We give the first combinatorial approximation algorithm for Maxcut that beats the trivial 0.5 factor by a constant. The main partitioning procedure is very intuitive, natural, and easily described. It essentially performs a number of random walks and aggregates the information to provide the partition. We can control the running time to get an approximation factor-running time tradeoff. We show that for any constant b > 1.5, there is an O(n^ b ) algorithm that outputs a (0.5+delta)-approximation for Maxcut, where delta = delta(b) is some positive constant. One of the components of our algorithm is a weak local graph partitioning procedure that may be of independent interest. Given a starting vertex @math and a conductance parameter phi, unless a random walk of length ell = O(log n) starting from i mixes rapidly (in terms of phi and ell), we can find a cut of conductance at most phi close to the vertex. The work done per vertex found in the cut is sublinear in n.
Trevisan @cite_12 also uses random walks to give approximation algorithms for (as a special case of unique games), although the algorithm only deals with the case when is @math . The property tester for bipartiteness in sparse graphs by Goldreich and Ron @cite_19 is a sublinear time procedure that uses random walks to distinguish graphs where @math from @math . The algorithm, however, does not actually give an approximation to . There is a similarity in flavor to Dinur's proof of the PCP theorem @cite_13 , which uses random walks and majority votes for gap amplification of CSPs. Our algorithm might be seen as some kind of belief propagation, where messages about labels are passed around. For the special case of cubic and maximum degree @math graphs, there has been a study of combinatorial algorithms for @cite_3 @cite_18 @cite_22 . These are based on graph theoretic properites and very different from our algorithms. Combinatorial algorithms for CSP (constraint satisfaction problems) based on LP relaxations have been studied in @cite_10 .
@cite_18 @cite_22 @cite_10 @cite_3 @cite_19 @cite_13 @cite_12
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Let @math be a nontrivial @math -ary predicate. Consider a random instance of the constraint satisfaction problem @math on @math variables with @math constraints, each being @math applied to @math randomly chosen literals. Provided the constraint density satisfies @math , such an instance is unsatisfiable with high probability. The problem is to efficiently find a proof of unsatisfiability. We show that whenever the predicate @math supports a @math - probability distribution on its satisfying assignments, the sum of squares (SOS) algorithm of degree @math (which runs in time @math ) refute a random instance of @math . In particular, the polynomial-time SOS algorithm requires @math constraints to refute random instances of CSP @math when @math supports a @math -wise uniform distribution on its satisfying assignments. Together with recent work of [LRS15], our result also implies that polynomial-size semidefinite programming relaxation for refutation requires at least @math constraints. Our results (which also extend with no change to CSPs over larger alphabets) subsume all previously known lower bounds for semialgebraic refutation of random CSPs. For every constraint predicate @math , they give a three-way hardness tradeoff between the density of constraints, the SOS degree (hence running time), and the strength of the refutation. By recent algorithmic results of [AOW15] and [RRS16], this full three-way tradeoff is , up to lower-order factors. Random constraint satisfaction problems (CSPs) are known to exhibit threshold phenomena: given a uniformly random instance of a CSP with @math variables and @math clauses, there is a value of @math beyond which the CSP will be unsatisfiable with high probability. Strong refutation is the problem of certifying that no variable assignment satisfies more than a constant fraction of clauses; this is the natural algorithmic problem in the unsatisfiable regime (when @math ). Intuitively, strong refutation should become easier as the clause density @math grows, because the contradictions introduced by the random clauses become more locally apparent. For CSPs such as @math -SAT and @math -XOR, there is a long-standing gap between the clause density at which efficient strong refutation algorithms are known, @math , and the clause density at which instances become unsatisfiable with high probability, @math . In this paper, we give spectral and sum-of-squares algorithms for strongly refuting random @math -XOR instances with clause density @math in time @math or in @math rounds of the sum-of-squares hierarchy, for any @math and any integer @math . Our algorithms provide a smooth transition between the clause density at which polynomial-time algorithms are known at @math , and brute-force refutation at the satisfiability threshold when @math . We also leverage our @math -XOR results to obtain strong refutation algorithms for SAT (or any other Boolean CSP) at similar clause densities. Our algorithms match the known sum-of-squares lower bounds due to Grigoriev and Schonebeck, up to logarithmic factors. Additionally, we extend our techniques to give new results for certifying upper bounds on the injective tensor norm of random tensors. par>We prove some non-approximability results for restrictions of basic combinatorial optimization problems to instances of bounded “degreeror bounded “width.” Specifically: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor @math , unless @math . H stad [18] proved that the problem is approximable to within a factor @math in polynomial time, and that is hard to approximate to within a factor @math . Our result uses a new randomized reduction from general instances of Max 3SAT to bounded-occurrences instances. The randomized reduction applies to other Max SNP problems as well. We observe that the Set Cover problem on instances where each set has size at most B is hard to approximate to within a factor @math unless @math . The result follows from an appropriate setting of parameters in Feige's reduction [11]. This is essentially tight in light of the existence of @math -approximate algorithms [20, 23, 9] We present a new PCP construction, based on applying parallel repetition to the inner verifier,'' and we provide a tight analysis for it. Using the new construction, and some modifications to known reductions from PCP to Hitting Set, we prove that Hitting Set with sets of size B is hard to approximate to within a factor @math . The problem can be approximated to within a factor B [19], and it is the Vertex Cover problem for B =2. The relationship between hardness of approximation and set size seems to have not been explored before. We observe that the Independent Set problem on graphs having degree at most B is hard to approximate to within a factor @math , unless P = NP . This follows from a comination of results by Clementi and Trevisan [28] and Reingold, Vadhan and Wigderson [27]. It had been observed that the problem is hard to approximate to within a factor @math unless P = NP [1]. An algorithm achieving factor @math is also known [21, 2, 30, 16 . Assuming that NP @math @math BPTIME( @math ), we show that graph min-bisection, dense @math -subgraph, and bipartite clique have no polynomial time approximation scheme (PTAS). We give a reduction from the minimum distance of code (MDC) problem. Starting with an instance of MDC, we build a quasi-random probabilistically checkable proof (PCP) that suffices to prove the desired inapproximability results. In a quasi-random PCP, the query pattern of the verifier looks random in a certain precise sense. Among the several new techniques we introduce, the most interesting one gives a way of certifying that a given polynomial belongs to a given linear subspace of polynomials. As is important for our purpose, the certificate itself happens to be another polynomial, and it can be checked probabilistically by reading a constant number of its values.