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How to Round Any CSP
"... A large number of interesting combinatorial optimization ..."
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A large number of interesting combinatorial optimization
Every 2CSP Allows Nontrivial Approximation
"... We use semidefinite programming to prove that any constraint satisfaction problem in two variables over any domain allows an efficient approximation algorithm that does better than picking a random assignment. Specifically we consider the case when each variable can take values in [d] and that each ..."
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We use semidefinite programming to prove that any constraint satisfaction problem in two variables over any domain allows an efficient approximation algorithm that does better than picking a random assignment. Specifically we consider the case when each variable can take values in [d] and that each constraint rejects t out of the d2 possible input pairs. Then, for some universal constant c, wecan,in probabilistic polynomial time, find an assignment whose objective value is, in expectation, within a factor 1 − t d2 + ct d4 log d of optimal, improving on the trivial bound of 1 − t/d².
Every Permutation CSP of Arity 3 is Approximation Restitant
 In 24th IEEE CCC
, 2009
"... Abstract—A permutation constraint satisfaction problem (permCSP) of arity k is specified by a subset Λ ⊆ Sk of permutations on {1, 2,..., k}. An instance of such a permCSP consists of a set of variables V and a collection of constraints each of which is an ordered ktuple of V. The objective is to f ..."
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Abstract—A permutation constraint satisfaction problem (permCSP) of arity k is specified by a subset Λ ⊆ Sk of permutations on {1, 2,..., k}. An instance of such a permCSP consists of a set of variables V and a collection of constraints each of which is an ordered ktuple of V. The objective is to find a global ordering σ of the variables that maximizes the number of constraint tuples whose ordering (under σ) follows a permutation in Λ. This is just the natural extension of constraint satisfaction problems over finite domains (such as Boolean CSPs) to the world of ordering problems. The simplest permCSP corresponds to the case when Λ consists of the identity permutation on two variables. This is just the Maximum Acyclic Subgraph (MAS) problem. It was recently shown that the MAS problem is UniqueGames hard to approximate within a factor better than the trivial 1/2 achieved by a random ordering [6]. Building on this work, in this paper we show that for every permCSP of arity 3, beating the random ordering is UniqueGames hard. The result is in fact stronger: we show that for every Λ ⊆ Π ⊆ S3, given an instance of permCSP(Λ) that is almostsatisfiable, it is hard to find an ordering that satisfies more than Π 6 + ε of the constraints even under the relaxed constraint Π (for arbitrary ε> 0). A special case of our result is that the Betweenness problem is hard to approximate beyond a factor 1/3. Interestingly, for satisfiable instances of Betweenness, a factor 1/2 approximation algorithm is known. Thus, every permutation CSP of arity up to 3 resists approximation beyond the trivial random ordering threshold. In contrast, for Boolean CSPs, there are both approximation resistant and nontrivially approximable CSPs of arity 3. Keywordshardness of approximation; betweenness; permutation constraint satisfaction problems; approximation resistance I.
On the Approximation Resistance of a Random Predicate
"... A predicate is approximation resistant if no probabilistic polynomial time approximation algorithm can do significantly better then the naive algorithm that picks an assignment uniformly at random. Assuming that the Unique Games Conjecture is true we prove that most Boolean predicates are approxima ..."
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A predicate is approximation resistant if no probabilistic polynomial time approximation algorithm can do significantly better then the naive algorithm that picks an assignment uniformly at random. Assuming that the Unique Games Conjecture is true we prove that most Boolean predicates are approximation resistant.
BEATING THE RANDOM ORDERING IS HARD: EVERY ORDERING CSP IS APPROXIMATION RESISTANT
, 2011
"... We prove that, assuming the Unique Games conjecture (UGC), every problem in the class of ordering constraint satisfaction problems (OCSPs) where each constraint has constant arity is approximation resistant. In other words, we show that if ρ is the expected fraction of constraints satisfied by a ra ..."
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We prove that, assuming the Unique Games conjecture (UGC), every problem in the class of ordering constraint satisfaction problems (OCSPs) where each constraint has constant arity is approximation resistant. In other words, we show that if ρ is the expected fraction of constraints satisfied by a random ordering, then obtaining a ρ ′ approximation for any ρ ′>ρis UGhard. For the simplest OCSP, the Maximum Acyclic Subgraph (MAS) problem, this implies that obtaining a ρapproximation for any constant ρ>1/2 is UGhard. Specifically, for every constant ε>0the following holds: given a directed graph G that has an acyclic subgraph consisting of a fraction (1−ε) of its edges, it is UGhard to find one with more than (1/2 +ε) of its edges. Note that it is trivial to find an acyclic subgraph with 1/2 the edges by taking either the forward or backward edges in an arbitrary ordering of the vertices of G. The MAS problem has been well studied, and beating the random ordering for MAS has been a basic open problem. An OCSP of arity k is specified by a subset Π ⊆ Sk of permutations on {1, 2,...,k}. An instance of such an OCSP is a set V and a collection of constraints, each of which is an ordered ktuple of V. The objective is to find a global linear ordering of V while maximizing the number of constraints ordered as in Π. A random ordering of V is expected to satisfy a ρ = Π
On the Advantage over Random for Maximum Acyclic Subgraph
"... In this paper we present a new approximation algorithm for the MAX ACYCLIC SUBGRAPH problem. Given an instance where the maximum acyclic subgraph contains 1/2+δ fraction of all edges, our algorithm finds an acyclic subgraph with 1/2 + Ω(δ / logn) fraction of all edges. 1 ..."
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In this paper we present a new approximation algorithm for the MAX ACYCLIC SUBGRAPH problem. Given an instance where the maximum acyclic subgraph contains 1/2+δ fraction of all edges, our algorithm finds an acyclic subgraph with 1/2 + Ω(δ / logn) fraction of all edges. 1
Fast SDP Algorithms for Constraint Satisfaction Problems
"... The class of constraint satisfactions problems (CSPs) captures many fundamental combinatorial optimization problems such as Max Cut, Max qCut, Unique Games, and Max kSat. Recently, Raghavendra (STOC‘08) identified a simple semidefinite programming relaxation that gives the best possible approximat ..."
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Cited by 12 (4 self)
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The class of constraint satisfactions problems (CSPs) captures many fundamental combinatorial optimization problems such as Max Cut, Max qCut, Unique Games, and Max kSat. Recently, Raghavendra (STOC‘08) identified a simple semidefinite programming relaxation that gives the best possible approximation for any CSP, assuming the Unique Games Conjecture. Raghavendra and Steurer (FOCS‘09) showed that, independent of the truth of the Unique Games Conjecture, the integrality gap of this relaxation cannot be improved even by adding a large class of valid inequalities. We present an algorithm that finds an approximately optimal solution to this relaxation in nearlinear time. Combining this algorithm with a rounding scheme of Raghavendra and Steurer (FOCS‘09) leads to an approximation algorithm for any CSP that runs in nearlinear time and has an approximation guarantee that matches the integrality gap, which is optimal assuming the Unique Games Conjecture.
On the Usefulness of Predicates
, 2012
"... Motivated by the pervasiveness of strong inapproximability results for MaxCSPs, we introduce a relaxed notion of an approximate solution of a MaxCSP. In this relaxed version, loosely speaking, the algorithm is allowed to replace the constraints of an instance by some other (possibly realvalued) c ..."
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Motivated by the pervasiveness of strong inapproximability results for MaxCSPs, we introduce a relaxed notion of an approximate solution of a MaxCSP. In this relaxed version, loosely speaking, the algorithm is allowed to replace the constraints of an instance by some other (possibly realvalued) constraints, and then only needs to satisfy as many of the new constraints as possible. To be more precise, we introduce the following notion of a predicate P being useful for a (realvalued) objective Q: given an almost satisfiable MaxP instance, there is an algorithm that beats a random assignment on the corresponding MaxQ instance applied to the same sets of literals. The standard notion of a nontrivial approximation algorithm for a MaxCSP with predicate P is exactly the same as saying that P is useful for P itself. We say that P is useless if it is not useful for any Q. This turns out to be equivalent to the following pseudorandomness property: given an almost satisfiable instance of MaxP it is hard to find an assignment such that the induced distribution on kbit strings defined by the instance is not essentially uniform. Under the Unique Games Conjecture, we give a complete and simple characterization of useful MaxCSPs defined by a predicate: such a MaxCSP is useless if and only if there is a pairwise independent distribution supported on the satisfying assignments of the predicate. It is natural to also consider the case when no negations are allowed in the CSP instance, and we derive a similar complete characterization (under the UGC) there as well. Finally, we also include some results and examples shedding additional light on the approximability of certain MaxCSPs.
Approximating Linear Threshold Predicates
"... We study constraint satisfaction problems on the domain {−1, 1}, where the given constraints are homogeneous linear threshold predicates. That is, predicates of the form sgn(w1x1 + · · · + wnxn) for some positive integer weights w1,..., wn. Despite their simplicity, current techniques fall short ..."
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We study constraint satisfaction problems on the domain {−1, 1}, where the given constraints are homogeneous linear threshold predicates. That is, predicates of the form sgn(w1x1 + · · · + wnxn) for some positive integer weights w1,..., wn. Despite their simplicity, current techniques fall short of providing a classification of these predicates in terms of approximability. In fact, it is not easy to guess whether there exists a homogeneous linear threshold predicate that is approximation resistant or not. The focus of this paper is to identify and study the approximation curve of a class of threshold predicates that allow for nontrivial approximation. Arguably the simplest such predicate is the majority predicate sgn(x1 + · · · + xn), for which we obtain an almost complete understanding of the asymptotic approximation curve, assuming the Unique Games Conjecture. Our techniques extend to a more general class of “majoritylike” predicates and we obtain parallel results for them. In order to classify these predicates, we introduce the notion of Chowrobustness that might be of independent interest.