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

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

inequality systems and minimax theorems abstract this paper is a comprehensive study of the results about the consistency of inequality systems the minimax theorems are the main tools used in proceeding to this survey most of the known results are quoted and some of them are weakened or generalized to infinite dimensional spaces applications to programming are given

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

multiobject behavior recognition by event driven selective attention method this paper presents a multiobject behaviour recognition approach based on assumption generation and verification i e feasible assumptions about the present behaviors consistent with the input image and behavior models are dynamically generated and verified by finding their supporting evidence in input images this can be realized by an architecture called the selective attention model which consists of a state dependent event detector and an event sequence analyzer the former detects image variation event in a limited image region focusing region which is not affected by occlusions and outliers the latter analyzes sequences of detected events and activates all feasible states representing assumptions about multiobject behaviors we further extend the system by introducing colored token propagation to discriminate different objects in state space and integration of multiviewpoint image sequences to disambiguate the single view recognition results extensive experiments of human behavior recognition in real world environments demonstrate the soundness and robustness of our architecture

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

hard constraint satisfaction problems have hard gaps at location 1 an instance of the maximum constraint satisfaction problem max csp is a finite collection of constraints on a set of variables and the goal is to assign values to the variables that maximises the number of satisfied constraints max csp captures many well known problems such as maxk sat and max cut and is consequently np hard thus it is natural to study how restrictions on the allowed constraint types or constraint language affect the complexity and approximability of max csp the pcp theorem is equivalent to the existence of a constraint language for which max csp has a hard gap at location 1 i e it is np hard to distinguish between satisfiable instances and instances where at most some constant fraction of the constraints are satisfiable all constraint languages for which the csp problem i e the problem of deciding whether all constraints can be satisfied is currently known to be np hard have a certain algebraic property we prove that any constraint language with this algebraic property makes max csp have a hard gap at location 1 which in particular implies that such problems cannot have a ptas unless p np we then apply this result to max csp restricted to a single constraint type this class of problems contains for instance max cut and max dicut assuming p np we show that such problems do not admit ptas except in some trivial cases our results hold even if the number of occurrences of each variable is bounded by a constant finally we give some applications of our results

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

resolution and linear cnf formulas improved n 3 maxsat algorithms abstract we study techniques for solving the maxsat problem on instances in which the variable degree is bounded by 3 the problem is np hard we show how resolution principle can be applied that converts an instance into an equivalent instance in which the cnf formula becomes a linear cnf formula we then show how more efficient branching strategies can be applied on linear cnf formulas as applications we present two algorithms one of running time o 1 194 k that solves the parameterized version of the problem and the other of running time o 1 237 n that solves the optimization version of the problem both significantly improving previous best upper bounds

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

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

an assumption based tms abstract this paper presents a new view of problem solving motivated by a new kind of truth maintenance system unlike previous truth maintenance systems which were based on manipulating justifications this truth maintenance system is in addition based on manipulating assumption sets as a consequence it is possible to work effectively and efficiently with inconsistent information context switching is free and most backtracking and all retraction is avoided these capabilities motivate a different kind of problem solving architecture in which multiple potential solutions are explored simultaneously this architecture is particularly well suited for tasks where a reasonable fraction of the potential solutions must be explored

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

inequality systems and optimization abstract in this paper new versions of solvability theorems are given for general inequality systems involving nonconvex functions and noncompact sets the results are applied to constrained optimization problems with an infinite number of constraints and to minimax problems on noncompact sets

on greedy algorithms in coding theory we study a wide class of problems in coding theory for which we consider two different formulations in terms of incidence matrices and in terms of hypergraphs these problems are dealt with using a greedy algorithm due to stein 1974 and lovasz 1975 some examples including constructing covering codes codes for conflict resolution separating systems source encoding with distortion etc are given a unified treatment under certain conditions derandomization can be performed leading to an essential reduction in the complexity of the constructions

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

computing unsatisfiable k sat instances with few occurrences per variable k s sat is the propositional satisfiability problem restricted to instances where each clause has exactly k distinct literals and every variable occurs at most s times it is known that there exists an exponential function f such that for s f k all k s sat instances are satisfiable but k f k 1 sat is already np complete k 3 exact values of f are only known for k 3 and 4 and it is open whether f is computable we introduce a computable function f1 which bounds f from above and determine the values of f1 by means of a calculus of integer sequences this new approach enables us to improve the best known upper bounds for f k generalizing the known constructions for unsatisfiable k s sat instances for small k

playing mastermind with many colors we analyze the general version of the classic guessing game mastermind with n positions and k colors since the case k l n1 e e 0 a constant is well understood we concentrate on larger numbers of colors for the most prominent case k e n our results imply that codebreaker can find the secret code with o nlog log n guesses this bound is valid also when only black answer pegs are used it improves the o nlog n bound first proven by chvatal we also show that if both black and white answer pegs are used then the o nlog log n bound holds for up to n2log log n colors these bounds are almost tight as the known lower bound of n shows unlike for k l n1 e simply guessing at random until the secret code is determined is not sufficient in fact we show that an optimal nonadaptive strategy deterministic or randomized needs nlog n guesses

typical rounding problems the linear discrepancy problem is to round a given 0 1 vector x to a binary vector y such that the rounding error with respect to a linear form is small i e such that a x y is small for some given matrix a the combinatorial discrepancy problem is the special case of x 1 2 1 2 t a famous result of beck and spencer math programming 30 1984 88 as well as lovasz et al european j combin 7 1986 151 shows that the linear discrepancy problem is not much harder than this special case any linear discrepancy problem can be solved with at most twice the maximum rounding error among the discrepancy problems of the submatrices of a in this paper we strengthen this result for the common situation that the discrepancy of submatrices having n0 columns is bounded by cn0 for some c 0 0 1 in this case we improve the constant by which the general problem is harder than the discrepancy one from 2 down to 2 2 3 we also find that a random vector has expected linear discrepancy 2 1 2 cn only hence in the typical situation that the discrepancy is decreasing for smaller matrices the linear discrepancy problem is even less difficult compared to the discrepancy one than assured by previous results we also obtain the bound lindisc a x 2 2 21 1 c x 1 our proofs use a reduction to pusher chooser games

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

nonasymptotic necessary conditions for nonsmooth infinite optimization problems abstract for nonsmooth infinite optimization problems with inequality constraints nonasymptotic conditions of f john and karush kuhn tucker type in terms of approximate derivatives are established the lagrange multiplier being a nonnegative radon measure on the index set which is assumed to be a compact hausdorff space the result is obtained with the aid of a farkas type theorem for infinite systems of nonlinear inequalities

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

randomization derandomization and antirandomization three games abstract three games are given between two players paul and carole with a common theme in each round paul does a split and carole chooses random play by carole allows a bound for the game value through derandomization this becomes a deterministic strategy for carole minimizing a weight function paul can use that same weight function to give a bound for the game value in the other direction

six standard deviations suffice given n sets on n elements it is shown that there exists a two coloring such that all sets have discrepancy at most knl 2 k an absolute constant this improves the basic probabilistic method with which k c ln n l 2 the result is extended to n finite sets of arbitrary size probabilistic techniques are melded with the pigeonhole principle an alternate proof of the existence of rudin shapiro functions is given showing that they are exponential in number given n linear forms in n variables with all coefficients in 1 1 it is shown that initial values pl p e 0 1 may be approximated by el e 0 1 so that the forms have small error 1 we state our main result first in the language of linear forms theorem 1 let 1 1 li xl x l ailxl xe ai txn 1 i n be n linear forms in n variables with all laijl 1 then there exist e1 esl e 1 1 such that 1 2 li el e l ks for all i 1 i n here k is an absolute constant when all aij e 0 1 we may consider e1 aij as the incidence matrix for a family of n sets on n elements that is we may set ai j aij 1 given a two coloring say red and blue of 1 n the discrepancy disc x of a set x c 1 n is defined as the number of red points in x minus the number of blue points in x if we interpret si 1 as meaning i is to be colored red and ej 1 as blue then we obtain the following result corollary 2 let al an c 1 n then there exists a two coloring of 1 n so that 1 3 disc aj kw foralli 1 i n received by the editors july 1 1984 1980 muthemcltics subject clclssific ltion primary 05b20 secondary 41a28 42a61 this work was initiated while the author was an irex exchange fellow at the mathematical institute budapest and completed with support of the national science foundation for the many kindnesses and for the creative research environment provided by institute staff and colleagues koszonom szepen t1985 american mathematical society 0002 9947 85 1 00 25 per page

on some combinatorial questions in finite dimensional spaces abstract bounds are given for supinf p i 1 i where sup is taken over all set systems v 1 v p of r n with 0 p i 1 conv v i and sup v v i 1 for i 1 p and inf is taken over all possible choices i v i for i 1 p another similar problem is considered the bounds are sharp

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

beyond tracking modelling activity and understanding behaviour in this work we present a unified bottom up and top down automatic model selection based approach for modelling complex activities of multiple objects in cluttered scenes an activity of multiple objects is represented based on discrete scene events and their behaviours are modelled by reasoning about the temporal and causal correlations among different events this is significantly different from the majority of the existing techniques that are centred on object tracking followed by trajectory matching in our approach object independent events are detected and classified by unsupervised clustering using expectation maximisation em and classified using automatic model selection based on schwarz s bayesian information criterion bic dynamic probabilistic networks dpns are formulated for modelling the temporal and causal correlations among discrete events for robust and holistic scene level behaviour interpretation in particular we developed a dynamically multi linked hidden markov model dml hmm based on the discovery of salient dynamic interlinks among multiple temporal processes corresponding to multiple event classes a dml hmm is built using bic based factorisation resulting in its topology being intrinsically determined by the underlying causality and temporal order among events extensive experiments are conducted on modelling activities captured in different indoor and outdoor scenes our experimental results demonstrate that the performance of a dml hmm on modelling group activities in a noisy and cluttered scene is superior compared to those of other comparable dynamic probabilistic networks including a multi observation hidden markov model mohmm a parallel hidden markov model pahmm and a coupled hidden markov model chmm

one more occurrence of variables makes satisfiability jump from trivial to np complete a boolean formula in a conjunctive normal form is called a k s formula if every clause contains exactly k variables and every variable occurs in at most s clauses the k s text sat problem is the satisfiability problem restricted to k s formulas it is proved that for every k geqslant 3 there is an integer f k such that k s text sat is trivial for s leqslant f k because every k s formula is satisfiable and is np complete for s geqslant f k 1 moreover f k grows exponentially with k namely lfloor 2 k ek rfloor leqslant f k leqslant 2 k 1 2 k 4 1 for k geqslant 4

boosted string representation and its application to video surveillance this paper presents a new behavior analysis system for analyzing human movements via a boosted string representation first of all we propose a triangulation based method to transform each action sequence into a set of symbols then an action sequence can be interpreted and analyzed using this string representation to analyze action sequences with this string representation three practical problems should be tackled usually an action sequence has different temporal scaling changes different initial states and symbol converting errors traditional methods like hidden markov models and finite state machines have limited abilities to deal with the above problems since many unknown states should be constructed and initialized to tackle the problems a novel string hypothesis generator is then proposed for generating a bank of string features from which different invariant features can be learned for classifying behaviors more accurately to learn the invariant features the adaboost algorithm is used and modified to train a strong classifier from the set of string hypotheses so that multiple human action events can be well classified in addition a forward classification scheme is proposed to classify all input action sequences more accurately even though they have various scaling changes and coding errors experimental results prove that the proposed method is a robust accurate and powerful tool for human movement analysis

ulam s searching game with a fixed number of lies abstract paul tries to find an unknown x from l to n by asking q yes no questions in response carole may lie up to k times for k fixed and n q sufficiently large necessary and sufficient conditions are given for paul to win

the complexity of approximately counting in 2 spin systems on k uniform bounded degree hypergraphs one of the most important recent developments in the complexity of approximate counting is the classification of the complexity of approximating the partition functions of antiferromagnetic 2 spin systems on bounded degree graphs this classification is based on a beautiful connection to the so called uniqueness phase transition from statistical physics on the infinite regular tree our objective is to study the impact of this classification on unweighted 2 spin models on k uniform hypergraphs as has already been indicated by yin and zhao the connection between the uniqueness phase transition and the complexity of approximate counting breaks down in the hypergraph setting nevertheless we show that for every non trivial symmetric k ary boolean function f there exists a degree bound 0 so that for all 0 the following problem is np hard given a k uniform hypergraph with maximum degree at most approximate the partition function of the hypergraph 2 spin model associated with f it is np hard to approximate this partition function even within an exponential factor by contrast if f is a trivial symmetric boolean function e g any function f that is excluded from our result then the partition function of the corresponding hypergraph 2 spin model can be computed exactly in polynomial time

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

dnf tautologies with a limited number of occurences of every variable it is known that every dnf tautology with all monomials of length k contains a variable with at least 2k k occurrences it is not known however if this bound is tight i e if there are tautologies with at most o 2k k occurrences of every variable dnf tautologies with 2k monomials of length k and with at most 2k k occurrences of every variable where log34 1 0 26 are presented this has the following consequence let k s sat be k sat restricted to instances with at most s occurrences of every variable it is known that for every k there is an sk such that k sk sat is np complete and k sk 1 sat is trivial in the sense that every instance has positive answer the above result implies that sk 2k k this improves the previously known bound sk 13642k for all k 6

european tenure games we study a variant of the tenure game introduced by spencer theoret comput sci 131 1994 415 in this version faculty is not fired but downgraded to the lowest rung instead for the upper bound we give a potential function argument showing that the value d of the game starting with d faculty on the first rung satisfies d log2d log2 log2d 1 98 we prove a nearly matching lower bound of log2d log2 log2d using a strategy that can be interpreted as an antirandomization of spencer s original game for d tending to infinity these bounds improve to log2d log2 log2d 1 o 1 d log2d log2 log2d 1 73 o 1 in particular the set of all d n such that the value of the game is precisely log2d log2 log2d 1 has lower density greater than 1 5

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

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

uniform sampling through the lovasz local lemma we propose a new algorithmic framework called partial rejection sampling to draw samples exactly from a product distribution conditioned on none of a number of bad events occurring our framework builds new connections between the variable framework of the lovasz local lemma and some classical sampling algorithms such as the cycle popping algorithm for rooted spanning trees among other applications we discover new algorithms to sample satisfying assignments of k cnf formulas with bounded variable occurrences