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

identifying codes and covering problems the identifying code problem for a given graph involves finding a minimum set of vertices whose neighborhoods uniquely overlap at any given graph vertex initially introduced in 1998 this problem has demonstrated its fundamental nature through a wide variety of applications such as fault diagnosis location detection and environmental monitoring in addition to deep connections to information theory superimposed and covering codes and tilings this work establishes efficient reductions between the identifying code problem and the well known set covering problem resulting in a tight hardness of approximation result and novel provably tight polynomial time approximations the main results are also extended to r robust identifying codes and analogous set 2r 1 multicover problems finally empirical support is provided for the effectiveness of the proposed approximations including good constructions for well known topologies such as infinite two dimensional grids

approximation algorithms for the test cover problem in the test cover problem a set of m items is given together with a collection of subsets called tests a smallest subcollection of tests is to be selected such that for each pair of items there is a test in the selection that contains exactly one of the two items it is known that the problem is np hard and that the greedy algorithm has a performance ratio o log m we observe that unless p np no polynomial time algorithm can do essentially better for the case that each test contains at most k items we give an o log k approximation algorithm we pay special attention to the case that each test contains at most two items a strong relation with a problem of packing paths in a graph is established which implies that even this special case is np hard we prove apx hardness of both problems derive performance guarantees for greedy algorithms and discuss the performance of a series of local improvement heuristics

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

algebraic methods for interactive proof systems a new algebraic technique for the construction of interactive proof systems is presented our technique is used to prove that every language in the polynomial time hierarchy has an interactive proof system this technique played a pivotal role in the recent proofs that ip pspace 28 and that mip nexp 4

improved approximation algorithms for the uncapacitated facility location problem we consider the uncapacitated facility location problem in this problem there is a set of locations at which facilities can be built a fixed cost fi is incurred if a facility is opened at location i furthermore there is a set of demand locations to be serviced by the opened facilities if the demand location j is assigned to a facility at location i then there is an associated service cost proportional to the distance between i and j cij the objective is to determine which facilities to open and an assignment of demand points to the opened facilities so as to minimize the total cost we assume that the distance function c is symmetric and satisfies the triangle inequality for this problem we obtain a 1 2 e approximation algorithm where 1 2 e approx 1 736 which is a significant improvement on the previously known approximation guarantees the algorithm works by rounding an optimal fractional solution to a linear programming relaxation our techniques use properties of optimal solutions to the linear program randomized rounding as well as a generalization of the decomposition techniques of shmoys tardos and aardal proceedings of the 29th acm symposium on theory of computing el paso tx 1997 pp 265 274

parameterized and approximation complexity of partial vc dimension abstract we introduce the problem partial vc dimension that asks given a hypergraph h x e and integers k and l whether one can select a set c x of k vertices of h such that the set e c e e of distinct hyperedge intersections with c has size at least l the sets e c define equivalence classes over e partial vc dimension is a generalization of vc dimension which corresponds to the case l 2 k and of distinguishing transversal which corresponds to the case l e the latter is also known as test cover in the dual hypergraph we also introduce the associated fixed cardinality maximization problem max partial vc dimension that aims at maximizing the number of equivalence classes induced by a solution set of k vertices we study the algorithmic complexity of partial vc dimension and max partial vc dimension both on general hypergraphs and on more restricted instances in particular neighborhood hypergraphs of graphs

a class of the first order impulsive dynamic equations on time scales abstract in this paper the first order impulsive dynamic equations on time scales is considered the existence and uniqueness of solution for a class of nonlinear impulsive dynamic equations on time scales is proved

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

np completeness and apx completeness of restrained domination in graphs let g v e be a simple graph a vertex set email protected v is a restrained dominating set if every vertex not in s is adjacent to a vertex in s and to a vertex in v s in this paper we investigate the np completeness of the restrained domination problem in planar graphs and split graphs meanwhile it is proved that the restrained domination problem is apx complete for bounded degree graphs

a threshold of ln n for approximating set cover given a collection f of subsets of s 1 n setcover is the problem of selecting as few as possiblesubsets from f such that their union covers s and maxk cover is the problem of selecting k subsets from f such that their union has maximum cardinality both these problems arenp hard we prove that 1 o 1 ln n is a threshold below which setcover cannot be approximated efficiently unless np has slightlysuperpolynomial time algorithms this closes the gap up to low orderterms between the ratio of approximation achievable by the greedyalogorithm which is 1 o 1 lnn and provious results of lund and yanakakis that showed hardness ofapproximation within a ratio of log 2 n 2s0 72 ln n for max k cover we show an approximationthreshold of 1 1 e up tolow order terms under assumption that p np

approximating the spanning star forest problem and its application to genomic sequence alignment this paper studies the algorithmic issues of the spanning star forest problem we prove the following results 1 there is a polynomial time approximation scheme for planar graphs 2 there is a polynomial time frac 3 5 approximation algorithm for graphs 3 it is np hard to approximate the problem within ratio frac 259 260 epsilon for graphs 4 there is a linear time algorithm to compute the maximum star forest of a weighted tree 5 there is a polynomial time frac 1 2 approximation algorithm for weighted graphs we also show how to apply this spanning star forest model to aligning multiple genomic sequences over a tandem duplication region

mixed integer programming with convex concave constraints fixed parameter tractability and applications to multicovering and voting abstract a classic result of lenstra math oper res 1983 says that an integer linear program can be solved in fixed parameter tractable fpt time for the parameterization by the number of variables we extend this result by incorporating piecewise linear convex or concave functions to our mixed integer programs this general technique allows us to analyze the parameterized complexity of a number of classic np hard computational problems in particular we prove that weighted set multicover is in fpt when parameterized by the number of elements to cover and that there exists an fpt time approximation scheme for multiset multicover for the same parameter this is our most technical result further we use our general technique to prove that a number of problems from computational social choice e g problems related to bribery and control in elections are in fpt when parameterized by the number of candidates for bribery this resolves a nearly 10 year old family of open problems and for weighted electoral control of approval voting this improves some previously known xp memberships to fpt memberships

fast approximation algorithms for the one warehouse multi retailer problem under general cost structures and capacity constraints we consider a well studied multi echelon deterministic inventory control problem known in the literature as the one warehouse multi retailer owmr problem we propose a simple and fast 2 approximation algorithm for this np hard problem by recombining the solutions of single echelon relaxations at the warehouse and at the retailers we then show that our approach remains valid under quite general assumptions on the cost structures and under capacity constraints at some retailers in particular we present the first approximation algorithms for the owmr problem with nonlinear holding costs truckload discount on procurement costs or with capacity constraints at some retailers in all cases the procedure is purely combinatorial and can be implemented to run in low polynomial time

approximation algorithms for the partition vertex cover problem we consider a natural generalization of the partial vertex cover problem here an instance consists of a graph g v e a cost function c v z a partition p 1 p r of the edge set e and a parameter k i for each partition p i the objective is to find a minimum cost set of vertices which cover at least k i edges from the partition p i we call this the partition vc problem in this paper we give matching upper and lower bound on the approximability of this problem our algorithm is based on a novel lp relaxation for this problem this lp relaxation is obtained by adding knapsack cover inequalities to a natural lp relaxation of the problem we show that this lp has integrality gap of o log r where r is the number of sets in the partition of the edge set we also extend our result to more general settings for example we consider a problem where additionally edges have profits and we would like to pick a minimum cost set of vertices which cover edges of total profit at least i for each partition p i we call this the knapsack partition vertex cover problem we give an o log r approximation algorithm for this problem as well

some optimal inapproximability results we prove optimal up to an arbitrary e 0 inapproximability results for max e k sat for k 3 maximizing the number of satisfied linear equations in an over determined system of linear equations modulo a prime p and set splitting as a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously in particular for max e2 sat max cut max di cut and vertex cover

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

structural analysis of subjective categorical data a general approach to the analysis of subjective categorical data is considered in which agreement matrices of two or more raters are directly expressed in terms of error and agreement parameters the method provides focused analyses of ratings from several raters for whom ratings have measurement error distributions that may induce bias in the evaluation of substantive questions of interest each rater s judgment process is modeled as a mixture of two components an error variable that is unique for the rater in question as well as an agreement variable that operationalizes the true values of the units of observation the statistical problems of identification estimation and testing of such measurement models are discussed

existence results for dynamic inclusions on time scales with nonlocal initial conditions this paper is mainly concerned with the existence of solutions for first order dynamic inclusions on time scales with nonlocal initial conditions by using bohnenblust karlin s fixed point theorem and leray schauder nonlinear alternative for multivalued maps some sufficient conditions are established an example is also included to illustrate our results

a best possible heuristic for the k center problem in this paper we present a 2 approximation algorithm for the k center problem with triangle inequality this result is best possible since for any 2 the existence of approximation algorithm would imply that p np it should be noted that no approximation algorithm for any constant has been reported to date linear programming duality theory provides interesting insight to the problem and enables us to derive in o e log e time a solution with value no more than twice the k center optimal value a by product of the analysis is an o e algorithm that identifies a dominating set in g2 the square of a graph g the size of which is no larger than the size of the minimum dominating set in the graph g the key combinatorial object used is called a strong stable set and we prove the np completeness of the corresponding decision problem

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

ip pspace in this paper it is proven that when both randomization and interaction are allowed the proofs that can be verified in polynomial time are exactly those proofs that can be generated with polynomial space

nagumo type existence results for second order nonlinear dynamic bvps abstract in this paper we present a nagumo condition for time scales equations that includes as a particular case a type of the classical one for differential equations involving a certain restriction the established nagumo condition further allows us to prove the existence of at least one solution lying between a pair of lower and upper solutions of a nonlinear second order dynamic equation with nonlinear functional boundary value conditions

toward synchronization in networks with nonlinear dynamics a submodular optimization framework synchronization underlies phenomena including memory and perception in the brain coordinated motion of animal flocks and stability of the power grid these synchronization phenomena are often modeled through networks of phase coupled oscillating nodes heterogeneity in the node dynamics however may prevent such networks from achieving the required level of synchronization in order to guarantee synchronization external inputs can be used to pin a subset of nodes to a reference frequency while the remaining nodes are steered toward synchronization via local coupling in this paper we present a submodular optimization framework for selecting a set of nodes to act as external inputs in order to achieve synchronization from a desired set of initial states we derive threshold based sufficient conditions for synchronization and then prove that these conditions are equivalent to constraints on monotone submodular functions over partition matroids based on this connection we map the sufficient conditions for synchronization to constraints on submodular functions leading to efficient algorithms with provable optimality bounds for selecting input nodes we illustrate our approach via numerical studies of synchronization in power systems

existence and approximation of solutions for first order discontinuous difference equations with nonlinear global conditions in the presence of lower and upper solutions abstract this paper is devoted to the study of the existence of solutions of first order difference equations verifying nonlinear conditions that involve the global behavior of the solution we prove that the existence of lower and upper solutions warrants the existence of such solutions lying in the sector formed by the mentioned functions we also can prove that some classical results for differential equations are not true in general for this case

on the parameterized complexity of vertex cover and edge cover with connectivity constraints abstract we investigate the effect of certain natural connectivity constraints on the parameterized complexity of two fundamental graph covering problems namely vertex cover and edge cover specifically we impose the additional requirement that each connected component of a solution have at least t vertices resp edges from the solution for a fixed positive integer t and call the problem t total vertex cover resp t total edge cover in both cases the parameter k is the size of the solution we show that both problems remain fixed parameter tractable with these restrictions with running times of the form o c k for some constant c 0 in each case where the o notation hides polynomial factors for each fixed t 2 t total vertex cover has no polynomial kernel unless conp np poly for each fixed t 2 t total edge cover has a linear vertex kernel of size t 1 t k these results significantly improve earlier work on these problems we illustrate the utility of the technique used to solve t total vertex cover by applying it to derive an o c k time fpt algorithm for the t total edge dominating set problem our no poly kernel result for t total vertex cover and the known np hardness result for t total edge cover are in stark contrast to the fact that vertex cover has a 2 k vertex kernel and that edge cover is solvable in polynomial time this illustrates how even the slightest connectivity requirement results in a drastic change in the tractability of problems the curse of connectivity

separating codes and traffic monitoring this paper studies the problem of traffic monitoring which consists of differentiating a set of walks on a directed graph by placing sensors on as few arcs as possible the problem of characterising a set of individuals by testing as few attributes as possible is already well known but traffic monitoring presents new challenges that the previous models of separation fall short from modelling such as taking into account the multiplicity and order of the arcs in a walk we introduce a new and stronger model of separation based on languages that generalises the traffic monitoring problem we study three subproblems with practical applications and develop methods to solve them by combining integer linear programming separating codes and language theory

nominal scale agreement among observers experiments are considered where each of a sample of subjects is assigned to one of c categories separately by each of a fixed or varying group of observers building on earlier publications general procedures are proposed to analyze agreements and disagreements among observers in the case of a varying group of observers it is shown that it is not necessary to demand a constant number of observers per subject in the case of a fixed group of observers the problem of missing data is considered the procedures are illustrated within the context of two clinical diagnosis examples in the first example it is investigated which categories are relatively hard to distinguish from one another a new theorem is applied that shows a useful property of the statistic kappa in the second example it is investigated if a subgroup of observers can be found with a significantly higher degree of interobserver agreement

algorithm and hardness results on neighborhood total domination in graphs abstract a set d v of a graph g v e is called a neighborhood total dominating set of g if d is a dominating set and the subgraph of g induced by the open neighborhood of d has no isolated vertex given a graph g min ntds is the problem of finding a neighborhood total dominating set of g of minimum cardinality the decision version of min ntds is known to be np complete for bipartite graphs and chordal graphs in this paper we extend this np completeness result to undirected path graphs chordal bipartite graphs and planar graphs we also present a linear time algorithm for computing a minimum neighborhood total dominating set in proper interval graphs we show that for a given graph g v e min ntds cannot be approximated within a factor of 1 e log v unless np dtime v o log log v and can be approximated within a factor of o log where is the maximum degree of the graph g finally we show that min ntds is apx complete for graphs of degree at most 3

self improved gaps almost everywhere for the agnostic approximation of monomials given a learning sample we focus on the hardness of finding monomials having low error inside the interval bounded below by the smallest error achieved by a monomial the best rule and bounded above by the error of the default class the poorest rule it is well known that when its lower bound is zero it is an easy task to find in linear time a monomial with zero error what we prove is that when this bound is not zero regardless of the location of the default class in 0 1 2 it becomes a huge complexity burden to beat significantly the default class in fact under some complexity theoretical assumptions it may already be hard to beat the trivial approximation ratios even when relaxing the time complexity constraint to be quasi polynomial or sub exponential our results also hold with uniform weights over the examples

inapproximability results for equations over infinite groups an equation over a group g is an expression of form w 1 w k 1 g where each w i is either a variable an inverted variable or a group constant and 1 g denotes the identity element such an equation is satisfiable if there is a setting of the variables to values in g such that the equality is realized engebretsen et al 2002 10 in this paper we study the problem of simultaneously satisfying a family of equations over an infinite group g let eq g k denote the problem of determining the maximum number of simultaneously satisfiable equations in which each equation has occurrences of exactly k different variables when g is an infinite cyclic group we show that it is np hard to approximate eq 1 g 3 to within 48 47 e where eq 1 g 3 denotes the special case of eq g 3 in which a variable may only appear once in each equation it is np hard to approximate eq 1 g 2 to within 30 29 e it is np hard to approximate the maximum number of simultaneously satisfiable equations of degree at most d to within d e for any e for any k 4 it is np hard to approximate eq g k within any constant factor these results extend hastad s results hastad 2001 17 and results of engebretsen et al 2002 10 who established the inapproximability results for equations over finite abelian groups and any finite groups respectively

approximation algorithms for solving the constrained arc routing problem in mixed graphs abstract given a mixed graph g with vertex set v let e and a denote the sets of edges and arcs respectively we use q and z to denote the sets of positive rational numbers and positive integers respectively for any connected mixed graph g v e a w l u with a length function w e a q and two integer functions l u e a z satisfying l e u e for each e e a we are asked to determine a minimum length tour t traversing each e e a at least l e and at most u e times this new constrained arc routing problem generalizes the mixed chinese postman problem let n v and m e a denote the number of vertices and edges including arcs respectively using network flow techniques we design a 1 1 l 0 approximation algorithm in time o n 2 m 3 log n to solve this constrained arc routing problem such that l e u e holds for each edge e e l e u e holds for each arc e a and l 0 min l e e e in addition we present two optimal combinatorial algorithms in times o n 3 and o nm 2 log n to solve this problem for the cases a and e respectively

bounding the payment of approximate truthful mechanisms in a stacs 2003 paper talwar analyzes the overpayment the vcg mechanism incurs for ensuring truthfulness in auctions among other results he studies k set cover given a universe u and a collection of sets s 1 s 2 s m each having a cost c s i and at most k elements of u find a minimum cost subcollection a cover whose union equals u and shows that the payment of the vcg mechanism is at most k c opt where opt is the best cover disjoint from the optimum cover opt the vcg mechanism requires finding an optimum cover for k 3 k set cover is known to be np hard and thus truthful mechanisms based on approximation algorithms are desirable we show that the payment incurred by two mechanisms based on approximation algorithms including the greedy algorithm is bounded by k 1 c opt k c opt the same approximation algorithms have payment bounded by k c opt c opt when applied to more general set systems which include k polymatroid cover a problem related to steiner tree computations if q is such that an element in a k set cover instance appears in at most q sets we show that the total payment based on our algorithms is bounded by q k 2 times the total payment of the vcg mechanism

approximation algorithms for inventory problems with submodular or routing costs we consider the following two deterministic inventory optimization problems with non stationary demands submodular joint replenishment problem this involves multiple item types and a single retailer who faces demands over a finite planning horizon of t periods in each time period any subset of item types can be ordered incurring a joint ordering cost which is submodular moreover items can be held in inventory while incurring a holding cost the objective is to find a sequence of orders that satisfies all demands and minimizes the total ordering and holding costs inventory routing problem this involves a single depot that stocks items and multiple retailer locations facing demands over a finite planning horizon of t periods in each time period any subset of locations can be visited using a vehicle originating from the depot there is also cost incurred for holding items at any retailer the objective here is to satisfy all demands while minimizing the sum of routing and holding costs we present a unified approach that yields mathcal o left frac log t log log t right ologtloglogt factor approximation algorithms for both problems when the holding costs are polynomial functions a special case is the classic linear holding cost model wherein this is the first sub logarithmic approximation ratio for either problem

set graphs ii complexity of set graph recognition and similar problems a graph g is said to be a set graph if it admits an acyclic orientation that is also extensional in the sense that the out neighborhoods of its vertices are pairwise distinct equivalently a set graph is the underlying graph of the digraph representation of a hereditarily finite set in this paper we continue the study of set graphs and related topics focusing on computational complexity aspects we prove that set graph recognition is np complete even when the input is restricted to bipartite graphs with exactly two leaves the problem remains np complete if in addition we require that the extensional acyclic orientation be also slim that is that the digraph obtained by removing any arc from it is not extensional our approach in fact allows us to also show that the counting variants of the above problems are p complete and prove similar complexity results for problems related to a generalization of extensional acyclic digraphs the so called hyper extensional digraphs which were proposed by aczel to describe hypersets our proofs are based on reductions from variants of the hamiltonian path problem we also consider a variant of the well known notion of a separating code in a digraph the so called open out separating code and show that it is np complete to determine whether an input extensional acyclic digraph contains an open out separating code of given size

a constant approximation algorithm for the one warehouse multiretailer problem deterministic inventory theory provides streamlined optimization models that attempt to capture trade offs in managing the flow of goods through a supply chain we will consider two well studied deterministic inventory models called the one warehouse multiretailer owmr problem and its special case the joint replenishment problem jrp and give approximation algorithms with worst case performance guarantees that is for each instance of the problem our algorithm produces a solution with cost that is guaranteed to be at most 1 8 times the optimal cost this is called a 1 8 approximation algorithm our results are based on an lp rounding approach we provide the first constant approximation algorithm for the owmr problem and improve the previous results for the jrp

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 local lemma is asymptotically tight for sat the local lemma is a fundamental tool of probabilistic combinatorics and theoretical computer science yet there are hardly any natural problems known where it provides an asymptotically tight answer the main theme of our article is to identify several of these problems among them a couple of widely studied extremal functions related to certain restricted versions of the k sat problem where the local lemma does give essentially optimal answers as our main contribution we construct unsatisfiable k cnf formulas where every clause has k distinct literals and every variable appears in at most 2 e o 1 2k k clauses the lopsided local lemma applied with an assignment of random values according to counterintuitive probabilities shows that this is asymptotically best possible the determination of this extremal function is particularly important as it represents the value where the corresponding k sat problem exhibits a complexity hardness jump from having every instance being a yes instance it becomes np hard just by allowing each variable to occur in one more clause the construction of our unsatisfiable cnf formulas is based on the binary tree approach of gebauer 2012 and thus the constructed formulas are in the class mu 1 of minimal unsatisfiable formulas having one more clause than variables the main novelty of our approach here comes in setting up an appropriate continuous approximation of the problem this leads us to a differential equation the solution of which we are able to estimate the asymptotically optimal binary trees are then obtained through a discretization of this solution the importance of the binary trees constructed is also underlined by their appearance in many other scenarios in particular they give asymptotically precise answers for seemingly unrelated problems like the european tenure game introduced by doerr 2004 and a search problem allowing a limited number of consecutive lies as yet another consequence we slightly improve the best known bounds on the maximum degree and maximum edge degree of a k uniform maker s win hypergraph in the neighborhood conjecture of beck

matching and weighted p2 packing algorithms and kernels parameterized algorithms and kernelization algorithms are presented for the weighted p 2 packing problem which is a generalization of the famous graph matching problem the parameterized algorithms are based on the following new techniques and observations 1 new study on structure relationship between graph matchings in general graphs and p 2 packings in bipartite graphs 2 an effective graph bi partitioning algorithm and 3 a polynomial time algorithm for a constrained weighted p 2 packing problem in bipartite graphs the kernelization algorithms are based on the following new techniques 1 the application of graph matching in kernelization 2 a crown reduction structure for weighted problems these techniques lead to randomized and deterministic parameterized algorithms that significantly improve the previous best upper bounds for the problem for both weighted and unweighted versions for the kernelization algorithm by using a weighted version of crown reduction a kernel of size o k 2 is presented where k is the given parameter of the problem

dynamic algorithms via the primal dual method abstract we develop a dynamic version of the primal dual method for optimization problems and apply it to obtain the following results 1 for the dynamic set cover problem we maintain an o f 2 approximately optimal solution in o f log m n amortized update time where f is the maximum frequency of an element n is the number of sets and m is the maximum number of elements in the universe at any point in time 2 for the dynamic b matching problem we maintain an o 1 approximately optimal solution in o log 3 n amortized update time where n is the number of nodes in the graph

combinatorics for smaller kernels the differential of a graph let g v e be a graph of order n and let b d be the set of vertices in v d that have a neighbor in the vertex set d the differential of d is defined as d b d d and the differential of a graph g written g is equal to max d d v if g is connected and n 3 g n 5 is known this immediately leads to a linear vertex kernel result in the terminology of parameterized complexity for the problem of deciding whether g k taking k as the parameter we then establish a new combinatorial result which establishes that g n 4 if g is a connected graph of order n 6 and if g contains no induced path of five vertices whose midpoint is a cut vertex and whose endpoints have degree one this technical combinatorial theorem can be used to derive an even smaller linear vertex kernel for general graphs also we show that the related maximization problem allows for a polynomial time factor 14 approximation algorithm

reliability and expected loss a unifying principle we provide a unified theoretical basis on which measures of data reliability may be derived or evaluated for both quantitative and qualitative data this approach evaluates reliability as the proportional reduction in loss prl that is attained in a sample by an optimal estimator the resulting measure is between 0 and 1 linearly related to expected loss and provides a direct way of contrasting the measured reliability in the sample with the least reliable and most reliable data generating cases the prl measure is a generalization of many of the commonly used reliability measures

efficient approximation algorithms for maximum coverage with group budget constraints abstract given a ground set u with a non negative weight w i for each i u a positive integer k and a collection of sets s which is partitioned into a family of disjoint groups g the goal of the maximum coverage problem with group budget constraints mcg is to select k sets from s such that the total weight of the union of the k sets is maximized and at most one set is selected from each group g g we first present an approximation algorithm with a factor 1 1 e and an exponential time via randomized linear programming rounding technique then we improve the time complexity of the algorithm to o m n 3 5 l k 3 5 q 7 l for s m u n and l being the length of the input by the key idea of modeling the selection of groups as computing a constrained flow in a corresponding auxiliary graph the algorithm is later shown can be extended to solve two generalizations of mcg last but not the least we present another algorithm with a time complexity o m n 3 5 l k 10 5 l and a slightly increased approximation ratio 1 e 1 1 mainly based on the idea of partition where 2 is a parameter tuning which can balance the time complexity and the ratio

the implicit hitting set approach to solve combinatorial optimization problems with an application to multigenome alignment we develop a novel framework the implicit hitting set approach for solving a class of combinatorial optimization problems the explicit hitting set problem is as follows given a set u and a family s of subsets of u find a minimum cardinality set that intersects hits every set in s in the implicit hitting set problem ihsp the family of subsets s is not explicitly listed its size is generally exponential in terms of the size of u instead it is given via a polynomial time oracle that verifies if a given set h is a hitting set or returns a set in s that is not hit by h many np hard problems can be straightforwardly formulated as implicit hitting set problems we show that the implicit hitting set approach is valuable in developing exact and heuristic algorithms for solving this class of combinatorial optimization problems specifically we provide a generic algorithmic strategy which combines efficient heuristics and exact methods to solve any ihsp given an instance of an ihsp the propose

strong lp formulations for scheduling splittable jobs on unrelated machines a natural extension of the makespan minimization problem on unrelated machines is to allow jobs to be partially processed by different machines while incurring an arbitrary setup time in this paper we present increasingly stronger lp relaxations for this problem and their implications on the approximability of the problem first we show that the straightforward lp extending the approach for the original problem has an integrality gap of 3 and yields an approximation algorithm of the same factor by applying a lift and project procedure we are able to improve both the integrality gap and the implied approximation factor to 1 phi 1 where phi is the golden ratio since this bound remains tight for the seemingly stronger machine configuration lp we propose a new job configuration lp that is based on an infinite continuum of fractional assignments of each job to the machines we prove that this lp has a finite representation and can be solved in polynomial time up to any accuracy interestingly we show that our problem cannot be approximated within a factor better than frac e e 1 approx 1 582 hbox unless mathcal p mathcal np ee 1 1 582 unlessp np which is larger than the inapproximability bound of 1 5 for the original problem

dynamic boundary value problems of the second order bernstein nagumo conditions and solvability abstract this article analyzes the solvability of second order nonlinear dynamic boundary value problems bvps on time scales new bernstein nagumo conditions are developed that guarantee an a priori bound on the delta derivative of potential solutions to the bvps under consideration topological methods are then employed to gain solvability

polynomial time approximation schemes for maximizing gross substitutes utility under budget constraints we consider the maximization of a gross substitutes utility function under budget constraints this problem naturally arises in applications such as exchange economies in mathematical economics and combinatorial auctions in algorithmic game theory we show that this problem admits a polynomial time approximation scheme ptas more generally we present a ptas for maximizing a discrete concave function called an m concave function under budget constraints our ptas is based on rounding an optimal solution of a continuous relaxation problem which is shown to be solvable in polynomial time by the ellipsoid method we also consider the maximization of the sum of two m concave functions under a single budget constraint this problem is a generalization of the budgeted max weight matroid intersection problem to the one with certain nonlinear objective functions we show that this problem also admits a ptas

tight approximation algorithms for maximum separable assignment problems a separable assignment problem sap is defined by a set of bins and a set of items to pack in each bin a value fij for assigning item j to bin i and a separate packing constraint for each bin i e for each bin a family of subsets of items that fit in to that bin the goal is to pack items into bins to maximize the aggregate value this class of problems includes the maximum generalized assignment problem gap 1 and a distributed caching problem dcp described in this paper given a approximation algorithm for finding the highest value packing of a single bin we give a polynomial time lp rounding based 1 1 e approximation algorithm a simple polynomial time local search 1 e approximation algorithm for any e 0 therefore for all examples of sap that admit an approximation scheme for the single bin problem we obtain an lp based algorithm with 1 1 e e approximation and a local search algorithm with e approximation guarantee furthermore for cases in which the subproblem admits a fully polynomial approximation scheme such as for gap the lp based algorithm analysis can be strengthened to give a guarantee of 1 1 e the best previously known approximation algorithm for gap is a approximation by shmoys and tardos and chekuri and khanna our lp algorithm is based on rounding a new linear programming relaxation with a provably better integrality gap to complement these results we show that sap and dcp cannot be approximated within a factor better than 1 1 e unless np dtime no log log n even if there exists a polynomial time exact algorithm for the single bin problem we extend the 1 1 e approximation algorithm to a constant factor approximation algorithms for a nonseparable assignment problem with applications in maximizing revenue for budget constrained combinatorial auctions and the adwords assignment problem we generalize the local search algorithm to yield a e approximation algorithm for the maximum k median problem with hard capacities

free bits pcps and nonapproximability towards tight results this paper continues the investigation of the connection between probabilistically checkable proofs pcps and the approximability of np optimization problems the emphasis is on proving tight nonapproximability results via consideration of measures such as the free bit complexity and the amortized free bit complexity of proof systems the first part of the paper presents a collection of new proof systems based on a new error correcting code called the long code we provide a proof system that has amortized free bit complexity of 2 epsilon implying that approximating maxclique within n frac13 e and approximating the chromatic number within n frac15 e are hard assuming np neq corp for any e 0 we also derive the first explicit and reasonable constant hardness factors for min vertex cover msat 2 and max cut and we improve the hardness factor for msat 3 we note that our nonapproximability factors for maxsnp problems are appreciably close to the values known to be achievable by polynomial time algorithms finally we note a general approach to the derivation of strong nonapproximability results under which the problem reduces to the construction of certain gadgets the increasing strength of nonapproximability results obtained via the pcp connection motivates us to ask how far this can go and whether pcps are inherent in any way the second part of the paper addresses this the main result is a reversal of the connection due to feige et al fglss connection j acm 43 1996 pp 268 292 where the latter had shown how to translate proof systems for np into np hardness of approximation results for maxclique we show how any np hardness of approximation result for maxclique yields a proof system for np roughly our result says that for any constant f if maxclique is np hard to approximate within n 1 1 f then np subseteq overline fpcp log f the latter being the class of languages possessing proofs of logarithmic randomness and amortized free bit complexity f this suggests that pcps are inherent to obtaining nonapproximability results furthermore the tight relation suggests that reducing the amortized free bit complexity is necessary for improving the nonapproximability results for maxclique the third part of our paper initiates a systematic investigation of the properties of pcp and fpcp free pcp as a function of the following various parameters randomness query complexity free bit complexity amortized free bit complexity proof size etc we are particularly interested in triviality results which indicate which classes are not powerful enough to capture np we also distill the role of randomized reductions in this area and provide a variety of useful transformations between proof checking complexity classes

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

almost tight bounds and existence theorems for single commodity confluent flows a flow of a commodity is said to be confluent if at any node all the flow of the commodity leaves along a single edge in this article we study single commodity confluent flow problems where we need to route given node demands to a single destination using a confluent flow single and multi commodity confluent flows arise in a variety of application areas most notably in networking in fact most flows in the internet are multi commodity confluent flows since internet routing is destination based we present near tight approximation algorithms hardness results and existence theorems for minimizing congestion in single commodity confluent flows the maximum edge congestion of a single commodity confluent flow occurs at one of the incoming edges of the destination therefore finding a minimum congestion confluent flow is equivalent to the following problem given a directed graph g with k sinks and non negative demands on all the nodes of g determine a confluent flow that routes every node demand to some sink such that the maximum congestion at a sink is minimized the main result of this article is a polynomial time algorithm for determining a confluent flow with congestion at most 1 p ln k in g if g admits a splittable flow with congestion at most 1 we complement this result in two directions first we present a graph g that admits a splittable flow with congestion at most 1 yet no confluent flow with congestion smaller than hk the kth harmonic number thus establishing tight upper and lower bounds to within an additive constant less than 1 second we show that it is np hard to approximate the congestion of an optimal confluent flow to within a factor of log2k 2 thus resolving the polynomial time approximability to within a multiplicative constant we also consider a demand maximization version of the problem we show that if g admits a splittable flow of congestion at most 1 then a variant of the congestion minimization algorithm yields a confluent flow in g with congestion at most 1 that satisfies 1 3 fraction of total demand we show that the gap between confluent flows and splittable flows is much smaller if the underlying graph is k connected in particular we prove that k connected graphs with k sinks admit confluent flows of congestion less than c p dmax where c is the congestion of the best splittable flow and dmax is the maximum demand of any node in g the proof of this existence theorem is non constructive and relies on topological techniques introduced by lovasz

general estimators for the reliability of qualitative data we study a proportional reduction in loss prl measure for the reliability of categorical data and consider the general case in which each ofn judges assigns a subject to one ofk categories this measure has been shown to be equivalent to a measure proposed by perreault and leigh for a special case when there are two equally competent judges and the correct category has a uniform prior distribution we consider a general framework where the correct category is assumed to have an arbitrary prior distribution and where classification probabilities vary by correct category judge and category of classification in this setting we consider prl reliability measures based on two estimators of the correct category the empirical bayes estimator and an estimator based on the judges consensus choice we also discuss four important special cases of the general model and study several types of lower bounds for prl reliability

deterministic versus randomized adaptive test cover in a combinatorial search problem with binary tests we are given a set of elements vertices and a hypergraph of possible tests hyperedges and the goal is to find an unknown target element using a minimum number of tests we explore the expected test number of randomized strategies our main results are that the ratio of the randomized and deterministic test numbers can be logarithmic in the number of elements that the optimal deterministic test number can be approximated in polynomial time only within a logarithmic factor whereas an approximation ratio 2 can be achieved in the randomized case and that optimal randomized strategies can be efficiently constructed at least for special classes of graphs

maximizing a monotone submodular function subject to a matroid constraint let f 2 x rightarrow cal r be a monotone submodular set function and let x cal i be a matroid we consider the problem rm max s in cal i f s it is known that the greedy algorithm yields a 1 2 approximation m l fisher g l nemhauser and l a wolsey math programming stud no 8 1978 pp 73 87 for this problem for certain special cases e g rm max s leq k f s the greedy algorithm yields a 1 1 e approximation it is known that this is optimal both in the value oracle model where the only access to f is through a black box returning f s for a given set s g l nemhauser and l a wolsey math oper res 3 1978 pp 177 188 and for explicitly posed instances assuming p neq np u feige j acm 45 1998 pp 634 652 in this paper we provide a randomized 1 1 e approximation for any monotone submodular function and an arbitrary matroid the algorithm works in the value oracle model our main tools are a variant of the pipage rounding technique of ageev and sviridenko j combin optim 8 2004 pp 307 328 and a continuous greedy process that may be of independent interest as a special case our algorithm implies an optimal approximation for the submodular welfare problem in the value oracle model j vondrak proceedings of the 38 th acm symposium on theory of computing 2008 pp 67 74 as a second application we show that the generalized assignment problem gap is also a special case although the reduction requires x to be exponential in the original problem size we are able to achieve a 1 1 e o 1 approximation for gap simplifying previously known algorithms additionally the reduction enables us to obtain approximation algorithms for variants of gap with more general constraints