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

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

point singularities and the singularity method in classical elastodynamics the object of the article is to elucidate the application of the singularity method in classical elastodynamics to this end solutions for higher order point singularities e g time harmonic concentrated moment centre of dilatation centre of rotation and force tensor are derived from the influence tensor by a method usually employed in the theory of electrostatics for the study of potential of a system of charged particles solutions of some elastodynamic problems concerning spherical cavities and rigid spherical inclusions are then derived by combining these point singularities

the approximability of constraint satisfaction problems we study optimization problems that may be expressed as boolean constraint satisfaction problems an instance of a boolean constraint satisfaction problem is given by m constraints applied to n boolean variables different computational problems arise from constraint satisfaction problems depending on the nature of the underlying constraints as well as on the goal of the optimization task here we consider four possible goals max csp min csp is the class of problems where the goal is to find an assignment maximizing the number of satisfied constraints minimizing the number of unsatisfied constraints max ones min ones is the class of optimization problems where the goal is to find an assignment satisfying all constraints with maximum minimum number of variables set to 1 each class consists of infinitely many problems and a problem within a class is specified by a finite collection of finite boolean functions that describe the possible constraints that may be used tight bounds on the approximability of every problem in max csp were obtained by creignou j comput system sci 51 1995 pp 511 522 in this work we determine tight bounds on the approximability i e the ratio to within which each problem may be approximated in polynomial time of every problem in max ones min csp and min ones combined with the result of creignou this completely classifies all optimization problems derived from boolean constraint satisfaction our results capture a diverse collection of optimization problems such as max 3 sat max cut max clique min cut nearest codeword etc our results unify recent results on the in approximability of these optimization problems and yield a compact presentation of most known results moreover these results provide a formal basis to many statements on the behavior of natural optimization problems that have so far been observed only empirically

series solutions of nano boundary layer flows by means of the homotopy analysis method we present here a similar solution for the nano boundary layer with nonlinear navier boundary condition three types of flows are considered i the flow past a wedge ii the flow in a convergent channel iii the flow driven by an exponentially varying outer flows the resulting differential equations are solved by the homotopy analysis method different from the perturbation methods the present method is independent of small physical parameters so that it is applicable for not only weak but also strong nonlinear flow phenomena numerical results are compared with the available exact results to demonstrate the validity of the present solution the effects of the slip length the index parameters n and m on the velocity profile and the tangential stress are investigated and discussed

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

approximate max flow min multi cut theorems and their applications consider the multicommodity flow problem in which the object is to maximize the sum of commodities routed we prove the following approximate max flow min multicut theorem dst frac mbox rm min multicut o log k leq mbox rm max flow leq mbox rm min multicut noindent where k is the number of commodities our proof is constructive it enables us to find a multicut within o log k of the max flow and hence also the optimal multicut in addition the proof technique provides a unified framework in which one can also analyse the case of flows with specified demands of leighton and rao and klein et al and thereby obtain an improved bound for the latter problem

the approximability of non boolean satisfiability problems and restricted integer programming in this paper we present improved approximation algorithms for two classes of maximization problems defined in barland et al j comput system sci 57 2 1998 144 our factors of approximation substantially improve the previous known results and are close to the best possible on the other hand we show that the approximation results in the framework of barland et al hold also in the parallel setting and thus we have a new common framework for both computational settings we prove almost tight non approximability results thus solving a main open question of barland et al we obtain the results through the constraint satisfaction problem over multi valued domains for which we develop approximation algorithms and show non approximability results our parallel approximation algorithms are based on linear programming and random rounding they are better than previously known sequential algorithms the non approximability results are based on new recent progress in the fields of probabilistically checkable proofs and multi prover one round proof systems

on syntactic versus computational views of approximability we attempt to reconcile the two distinct views of approximation classes syntactic and computational syntactic classes such as max snp permit structural results and have natural complete problems while computational classes such as apx allow us to work with classes of problems whose approximability is well understood our results provide a syntactic characterization of computational classes and give a computational framework for syntactic classes we compare the syntactically defined class max snp with the computationally defined class apx and show that every problem in apx can be placed i e has approximation preserving reduction to a problem in max snp our methods introduce a simple yet general technique for creating approximation preserving reductions which shows that any well approximable problem can be reduced in an approximation preserving manner to a problem which is hard to approximate to corresponding factors the reduction then follows easily from the recent nonapproximability results for max snp hard problems we demonstrate the generality of this technique by applying it to other classes such as max snp rmax 2 and min f pi 2 1 which have the clique problem and the set cover problem respectively as complete problems the syntactic nature of max snp was used by papadimitriou and yannakakis j comput system sci 43 1991 pp 425 440 to provide approximation algorithms for every problem in the class we provide an alternate approach to demonstrating this result using the syntactic nature of max snp we develop a general paradigm nonoblivious local search useful for developing simple yet efficient approximation algorithms we show that such algorithms can find good approximations for all max snp problems yielding approximation ratios comparable to the best known for a variety of specific max snp hard problems nonoblivious local search provably outperforms standard local search in both the degree of approximation achieved and the efficiency of the resulting algorithms

the complexity of minimal satisfiability problems a dichotomy theorem for a class of decision problems is a result asserting that certain problems in the class are solvable in polynomial time while the rest are np complete the first remarkable such dichotomy theorem was proved by schaefer in 1978 it concerns the class of generalized satisfiability problems sat s whose input is a cnf s formula i e a formula constructed from elements of a fixed set s of generalized connectives using conjunctions and substitutions by variables here we investigate the complexity of minimal satisfiability problems min sat s where s is a fixed set of generalized connectives the input to such a problem is a cnf s formula and a satisfying truth assignment the question is to decide whether there is another satisfying truth assignment that is strictly smaller than the given truth assignment with respect to the coordinate wise partial order on truth assignments minimal satisfiability problems were first studied by researchers in artificial intelligence while investigating the computational complexity of propositional circumscription the question of whether dichotomy theorems can be proved for these problems was raised at that time but was left open we settle this question affirmatively by establishing a dichotomy theorem for the class of all min sat s problems where s is a finite set of generalized connectives we also prove a dichotomy theorem for a variant of min sat s in which the minimization is restricted to a subset of the variables whereas the remaining variables may vary arbitrarily this variant is related to extensions of propositional circumscription and was first studied by cadoli moreover we show that similar dichotomy theorems hold also when some of the variables are assigned constant values finally we give simple criteria that tell apart the polynomial time solvable cases of these minimal satisfiability problems from the np complete ones

generalising submodularity and horn clauses tractable optimization problems defined by tournament pair multimorphisms the submodular function minimization problem sfm is a fundamental problem in combinatorial optimization and several fully combinatorial polynomial time algorithms have recently been discovered to solve this problem the most general versions of these algorithms are able to minimize any submodular function whose domain is a set of tuples over any totally ordered finite set and whose range includes both finite and infinite values in this paper we demonstrate that this general form of sfm is just one example of a much larger class of tractable discrete optimization problems defined by valued constraints these tractable problems are characterized by the fact that their valued constraints have an algebraic property which we call a tournament pair multimorphism this larger tractable class also includes the problem of satisfying a set of horn clauses horn sat as well as various extensions of this problem to larger finite domains

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

gadgets approximation and linear programming we present a linear programming based method for finding gadgets i e combinatorial structures reducing constraints of one optimization problem to constraints of another a key step in this method is a simple observation which limits the search space to a finite one using this new method we present a number of new computer constructed gadgets for several different reductions this method also answers a question posed by bellare goldreich and sudan siam j comput 27 1998 pp 804 915 of how to prove the optimality of gadgets linear programming duality gives such proofs the new gadgets when combined with recent results of ha stad proceedings of the 29th acm symposium on theory of computing 1997 pp 1 10 improve the known inapproximability results for max cut and max dicut showing that approximating these problems to within factors of 16 17 epsilon and 12 13 epsilon respectively is np hard for every epsilon 0 prior to this work the best known inapproximability thresholds for both problems were 71 72 m bellare o goldreich and m sudan siam j comput 27 1998 pp 804 915 without using the gadgets from this paper the best possible hardness that would follow from bellare goldreich and sudan and ha s tad is 18 19 we also use the gadgets to obtain an improved approximation algorithm for max3 sat which guarantees an approximation ratio of 801 this improves upon the previous best bound implicit from m x goemans and d p williamson j acm 42 1995 pp 1115 1145 u feige and m x goemans proceedings of the third israel symposium on theory of computing and systems 1995 pp 182 189 of 7704

completeness in approximation classes we introduce a formal framework for studying approximation properties of np optimization npo problems the classes we consider are those appearing in the literature namely the class of approximable problems within a constant e apx the class of problems having a polynomial time approximation scheme pas and the class of problems having a fully polynomial time approximation scheme fpas we define natural approximation preserving reductions and obtain completeness results for these classes a complete problem in a class can not have stronger approximation properties unless p np we also show that the degree structure of npo allows intermediate degrees that is if p np there are problems which are neither complete nor belong to a lower class

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

the power of linear programming for general valued csps let d called the domain be a fixed finite set and let gamma called the valued constraint language be a fixed set of functions of the form f d m to mathbb q cup infty where different functions might have different arity m we study the valued constraint satisfaction problem parametrized by gamma denoted by vcsp gamma these are minimization problems given by n variables and the objective function given by a sum of functions from gamma each depending on a subset of the n variables for example if d 0 1 and gamma contains all ternary 0 infty valued functions vcsp gamma corresponds to 3 sat more generally if gamma contains only 0 infty valued functions vcsp gamma corresponds to csp gamma if d 0 1 and gamma contains all ternary 0 1 valued functions vcsp gamma corresponds to min 3 sat in which the goal is to minimize the number of unsatisfied clauses in a 3 cnf instance finite valued constraint languages contain functions that take on only rational values and not infinite values our main result is a precise algebraic characterization of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation blp for a valued constraint language gamma blp is a decision procedure for gamma if and only if gamma admits a symmetric fractional polymorphism of every arity for a finite valued constraint language gamma blp is a decision procedure if and only if gamma admits a symmetric fractional polymorphism of some arity or equivalently if gamma admits a symmetric fractional polymorphism of arity 2 using these results we obtain tractability of several novel classes of problems including problems over valued constraint languages that are 1 submodular on arbitrary lattices 2 k submodular on arbitrary finite domains 3 weakly and hence strongly tree submodular on arbitrary trees

structure in approximation classes the study of the approximability properties of np hard optimization problems has recently made great advances mainly due to the results obtained in the field of proof checking the last important breakthrough proves the apx completeness of several important optimization problems and thus reconciles two distinct views of approximation classes syntactic and computational s khanna et al in proc 35th ieee symp on foundations of computer science ieee computer society press los alamitos ca 1994 pp 819 830 in this paper we obtain new results on the structure of several computationally defined approximation classes in particular after defining a new approximation preserving reducibility to be used for as many approximation classes as possible we give the first examples of natural npo complete problems and the first examples of natural apx intermediate problems moreover we state new connections between the approximability properties and the query complexity of npo problems

on simple and creative sets in np abstract two structurally defined types of np sets are studied k simple sets are defined and shown to exist in np other properties of these sets are investigated k creative sets as previously defined by joseph and young 1985 are next considered a new condition is given which implies that a set is k creative several previously considered np complete sets are proved to be k creative

periodic unidirectional flows of a viscoelastic fluid with the fractional maxwell model exact analytic solutions for the flows of viscoelastic fluids are constructed the flows are induced by general periodic oscillations of a plate based on the flow conditions described three flow situations are solved by fourier transform further the fractional calculus approach is taken into account in the constitutive relationship of viscoelastic fluid model finally some interesting flows caused by certain special oscillations are also established

on the hamming distance of constraint satisfaction problems in this paper we consider a new optimization problem called max hammingdistance f where f is a family of boolean constraints this problem consists in finding two satisfying assignments that differ in the maximum number of variable values in other words the problem looks for the maximum difference between two models of the constraints given in input we give a complete classification of the approximability properties of max hammingdistance f by using a specialization of the criteria introduced by schaefer in order to classify constraint satisfaction problems and subsequently used by khanna sudan trevisan and williamson to classify constraint satisfaction optimization problems

resolution for max sat max sat is the problem of finding an assignment minimizing the number of unsatisfied clauses in a cnf formula we propose a resolution like calculus for max sat and prove its soundness and completeness we also prove the completeness of some refinements of this calculus from the completeness proof we derive an exact algorithm for max sat and a time upper bound we also define a weighted max sat resolution like rule and show how to adapt the soundness and completeness proofs of the max sat rule to the weighted max sat rule finally we give several particular max sat problems that require an exponential number of steps of our max sat rule to obtain the minimal number of unsatisfied clauses of the combinatorial principle these results are based on the corresponding resolution lower bounds for those particular problems

complexity of generalized satisfiability counting problems abstract the class of generalized satisfiability problems introduced in 1978 by schaefer presents a uniform way of studying the complexity of satisfiability problems with special conditions the complexity of each decision and counting problem in this class depends on the involved logical relations in 1979 valiant defined the class p the class of functions definable as the number of accepting computations of a polynomial time nondeterministic turing machine clearly all satisfiability counting problems belong to this class p we prove a dichotomy theorem for generalized satisfiability counting problems that is if all logical relations involved in a generalized satisfiability counting problem are affine then the number of satisfying assignments of this problem can be computed in polynomial time otherwise this function is p complete this gives us a comparison between decision and counting generalized satisfiability problems we can determine exactly the polynomial satisfiability decision problems whose number of solutions can be computed in polynomial time and also the polynomial satisfiability decision problems whose counting counterparts are already p complete moreover taking advantage of a similar dichotomy result proved in 1978 by schaefer for generalized satisfiability decision problems we get as a corollary the implication that the counting counterpart of each np complete generalized satisfiability decision problem is p complete

the complexity of soft constraint satisfaction over the past few years there has been considerable progress in methods to systematically analyse the complexity of constraint satisfaction problems with specified constraint types one very powerful theoretical development in this area links the complexity of a set of constraints to a corresponding set of algebraic operations known as polymorphisms in this paper we extend the analysis of complexity to the more general framework of combinatorial optimisation problems expressed using various forms of soft constraints we launch a systematic investigation of the complexity of these problems by extending the notion of a polymorphism to a more general algebraic operation which we call a multimorphism we show that many tractable sets of soft constraints both established and novel can be characterised by the presence of particular multimorphisms we also show that a simple set of np hard constraints has very restricted multimorphisms finally we use the notion of multimorphism to give a complete classification of complexity for the boolean case which extends several earlier classification results for particular special cases

the complexity of conservative valued csps we study the complexity of valued constraint satisfaction problems vcsps parametrized by a constraint language a fixed set of cost functions over a finite domain an instance of the problem is specified by a sum of cost functions from the language and the goal is to minimize the sum under the unique games conjecture the approximability of finite valued vcsps is well understood see raghavendra 2008 however there is no characterization of finite valued vcsps let alone general valued vcsps that can be solved exactly in polynomial time thus giving insights from a combinatorial optimization perspective we consider the case of languages containing all possible unary cost functions in the case of languages consisting of only 0 valued cost functions i e relations such languages have been called conservative and studied by bulatov 2003 2011 and recently by barto 2011 since we study valued languages we call a language conservative if it contains all finite valued unary cost functions the computational complexity of conservative valued languages has been studied by cohen et al 2006 for languages over boolean domains by deineko et al 2008 for 0 1 valued languages a k a max csp and by takhanov 2010a for 0 valued languages containing all finite valued unary cost functions a k a min cost hom we prove a schaefer like dichotomy theorem for conservative valued languages if all cost functions in the language satisfy a certain condition specified by a complementary combination of stp and mjn multimorphisms then any instance can be solved in polynomial time via a new algorithm developed in this article otherwise the language is np hard this is the first complete complexity classification of general valued constraint languages over non boolean domains it is a common phenomenon that complexity classifications of problems over non boolean domains are significantly harder than the boolean cases the polynomial time algorithm we present for the tractable cases is a generalization of the submodular minimization problem and a result of cohen et al 2008 our results generalize previous results by takhanov 2010a and a subset of results by cohen et al 2006 and deineko et al 2008 moreover our results do not rely on any computer assisted search as in deineko et al 2008 and provide a powerful tool for proving hardness of finite valued and general valued languages

on the mhd flow of a second grade fluid in a porous channel the steady flow of a second grade fluid in a porous channel is considered the constitutive equations are those used for a second grade fluid the fluid is electrically conducting in the presence of a uniform magnetic field applied in the transverse direction to the flow it is shown that an analytical solution is possible by employing a homotopy analysis method ham the convergence of the obtained solution is also taken into account assessment for the influence of various parameters of interest on the velocity is undertaken

couette and poiseuille flows of an oldroyd 6 constant fluid with magnetic field abstract in this paper we develop a set of differential equations describing the steady flow of an oldroyd 6 constant magnetohydrodynamic fluid the fluid is electrically conducting in the presence of a uniform transverse magnetic field the developed non linear differential equation takes into account the effect of the material constants and the applied magnetic field we presented the solution for three types of steady flows namely i couette flow ii poiseuille flow and iii generalized couette flow homotopy analysis method ham is used to solve the non linear differential equation analytically it is found from the present analysis that for steady flow the obtained solutions are strongly dependent on the material constants non newtonian parameters which is different from the model of oldroyd 3 constant fluid numerical solutions are also given and compared with the solutions by ham

tight bounds and 2 approximation algorithms for integer programs with two variables per inequality the problem of integer programming in bounded variables over constraints with no more than two variables in each constraint is np complete even when all variables are binary this paper deals with integer linear minimization problems inn variables subject tom linear constraints with at most two variables per inequality and with all variables bounded between 0 andu for such systems a 2 approximation algorithm is presented that runs in time o mnu2 log un2m so it is polynomial in the input size if the upper boundu is polynomially bounded the algorithm works by finding first a super optimal feasible solution that consists of integer multiples of 1 2 that solution gives a tight bound on the value of the minimum it furthermore has an identifiable subset of integer components that retain their value in an integer optimal solution of the problem these properties are a generalization of the properties of the vertex cover problem the algorithm described is in particular a 2 approximation algorithm for the problem of minimizing the total weight of true variables among all truth assignments to the 2 satisfiability problem

lp sdp hierarchy lower bounds for decoding random ldpc codes random d v d c regular low density parity check ldpc codes where each variable is involved in d v parity checks and each parity check involves d c variables are well known to achieve the shannon capacity of the binary symmetric channel for sufficiently large d v and d c under exponential time decoding however polynomial time algorithms are only known to correct a much smaller fraction of errors one of the most powerful polynomial time algorithms with a formal analysis is the linear programming lp decoding algorithm of feldman et al which is known to correct an omega 1 d c fraction of errors in this paper we show that fairly powerful extensions of lp decoding based on the sherali adams and lasserre hierarchies fail to correct much more errors than the basic lp decoder in particular we show that 1 for any values of d v and d c a linear number of rounds of the sherali adams lp hierarchy cannot correct more than an o 1 d c fraction of errors on a random d v d c regular ldpc code and 2 for any value of d v and infinitely many values of d c a linear number of rounds of the lasserre sdp hierarchy cannot correct more than an o 1 d c fraction of errors on a random d v d c regular ldpc code our proofs use a new stretching and collapsing technique that allows us to leverage recent progress in the study of the limitations of lp sdp hierarchies for maximum constraint satisfaction problems max csps the problem then reduces to the construction of special balanced pairwise independent distributions for sherali adams and special cosets of balanced pairwise independent subgroups for lasserre our algebraic construction for the lasserre hierarchy is based on designing sets of points in mathbb f q d for q any power of 2 and d 2 3 with special hyperplane incidence properties constructions that may be of independent interest an intriguing consequence of our work is that expansion seems to be both the strength and the weakness of random regular ldpc codes some of our techniques are more generally applicable to a large class of boolean csps called min ones in particular for k hypergraph vertex cover we obtain an improved integrality gap of k 1 epsilon that holds after a linear number of rounds of the lasserre hierarchy for any k q 1 with q an arbitrary prime power the best previous gap for a linear number of rounds was equal to 2 epsilon and due to schoenebeck

the approximability of three valued max csp in the maximum constraint satisfaction problem max csp one is given a finite collection of possibly weighted constraints on overlapping sets of variables and the goal is to assign values from a given domain to the variables so as to maximize the number or the total weight for the weighted case of satisfied constraints this problem is np hard in general and therefore it is natural to study how restricting the allowed types of constraints affects the approximability of the problem it is known that every boolean that is two valued max csp with a finite set of allowed constraint types is either solvable exactly in polynomial time or else apx complete and hence can have no polynomial time approximation scheme unless p np it has been an open problem for several years whether this result can be extended to non boolean max csp which is much more difficult to analyze than the boolean case in this paper we make the first step in this direction by establishing this result for max csp over a three element domain moreover we present a simple description of all polynomial time solvable cases of our problem this description uses the well known algebraic combinatorial property of supermodularity we also show that every hard three valued max csp contains in a certain specified sense one of the two basic hard max csps which are the maximum k colorable subgraph problems for k 2 3

affine reductions for lps and sdps we define a reduction mechanism for lp and sdp formulations that degrades approximation factors in a controlled fashion our reduction mechanism is a minor restriction of classical hardness reductions requiring an additional independence assumption and it allows for reusing many hardness reductions that have been used to show inapproximability in the context of pcp theorems as a consequence we establish strong linear programming inapproximability for lps with a polynomial number of constraints for many problems in particular we obtain a frac 3 2 varepsilon inapproximability for answering an open question in chan et al proceedings of focs pp 350 359 2013 https doi org 10 1109 focs 2013 45 and prove an inapproximability factor of frac 1 2 varepsilon for bounded degree in the case of sdps we obtain inapproximability results for these problems relative to the sdp inapproximability of textsf maxcut moreover using our reduction framework we are able to reproduce various results for csps from chan et al proceedings of focs pp 350 359 2013 https doi org 10 1109 focs 2013 45 via simple reductions from max 2 xor

the complexity of finite valued csps we study the computational complexity of exact minimization of rational valued discrete functions let be a set of rational valued functions on a fixed finite domain such a set is called a finite valued constraint language the valued constraint satisfaction problem vcsp is the problem of minimizing a function given as a sum of functions from we establish a dichotomy theorem with respect to exact solvability for all finite valued constraint languages defined on domains of arbitrary finite size we show that every constraint language either admits a binary symmetric fractional polymorphism in which case the basic linear programming relaxation solves any instance of vcsp exactly or satisfies a simple hardness condition that allows for a polynomial time reduction from max cut to vcsp

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

max ones generalized to larger domains we study a family of problems called maximum solution where the objective is to maximize a linear goal function over the feasible integer assignments to a set of variables subject to a set of constraints when the domain is boolean i e restricted to 0 1 the maximum solution problem is identical to the well studied max ones problem and the approximability is completely understood for all restrictions on the underlying constraints s khanna m sudan l trevisan and d p williamson siam j comput 30 2001 pp 1863 1920 we continue this line of research by considering domains containing more than two elements we present two main results a complete classification for the approximability of all maximal constraint languages over domains of cardinality at most 4 and a complete classification of the approximability of the problem when the set of allowed constraints contains all permutation constraints under the assumption that a conjecture due to szczepara minimal clones generated by groupoids ph d thesis universite de montreal montreal qc 1996 holds we give a complete classification for all maximal constraint languages these classes of languages are well studied in universal algebra and computer science they have for instance been considered in connection with machine learning and constraint satisfaction our results are proved by using algebraic results from clone theory and the results indicate that this approach is very powerful for classifying the approximability of certain optimization problems

the complexity and approximability of finding maximum feasible subsystems of linear relations we study the combinatorial problem which consists given a system of linear relations of finding a maximum feasible subsystem that is a solution satisfying as many relations as possible the computational complexity of this general problem named max fls is investigated for the four types of relations and various constrained versions of max fls where a subset of relations must be satisfied or where the variables take bounded discrete values are also considered we establish the complexity of solving these problems optimally and whenever they are intractable we determine their degree of approximability max fls with or relations is np hard even when restricted to homogeneous systems with bipolar coefficients whereas it can be solved in polynomial time for relations with real coefficients the various np hard versions of max fls belong to different approximability classes depending on the type of relations and the additional constraints we show that the range of approximability stretches from apx complete problems which can be approximated within a constant but not within every constant unless p np to npo pb complete ones that are as hard to approximate as all np optimization problems with polynomially bounded objective functions while max fls with equations and integer coefficients cannot be approximated within pe for some e 0 where p is the number of relations the same problem over gf q for a prime q can be approximated within q but not within qe for some e 0 max fls with strict or nonstrict inequalities can be approximated within 2 but not within every constant factor our results also provide strong bounds on the approximability of two variants of max fls with and relations that arise when training perceptrons which are the building blocks of artificial neural networks and when designing linear classifiers

the computational structure of monotone monadic snp and constraint satisfaction a study through datalog and group theory this paper starts with the project of finding a large subclass of np which exhibits a dichotomy the approach is to find this subclass via syntactic prescriptions while the paper does not achieve this goal it does isolate a class of problems specified by monotone monadic snp without inequality which may exhibit this dichotomy we justify the placing of all these restrictions by showing essentially using ladner s theorem that classes obtained by using only two of the above three restrictions do not show this dichotomy we then explore the structure of this class we show that all problems in this class reduce to the seemingly simpler class csp we divide csp into subclasses and try to unify the collection of all known polytime algorithms for csp problems and extract properties that make csp problems np hard this is where the second part of the title a study through datalog and group theory comes in we present conjectures about this class which would end in showing the dichotomy

the stiffness of an elastic solid with an embedded nominally spherical inclusion subjected to a small arbitrary motion the article examines the problem of translation and rotation of a nominally slightly deformed spherical rigid inclusion embedded into an unbounded elastic medium to the first order in the small parameter characterizing the boundary perturbation explicit expressions are deduced for the induced displacement field as well as for the net force and net torque required to produce the applied translation and rotation

a note on magnetohydrodynamic flow of a power law fluid over a stretching sheet this paper presents a numerical study of the flow of an electrically conducting power law fluid in the presence of a uniform transverse magnetic field this flow is governed by the non linear differential equationn f n 1 f f 2 2nn 1ff mf 0where a prime denotes differentiation with respect to the similarity variable h and n m are respectively the power law index and the magnetic parameter in this work the numerical solutions are obtained using a runge kutta algorithm for high order initial value problems

on the homotopy analysis method for nonlinear problems a powerful easy to use analytic tool for nonlinear problems in general namely the homotopy analysis method is further improved and systematically described through a typical nonlinear problem i e the algebraically decaying viscous boundary layer flow due to a moving sheet two rules the rule of solution expression and the rule of coefficient ergodicity are proposed which play important roles in the frame of the homotopy analysis method and simplify its applications in science and engineering an explicit analytic solution is given for the first time with recursive formulas for coefficients this analytic solution agrees well with numerical results and can be regarded as a definition of the solution of the considered nonlinear problem

the approximability of max csp with fixed value constraints in the maximum constraint satisfaction problem max csp one is given a finite collection of possibly weighted constraints on overlapping sets of variables and the goal is to assign values from a given finite domain to the variables so as to maximize the number or the total weight for the weighted case of satisfied constraints this problem is np hard in general and therefore it is natural to study how restricting the allowed types of constraints affects the approximability of the problem in this article we show that any max csp problem with a finite set of allowed constraint types which includes all fixed value constraints i e constraints of the form x e a is either solvable exactly in polynomial time or else is apx complete even if the number of occurrences of variables in instances is bounded moreover we present a simple description of all polynomial time solvable cases of our problem this description relies on the well known algebraic combinatorial property of supermodularity

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

a rotated rigid ellipsoidal inclusion in an elastic medium a rigid ellipsoidal inclusion is embedded at arbitrary orientation in a homogeneous arbitrarily anisotropic elastic matrix and is translated infinitesimally by an externally imposed force we find directly the relation between the force and translation vectors and the stress strain and rotation concentrations over the ellipsoidal surface without having to solve the equations of equilibrium in the matrix or the fundamental ones of a point force we refer particularly then to a spheroid aligned along the axis of symmetry of a transversely isotropic matrix and subsequently to the full elastic field of a general ellipsoid in an isotropic matrix

a new technique of using homotopy analysis method for second order nonlinear differential equations abstract in this paper a new technique of homotopy analysis method nham is proposed for solving second order nonlinear differential equations this method improves the convergence of the series solution eliminates the unneeded terms and reduces time consuming in the standard homotopy analysis method ham the proposed provides an approximate solution by rewriting the second order nonlinear differential equation in the form of two first order differential equations the solution of these two differential equations is obtained as a power series solution this scheme is tested on four non linear exactly solvable differential equations three of the examples are initial value problems and the fourth is boundary value problem the results demonstrate reliability and efficiency of the algorithm developed