this paper we introduce the permanental nullity of a graph the multiplicity of the number zero in the permanental spectrum of a graph to study graphs determined by their permanental spectra first we determine all graphs of order n whose permanental nullities are n 2 n 3 n 4 and n 5 respectively then we show that all graphs with the permanental nullity n 2 n 3 or n 5 and all non bipartite graphs with the permanental nullity n 4 are determined by their permanental spectra in particular we prove that the complete bipartite graphs are determined by their permanental spectra

the gradient theory of phase transitions for systems with two potential wells on generalise la theorie du gradient des transitions de phase au cas a valeur vectoriel on considere la famille des perturbations e e u w u dx e 2 u 2 dx de la fonctionnelle non convexe e 0 u w u dx ou w r n r supporte deux phases et n 1

the complexity of dominating set in geometric intersection graphs abstract we study the parameterized complexity of the dominating set problem in geometric intersection graphs in one dimension we investigate intersection graphs induced by translates of a fixed pattern q that consists of a finite number of intervals and a finite number of isolated points we prove that dominating set on such intersection graphs is polynomially solvable whenever q contains at least one interval and whenever q contains no intervals and for any two point pairs in q the distance ratio is rational the remaining case where q contains no intervals but does contain an irrational distance ratio is shown to be np complete and contained in fpt when parameterized by the solution size in two and higher dimensions we prove that dominating set is contained in w 1 for intersection graphs of semi algebraic sets with constant description complexity so far this was only known for unit squares finally we establish w 1 hardness for a large class of intersection graphs

the inapproximability of non np hard optimization problems the inapproximability of non np hard optimization problems is investigated techniques are given to show that the problems log dominating set and log hypergraph vertex cover cannot be approximated to a constant ratio in polynomial time unless the corresponding np hard versions are also approximable in deterministic subexponential time a direct connection is established between non np hard problems and a pcp characterization of np reductions from the pcp characterization show that log clique is not approximable in polynomial time and max sparse sat does not have a fptas under the assumption that np dtime 2o n log n a number of nontrivial approximation preserving reductions are also presented making it possible to extend inapproximability results to more natural non np hard problems such as tournament dominating set and rich hypergraph vertex cover

the complexity of the matroid greedoid partition problem we show that the maximum matroid greedoid partition problem is np hard to approximate to within 1 2 e for any e 0 which matches the trivial factor 1 2 approximation algorithm the main tool in our hardness of approximation result is an extractor code with polynomial rate alphabet size and list size together with an efficient algorithm for list decoding we show that the recent extractor construction of guruswami umans and vadhan v guruswami c umans s p vadhan unbalanced expanders and randomness extractors from parvaresh vardy codes in ieee conference on computational complexity ieee computer society 2007 pp 96 108 can be used to obtain a code with these properties we also show that the parameterized matroid greedoid partition problem is fixed parameter tractable

an improved linear kernel for complementary maximal strip recovery simpler and smaller abstract we study the complementary maximal strip recovery problem cmsr where the given are two strings s 1 and s 2 of distinct letters each of which appears either in the positive form or the negative form the question is whether there are k letters whose deletion results in two matched strings string s 1 matches string s 2 if there are partitions of s 1 and s 2 such that each component of the partitions contains at least two letters and moreover for each component s 1 i of the partition of s 1 there is a unique component s 2 j in the partition of s 2 which is either equal to s 1 i or can be obtained from s 1 i by firstly reversing the order of the letters and then negating the letters the cmsr problem is known to be np hard and fixed parameter tractable with respect to k in particular a linear kernel of size 74 k 4 was developed based on 8 reduction rules very recently by imposing 3 new reduction rules to the previous kernelization the linear kernel has been improved to 58k we aim to simplify the kernelization yet obtain an improved kernel in particular we study 7 reduction rules which lead to a linear kernel of size 42 k 24

on the kernelization of split graph problems abstract a split graph is a graph whose vertices can be partitioned into a clique and an independent set we study numerous problems on split graphs namely the k vertex disjoint paths k cycle k path and k l stable set problems in the k vertex disjoint paths problem we are given a graph and k terminal pairs of vertices and are asked whether there is a set of k vertex disjoint paths linking these terminal pairs respectively in the k cycle k path problem we are given a graph and are asked whether there is a path cycle of length k the k l stable set problem takes a graph and an integer k as input and asks whether the graph has a subset of k vertices such that the distance between every two vertices in the subset is at least l 1 it is known that all the above problems are np complete on split graphs we derive a 4 k vertex kernel for the k vertex disjoint paths problem and an o k 2 vertex kernel for both the k path problem and the k cycle problem concerning the k l stable set problem for l 1 or l 3 the problem is polynomial time solvable on split graphs for l 2 we prove that the k l stable set problem is w 1 complete on split graphs with respect to k however if the given split graph contains no k 1 r as an induced subgraph and every vertex in the independent set of the split graph has degree at most d we derive a linear vertex kernel for the k 2 stable set problem where both r and d are constants

computing the permanental polynomials of graphs let m be an nn matrix with entries mij i j 1 2 n the permanent of m is defined to be per m i 1nmi i where the sum is taken over all permutations of 1 2 n the permanental polynomial of m is defined by per xinm where in is the identity matrix of size n in this paper we give recursive formulas for computing permanental polynomials of the laplacian matrix and the signless laplacian matrix of a graph respectively

preprocessing of intractable problems some computationally hard problems e g deduction in logical knowledge bases are such that part of an instance is known well before the rest of it and remains the same for several subsequent instances of the problem in these cases it is useful to preprocess off line this known part so as to simplify the remaining on line problem in this paper we investigate such a technique in the context of intractable i e np hard problems recent results in the literature show that not all np hard problems behave in the same way for some of them preprocessing yields polynomial time on line simplified problems we call them compilable while for other ones their compilability implies some consequences that are considered unlikely our primary goal is to provide a sound methodology that can be used to either prove or disprove that a problem is compilable to this end we define new models of computation complexity classes and reductions we find complete problems for such classes completeness meaning they are the less likely to be compilable we also investigate preprocessing that does not yield polynomial time on line algorithms but generically decreases complexity this leads us to define hierarchies of compilability that are the analog of the polynomial hierarchy a detailed comparison of our framework to the idea of parameterized tractability shows the differences between the two approaches

the intractability of computing the minimum distance of a code it is shown that the problem of computing the minimum distance of a binary linear code is np hard and the corresponding decision problem is np complete this result constitutes a proof of the conjecture of berlekamp mceliece and van tilborg 1978 extensions and applications of this result to other problems in coding theory are discussed

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

fixed parameter tractability and completeness ii on completeness for w 1 abstract for many fixed parameter problems that are trivially solvable in polynomial time such as k dominating set essentially no better algorithm is presently known than the one which tries all possible solutions other problems such as feedback vertex set exhibit fixed parameter tractability for each fixed k the problem is solvable in time bounded by a polynomial of degree c where c is a constant independent of k in a previous paper the w hierarchy of parameterized problems was defined and complete problems were identified for the classes w t for t 2 our main result shows that independent set is complete for w 1

the parametrized complexity of some fundamental problems in coding theory the parametrized complexity of a number of fundamental problems in the theory of linear codes and integer lattices is explored concerning codes the main results are that maximum likelihood decoding and weight distrubution are hard for the parametrized complexity class w 1 the np completeness of these two problems was established by berlekamp mceliece and van tilborg in 1978 using by means of a reduction from three dimensional matching on the other hand our proof of hardness for w 1 is based on a parametric polynomial time transformation from perfect code in graphs an immediate consequence of our results is that bounded distance decoding is likely to be hard for binary linear codes concerning lattices we address the theta series problem of determining for an integer lattice l given by a set of generators and a positive integer k whether there is a vector x in l of euclidean norm k we prove here for the first time that theta series is np complete and show that it is also hard for w 1 furthermore we prove that the nearest vector problem for integer lattices is hard for w 1 these problems are the counterparts of weight distrubution and maximum likelihood decoding for lattices relations between all these problems and combinatorial problems in graphs are discussed