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12
Semidefinite Programming and Integer Programming
"... We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems. ..."
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Cited by 47 (7 self)
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We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems.
Conditional hardness for approximate coloring
 In STOC 2006
, 2006
"... We study the APPROXIMATECOLORING(q, Q) problem: Given a graph G, decide whether χ(G) ≤ q or χ(G) ≥ Q (where χ(G) is the chromatic number of G). We derive conditional hardness for this problem for any constant 3 ≤ q < Q. For q ≥ 4, our result is based on Khot’s 2to1 conjecture [Khot’02]. For q = ..."
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Cited by 37 (12 self)
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We study the APPROXIMATECOLORING(q, Q) problem: Given a graph G, decide whether χ(G) ≤ q or χ(G) ≥ Q (where χ(G) is the chromatic number of G). We derive conditional hardness for this problem for any constant 3 ≤ q < Q. For q ≥ 4, our result is based on Khot’s 2to1 conjecture [Khot’02]. For q = 3, we base our hardness result on a certain ‘⊲< shaped ’ variant of his conjecture. We also prove that the problem ALMOST3COLORINGε is hard for any constant ε> 0, assuming Khot’s Unique Games conjecture. This is the problem of deciding for a given graph, between the case where one can 3color all but a ε fraction of the vertices without monochromatic edges, and the case where the graph contains no independent set of relative size at least ε. Our result is based on bounding various generalized noisestability quantities using the invariance principle of Mossel et al [MOO’05].
CSP Gaps and Reductions in the Lasserre Hierarchy
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 104 (2008)
, 2008
"... We study integrality gaps for SDP relaxations of constraint satisfaction problems, in the hierarchy of SDPs defined by Lasserre. Schoenebeck [25] recently showed the first integrality gaps for these problems, showing that for MAX kXOR, the ratio of the SDP optimum to the integer optimum may be as l ..."
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Cited by 21 (5 self)
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We study integrality gaps for SDP relaxations of constraint satisfaction problems, in the hierarchy of SDPs defined by Lasserre. Schoenebeck [25] recently showed the first integrality gaps for these problems, showing that for MAX kXOR, the ratio of the SDP optimum to the integer optimum may be as large as 2 even after Ω(n) rounds of the Lasserre hierarchy. We show that for the general MAX kCSP problem over binary domain, the ratio of SDP optimum to the value achieved by the optimal assignment, can be as large as 2 k /2k − ɛ even after Ω(n) rounds of the Lasserre hierarchy. For alphabet size q which is a prime, we give a lower bound of q k /q(q − 1)k − ɛ for Ω(n) rounds. The method of proof also gives optimal integrality gaps for a predicate chosen at random. We also explore how to translate gaps for CSP into integrality gaps for other problems using reductions, and establish SDP gaps for Maximum Independent Set, Approximate Graph Coloring, Chromatic Number and Minimum Vertex Cover. For Independent Set and Chromatic Number, we show integrality gaps of n/2 O( √ log nlog log n) even after 2
Approximating Maximum Subgraphs Without Short Cycles
"... Abstract MaxgGirth Subgraph is the problem of finding a maximum edge subgraph of G with girthat least g. This problem is NPhard, and (to the best of our knowledge) was first addressedalgorithmically by Pevzner et al. [Genome Research 2004] in the context of computational biology where it is shown ..."
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Cited by 6 (0 self)
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Abstract MaxgGirth Subgraph is the problem of finding a maximum edge subgraph of G with girthat least g. This problem is NPhard, and (to the best of our knowledge) was first addressedalgorithmically by Pevzner et al. [Genome Research 2004] in the context of computational biology where it is shown to be central in the study Genomic Sequencing. Pevzner et al.propose heuristics for MaxgGirth Subgraph that do not guarantee bounded approximationratios. In this work we initiate the study of MaxgGirth Subgraph in the context of approximation algorithms.
Orthogonal Vector Coloring
, 2010
"... A vector coloring of a graph is an assignment of a vector to each vertex where the presence or absence of an edge between two vertices dictates the value of the inner product of the corresponding vectors. In this paper, we obtain results on orthogonal vector coloring, where adjacent vertices must be ..."
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Cited by 2 (1 self)
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A vector coloring of a graph is an assignment of a vector to each vertex where the presence or absence of an edge between two vertices dictates the value of the inner product of the corresponding vectors. In this paper, we obtain results on orthogonal vector coloring, where adjacent vertices must be assigned orthogonal vectors. We introduce two vector analogues of list coloring along with their chromatic numbers and characterize all graphs that have (vector) chromatic number two in each case.
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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Cited by 2 (0 self)
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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.
Graph Coloring (1994, . . .
"... An independent set in an undirected graph G = (V, E) is a set of vertices that induce a subgraph which does not contain any edges. The size of the maximum independent set in G is denoted by α(G). For an integer k, a kcoloring of G is a function σ: V → [1... k] which assigns colors to the vertices o ..."
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An independent set in an undirected graph G = (V, E) is a set of vertices that induce a subgraph which does not contain any edges. The size of the maximum independent set in G is denoted by α(G). For an integer k, a kcoloring of G is a function σ: V → [1... k] which assigns colors to the vertices of G. A valid kcoloring of G is a coloring in which each color class is an independent set. The chromatic number χ(G) of G is the smallest k for which there exists a valid kcoloring of G. Finding χ(G) is a fundamental NPhard problem. Hence, when limited to polynomial time algorithms, one turns to the question of estimating the value of χ(G) or to the closely related problem of approximate coloring. Problem 1 (Approximate coloring). Input: Undirected graph G = (V, E). Output: A valid coloring of G with r · χ(G) colors, for some approximation ratio r ≥ 1.
New approximation guarantee for . . .
, 2006
"... We describe how to color every 3colorable graph with O(n0.2111) colors, thus improving an algorithm of Blum and Karger from almost a decade ago. Our analysis uses new geometric ideas inspired by the recent work of Arora, Rao, and Vazirani on SPARSEST CUT, and these ideas show promise of leading to ..."
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We describe how to color every 3colorable graph with O(n0.2111) colors, thus improving an algorithm of Blum and Karger from almost a decade ago. Our analysis uses new geometric ideas inspired by the recent work of Arora, Rao, and Vazirani on SPARSEST CUT, and these ideas show promise of leading to further improvements.
Combinatorial covering of . . .
"... We consider the problem of coloring a 3colorable graph in polynomial time using as few colors as possible. We present a combinatorial algorithm getting down to Õ(n4/11) colors. This is the first combinatorial improvement of Blum’s Õ(n3/8) bound from FOCS’90. Like Blum’s algorithm, our new algorithm ..."
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We consider the problem of coloring a 3colorable graph in polynomial time using as few colors as possible. We present a combinatorial algorithm getting down to Õ(n4/11) colors. This is the first combinatorial improvement of Blum’s Õ(n3/8) bound from FOCS’90. Like Blum’s algorithm, our new algorithm composes nicely with recent semidefinite approaches. The current best bound is O(n0.2072) colors by Chlamtac from FOCS’07. We now bring it down toO(n0.2038) colors.