Roman Bacik, Sanjeev Mahajan

The graph homomorphism problem is a canonical $NP$-complete problem.

It generalizes various other well-studied problems such as graph

coloring and finding cliques. To get a better insight into a

combinatorial problem, one often studies relaxations of the problem.

We define fractional homomorphisms and pseudo-homomorphisms ...
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Uriel Feige, Marek Karpinski, Michael Langberg

We design a $0.795$ approximation algorithm for the Max-Bisection problem

restricted to regular graphs. In the case of three regular graphs our

results imply an approximation ratio of $0.834$.

Amin Coja-Oghlan, Andreas Goerdt, André Lanka, Frank Schädlich

Abstract. It is known that random k-SAT formulas with at least

(2^k*ln2)*n random clauses are unsatisfiable with high probability. This

result is simply obtained by bounding the expected number of satisfy-

ing assignments of a random k-SAT instance by an expression tending

to 0 when n, the number of variables ...
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Moses Charikar, Konstantin Makarychev, Yury Makarychev

We present a c.k/2^k approximation algorithm for the Max k-CSP problem (where c > 0.44 is an absolute constant). This result improves the previously best known algorithm by Hast, which has an approximation guarantee of Omega(k/(2^k log k)). Our approximation guarantee matches the upper bound of Samorodnitsky and Trevisan up ... more >>>

Moses Charikar, Konstantin Makarychev, Yury Makarychev

In this note we present an approximation algorithm for MAX 2SAT that given a (1 - epsilon) satisfiable instance finds an assignment of variables satisfying a 1 - O(sqrt{epsilon}) fraction of all constraints. This result is optimal assuming the Unique Games Conjecture.

The best previously known result, due ... more >>>

Grant Schoenebeck, Luca Trevisan, Madhur Tulsiani

We study semidefinite programming relaxations of Vertex Cover arising from

repeated applications of the LS+ ``lift-and-project'' method of Lovasz and

Schrijver starting from the standard linear programming relaxation.

Goemans and Kleinberg prove that after one round of LS+ the integrality

gap remains arbitrarily close to 2. Charikar proves an integrality ...
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Konstantinos Georgiou, Avner Magen, Iannis Tourlakis

We prove that the integrality gap after tightening the standard LP relaxation for Vertex Cover with Omega(sqrt(log n/log log n)) rounds of the SDP LS+ system is 2-o(1).

more >>>Venkatesan Guruswami, Prasad Raghavendra

We study the approximability of the \maxcsp problem over non-boolean domains, more specifically over $\{0,1,\ldots,q-1\}$ for some integer $q$. We obtain a approximation algorithm that achieves a ratio of $C(q) \cdot k/q^k$ for some constant $C(q)$ depending only on $q$. Further, we extend the techniques of Samorodnitsky and Trevisan to ... more >>>

Madhur Tulsiani

We study integrality gaps for SDP relaxations of constraint satisfaction problems, in the hierarchy of SDPs defined by Lasserre. Schoenebeck recently showed the first integrality gaps for these

problems, showing that for MAX k-XOR, the ratio of the SDP optimum to the integer optimum may be as large as ...
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Boaz Barak, Moritz Hardt, Thomas Holenstein, David Steurer

We study the question of whether the value of mathematical programs such as

linear and semidefinite programming hierarchies on a graph $G$, is preserved

when taking a small random subgraph $G'$ of $G$. We show that the value of the

Goemans-Williamson (1995) semidefinite program (SDP) for \maxcut of $G'$ is

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Siavosh Benabbas, Konstantinos Georgiou, Avner Magen

We study the performance of the Sherali-Adams system for VERTEX COVER on graphs with vector

chromatic number $2+\epsilon$. We are able to construct solutions for LPs derived by any number of Sherali-Adams tightenings by introducing a new tool to establish Local-Global Discrepancy. When restricted to

$O(1/ \epsilon)$ tightenings we show ...
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Venkatesan Guruswami, Prasad Raghavendra, Rishi Saket, Yi Wu

The Unique Games conjecture (UGC) has emerged in recent years as the starting point for several optimal inapproximability results. While for none of these results a reverse reduction to Unique Games is known, the assumption of bijective projections in the Label Cover instance seems critical in these proofs. In this ... more >>>

Venkatesan Guruswami, Ali Kemal Sinop

We present an approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semidefinite objective functions and global linear constraints. This framework includes well known graph problems such as Minimum graph bisection, Edge expansion, Uniform sparsest cut, and Small Set expansion, as well as the Unique Games problem. These ... more >>>

Venkatesan Guruswami, Ali Kemal Sinop

Convex relaxations based on different hierarchies of

linear/semi-definite programs have been used recently to devise

approximation algorithms for various optimization problems. The

approximation guarantee of these algorithms improves with the number

of {\em rounds} $r$ in the hierarchy, though the complexity of solving

(or even writing down the solution for) ...
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Boaz Barak, Jonathan Kelner, David Steurer

We present a general approach to rounding semidefinite programming relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our approach is based on using the connection between these relaxations and the Sum-of-Squares proof system to transform a *combining algorithm* -- an algorithm that maps a distribution over solutions into a (possibly ... more >>>

Boaz Barak, David Steurer

In order to obtain the best-known guarantees, algorithms are traditionally tailored to the particular problem we want to solve. Two recent developments, the Unique Games Conjecture (UGC) and the Sum-of-Squares (SOS) method, surprisingly suggest that this tailoring is not necessary and that a single efficient algorithm could achieve best possible ... more >>>

Massimo Lauria, Jakob Nordström

We exhibit families of 4-CNF formulas over n variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) d but require SOS proofs of size n^{Omega(d)} for values of d = d(n) from constant all the way up to n^{delta} for some universal constant delta. This shows that ... more >>>

Boaz Barak, Samuel Hopkins, Jonathan Kelner, Pravesh Kothari, Ankur Moitra, Aaron Potechin

We prove that with high probability over the choice of a random graph $G$ from the Erd\H{o}s-R\'enyi distribution $G(n,1/2)$, the $n^{O(d)}$-time degree $d$ Sum-of-Squares semidefinite programming relaxation for the clique problem will give a value of at least $n^{1/2-c(d/\log n)^{1/2}}$ for some constant $c>0$.

This yields a nearly tight ...
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Ryan O'Donnell

Suppose we want to minimize a polynomial $p(x) = p(x_1, \dots, x_n)$, subject to some polynomial constraints $q_1(x), \dots, q_m(x) \geq 0$, using the Sum-of-Squares (SOS) SDP hierarachy. Assume we are in the "explicitly bounded" ("Archimedean") case where the constraints include $x_i^2 \leq 1$ for all $1 \leq i \leq ... more >>>

Boaz Barak, Pravesh Kothari, David Steurer

For every constant $\epsilon>0$, we give an $\exp(\tilde{O}(\sqrt{n}))$-time algorithm for the $1$ vs $1-\epsilon$ Best Separable State (BSS) problem of distinguishing, given an $n^2\times n^2$ matrix $M$ corresponding to a quantum measurement, between the case that there is a separable (i.e., non-entangled) state $\rho$ that $M$ accepts with probability $1$, ... more >>>