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Towards Sharp Inapproximability For Any 2CSP
"... We continue the recent line of work on the connection between semidefinite programmingbased approximation algorithms and the Unique Games Conjecture. Given any boolean 2CSP (or more generally, any nonnegative objective function on two boolean variables), we show how to reduce the search for a good ..."
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We continue the recent line of work on the connection between semidefinite programmingbased approximation algorithms and the Unique Games Conjecture. Given any boolean 2CSP (or more generally, any nonnegative objective function on two boolean variables), we show how to reduce the search for a good inapproximability result to a certain numeric minimization problem. The key objects in our analysis are the vector triples arising when doing clausebyclause analysis of algorithms based on semidefinite programming. Given a weighted set of such triples of a certain restricted type, which are “hard ” to round in a certain sense, we obtain a Unique Gamesbased inapproximability matching this “hardness ” of rounding the set of vector triples. Conversely, any instance together with an SDP solution can be viewed as a set of vector triples, and we show that we can always find an assignment to the instance which is at least as good as the “hardness ” of rounding the corresponding set of vector triples. We conjecture that the restricted type required for the hardness result is in fact no restriction, which would imply that these upper and lower bounds match exactly. This conjecture is supported by all existing results for specific 2CSPs. As an application, we show that MAX 2AND is hard to approximate within 0.87435. This improves upon the best previous hardness of αGW + ɛ ≈ 0.87856, and comes very close to matching the approximation ratio of the best algorithm known, 0.87401. It also establishes that balanced instances of MAX 2AND, i.e., instances in which each variable occurs positively and negatively equally often, are not the hardest to approximate, as these can be approximated within a factor αGW.
How to Round Any CSP
"... A large number of interesting combinatorial optimization ..."
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A large number of interesting combinatorial optimization
Improved approximation guarantees through higher levels of SDP hierarchies
 In Proceedings of the 11th International Workshop, APPROX
, 2008
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Convex Relaxations and Integrality Gaps
"... We discuss the effectiveness of linear and semidefinite relaxations in approximating the optimum for combinatorial optimization problems. Various hierarchies of these relaxations, such as the ones defined by Lovász and Schrijver [47], Sherali and Adams [55] and Lasserre [42] generate increasingly st ..."
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We discuss the effectiveness of linear and semidefinite relaxations in approximating the optimum for combinatorial optimization problems. Various hierarchies of these relaxations, such as the ones defined by Lovász and Schrijver [47], Sherali and Adams [55] and Lasserre [42] generate increasingly strong linear and semidefinite programming relaxations starting from a basic one. We survey some positive applications of these hierarchies, where their use yields improved approximation algorithms. We also discuss known lower bounds on the integrality gaps of relaxations arising from these hierarchies, demonstrating limits on the applicability of such hierarchies for certain optimization problems.
On the unique games conjecture
 In FOCS
, 2005
"... This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1 ..."
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Cited by 15 (1 self)
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This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1
New Tools for Graph Coloring
"... How to color 3 colorable graphs with few colors is a problem of longstanding interest. The best polynomialtime algorithm uses n0.2072 colors. There are no indications that coloring using say O(log n) colors is hard. It has been suggested that lift and project based SDP relaxations could be used to ..."
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How to color 3 colorable graphs with few colors is a problem of longstanding interest. The best polynomialtime algorithm uses n0.2072 colors. There are no indications that coloring using say O(log n) colors is hard. It has been suggested that lift and project based SDP relaxations could be used to design algorithms that use nɛ colors for arbitrarily small ɛ> 0. We explore this possibility in this paper and find some cause for optimism. While the case of general graphs is till open, we can analyse the Lasserre relaxation for two interesting families of graphs. For graphs with low threshold rank (a class of graphs identified in the recent paper of Arora, Barak and Steurer on the unique games problem), Lasserre relaxations can be used to find an independent set of size Ω(n) (i.e., progress towards a coloring with O(log n) colors) in nO(D) time, where D is the threshold rank of the graph. This algorithm is inspired by recent work of Barak, Raghavendra, and Steurer on using Lasserre Hierarchy for unique games. The algorithm can also be used to show that known integrality gap instances for SDP relaxations like strict vector chromatic number cannot survive a few rounds of Lasserre lifting, which also seems reason for optimism. For distance transitive graphs of diameter ∆, we can show how to color them using O(log n) colors in n2O(∆) time. This family is interesting because the family of graphs of diameter O(1/ɛ) is easily seen to be complete for coloring with nɛ colors. The distancetransitive property implies that the graph “looks” the same in all neighborhoods.
Linear Index Coding via Semidefinite Programming
"... In the index coding problem, introduced by Birk and Kol (INFOCOM, 1998), the goal is to broadcast an n bit word to n receivers (one bit per receiver), where the receivers have side information represented by a graph G. The objective is to minimize the length of a codeword sent to all receivers which ..."
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In the index coding problem, introduced by Birk and Kol (INFOCOM, 1998), the goal is to broadcast an n bit word to n receivers (one bit per receiver), where the receivers have side information represented by a graph G. The objective is to minimize the length of a codeword sent to all receivers which allows each receiver to learn its bit. For linear index coding, the minimum possible length is known to be equal to a graph parameter called minrank (BarYossef et al., FOCS, 2006). We show a polynomial time algorithm that, given an n vertex graph G with minrank k, finds a linear index code for G of length Õ(n f (k)), where f (k) depends only on k. For example, for k = 3 we obtain f (3) ≈ 0.2574. Our algorithm employs a semidefinite program (SDP) introduced by Karger, Motwani and Sudan (J. ACM, 1998) for graph coloring and its refined analysis due to Arora, Chlamtac and Charikar (STOC, 2006). Since the SDP we use is not a relaxation of the minimization problem we consider, a crucial component of our analysis is an upper bound on the objective value of the SDP in terms of the minrank. At the heart of our analysis lies a combinatorial result which may be of independent interest. Namely, we show an exact expression for the maximum possible value of the Lovász ϑfunction of a graph with minrank k. This yields a tight gap between two classical upper bounds on the Shannon capacity of a graph.
Hypergraph list coloring and Euclidean Ramsey Theory
, 2010
"... A hypergraph is simple if it has no two edges sharing more than a single vertex. It is slist colorable (or schoosable) if for any assignment of a list of s colors to each of its vertices, there is a vertex coloring assigning to each vertex a color from its list, so that no edge is monochromatic. W ..."
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A hypergraph is simple if it has no two edges sharing more than a single vertex. It is slist colorable (or schoosable) if for any assignment of a list of s colors to each of its vertices, there is a vertex coloring assigning to each vertex a color from its list, so that no edge is monochromatic. We prove that for every positive integer r, there is a function dr(s) such that no runiform simple hypergraph with average degree at least dr(s) is slistcolorable. This extends a similar result for graphs, due to the first author, but does not give as good estimates of dr(s) as are known for d2(s), since our proof only shows that for each fixed r ≥ 2, dr(s) ≤ 2 crsr−1. We use the result to prove that for any finite set of points X in the plane, and for any finite integer s, one can assign a list of s distinct colors to each point of the plane, so that any coloring of the plane that colors each point by a color from its list contains a monochromatic isometric copy of X.
Distributed Coloring Depending on the Chromatic Number or the Neighborhood Growth
"... Abstract 1 We deterministically compute a ∆+1 coloring in time O(∆5c+2·(∆5) 2/c /(∆1) ɛ + (∆1) ɛ + log ∗ n) and O(∆5c+2 · (∆5) 1/c / ∆ ɛ + ∆ ɛ + (∆5) d log ∆5 log n) for arbitrary constants d, ɛ and arbitrary constant integer c, where ∆i is defined as the maximal number of nodes within distance i fo ..."
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Abstract 1 We deterministically compute a ∆+1 coloring in time O(∆5c+2·(∆5) 2/c /(∆1) ɛ + (∆1) ɛ + log ∗ n) and O(∆5c+2 · (∆5) 1/c / ∆ ɛ + ∆ ɛ + (∆5) d log ∆5 log n) for arbitrary constants d, ɛ and arbitrary constant integer c, where ∆i is defined as the maximal number of nodes within distance i for a node and ∆: = ∆1. Our greedy algorithm improves the stateoftheart ∆+1 coloring algorithms for a large class of graphs, e.g. graphs of moderate neighborhood growth. We also state and analyze a randomized coloring algorithm in terms of the chromatic number, the run time and the used colors. If ∆ ∈ Ω(log 1+1 / log ∗ n n) and χ ∈ O(∆ / log 1+1 / log ∗ n n) then our algorithm executes in time O(log χ + log ∗ n) with high probability. For graphs of polylogarithmic chromatic number the analysis reveals an exponential gap compared to the fastest ∆ + 1 coloring algorithm running in time O(log ∆ + √ log n). The algorithm works without knowledge of χ and uses less than ∆ colors, i.e., (1 − 1/O(χ)) ∆ with high probability. To the best of our knowledge this is the first distributed algorithm for (such) general graphs taking the chromatic number χ into account. 1