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15
SDP gaps and UGChardness for MaxCutGain
, 2008
"... Given a graph with maximum cut of (fractional) size c, the Goemans–Williamson semidefinite programming (SDP)based algorithm is guaranteed to find a cut of size at least.878 · c. However this guarantee becomes trivial when c is near 1/2, since making random cuts guarantees a cut of size 1/2 (i.e., ..."
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Cited by 25 (4 self)
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Given a graph with maximum cut of (fractional) size c, the Goemans–Williamson semidefinite programming (SDP)based algorithm is guaranteed to find a cut of size at least.878 · c. However this guarantee becomes trivial when c is near 1/2, since making random cuts guarantees a cut of size 1/2 (i.e., half of all edges). A few years ago, Charikar and Wirth (analyzing an algorithm of Feige and Langberg) showed that given a graph with maximum cut 1/2 + ε, one can find a cut of size 1/2 + Ω(ε/log(1/ε)). The main contribution of our paper is twofold: 1. We give a natural and explicit 1/2 + ε vs. 1/2 + O(ε/log(1/ε)) integrality gap for the MaxCut SDP based on Euclidean space with the Gaussian probability distribution. This shows that the SDProunding algorithm of CharikarWirth is essentially best possible. 2. We show how this SDP gap can be translated into a Long Code test with the same parameters. This implies that beating the CharikarWirth guarantee with any efficient algorithm is NPhard, assuming the Unique Games Conjecture (UGC). This result essentially settles the asymptotic approximability of MaxCut, assuming UGC. Building on the first contribution, we show how “randomness reduction ” on related SDP gaps for the QuadraticProgramming problem lets us make the Ω(log(1/ε)) gap as large as Ω(logn) for nvertex graphs. In addition to optimally answering an open question
The Grothendieck constant is strictly smaller than Krivine’s bound
 IN 52ND ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE. PREPRINT AVAILABLE AT HTTP://ARXIV.ORG/ABS/1103.6161
, 2011
"... The (real) Grothendieck constant KG is the infimum over those K ∈ (0, ∞) such that for every m, n ∈ N and every m × n real matrix (aij) we have m ∑ n∑ m ∑ n∑ aij〈xi, yj 〉 � K max aijεiδj. max {xi} m i=1,{yj}n j=1 ⊆Sn+m−1 i=1 j=1 {εi} m i=1,{δj}n j=1⊆{−1,1} i=1 j=1 2 log(1+ √ 2) The classical Groth ..."
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Cited by 17 (2 self)
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The (real) Grothendieck constant KG is the infimum over those K ∈ (0, ∞) such that for every m, n ∈ N and every m × n real matrix (aij) we have m ∑ n∑ m ∑ n∑ aij〈xi, yj 〉 � K max aijεiδj. max {xi} m i=1,{yj}n j=1 ⊆Sn+m−1 i=1 j=1 {εi} m i=1,{δj}n j=1⊆{−1,1} i=1 j=1 2 log(1+ √ 2) The classical Grothendieck inequality asserts the nonobvious fact that the above inequality does hold true for some K ∈ (0, ∞) that is independent of m, n and (aij). Since Grothendieck’s 1953 discovery of this powerful theorem, it has found numerous applications in a variety of areas, but despite attracting a lot of attention, the exact value of the Grothendieck constant KG remains a mystery. The last progress on this problem was in π 1977, when Krivine proved that KG � and conjectured that his bound is optimal. Krivine’s conjecture has been restated repeatedly since 1977, resulting in focusing the subsequent research on the search for examples of matrices (aij) which exhibit (asymptotically, as m, n → ∞) a lower bound on KG that matches Krivine’s bound. Here we obtain an improved Grothendieck inequality that holds for all matrices (aij) and yields a bound KG < π 2 log(1+ √ 2) − ε0 for some effective constant ε0> 0. Other than disproving Krivine’s conjecture, and along the way also disproving an intermediate conjecture of König that was made in 2000 as a step towards Krivine’s conjecture, our main contribution is conceptual: despite dealing with a binary rounding problem, random 2dimensional projections, when combined with a careful partition of R 2 in order to round the projected vectors to values in {−1, 1}, perform better than the ubiquitous random hyperplane technique. By establishing the usefulness of higher dimensional rounding schemes, this fact has consequences in approximation algorithms. Specifically, it yields the best known polynomial time approximation algorithm for the FriezeKannan Cut Norm problem, a generic and wellstudied optimization problem with many applications.
Grothendieck Inequalities for Semidefinite Programs with Rank Constraint
, 2010
"... Grothendieck inequalities are fundamental inequalities which are frequently used in many areas of mathematics and computer science. They can be interpreted as upper bounds for the integrality gap between two optimization problems: A difficult semidefinite program with rank1 constraint and its easy ..."
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Cited by 12 (3 self)
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Grothendieck inequalities are fundamental inequalities which are frequently used in many areas of mathematics and computer science. They can be interpreted as upper bounds for the integrality gap between two optimization problems: A difficult semidefinite program with rank1 constraint and its easy semidefinite relaxation where the rank constrained is dropped. For instance, the integrality gap of the GoemansWilliamson approximation algorithm for MAX CUT can be seen as a Grothendieck inequality. In this paper we consider Grothendieck inequalities for ranks greater than 1 and we give one application in statistical mechanics: Approximating ground states in the nvector model.
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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This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1
Grothendiecktype inequalities in combinatorial optimization
 COMM. PURE APPL. MATH
, 2011
"... We survey connections of the Grothendieck inequality and its variants to combinatorial optimization and computational complexity. ..."
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Cited by 9 (3 self)
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We survey connections of the Grothendieck inequality and its variants to combinatorial optimization and computational complexity.
SumofSquares Proofs and the Quest toward Optimal Algorithms
"... Abstract. In order to obtain the bestknown guarantees, algorithms are traditionally tailored to the particular problem we want to solve. Two recent developments, the Unique Games Conjecture (UGC) and the SumofSquares (SOS) method, surprisingly suggest that this tailoring is not necessary and that ..."
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Cited by 5 (0 self)
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Abstract. In order to obtain the bestknown guarantees, algorithms are traditionally tailored to the particular problem we want to solve. Two recent developments, the Unique Games Conjecture (UGC) and the SumofSquares (SOS) method, surprisingly suggest that this tailoring is not necessary and that a single efficient algorithm could achieve best possible guarantees for a wide range of different problems. The Unique Games Conjecture (UGC) is a tantalizing conjecture in computational complexity, which, if true, will shed light on the complexity of a great many problems. In particular this conjecture predicts that a single concrete algorithm provides optimal guarantees among all efficient algorithms for a large class of computational problems. The SumofSquares (SOS) method is a general approach for solving systems of polynomial constraints. This approach is studied in several scientific disciplines, including real algebraic geometry, proof complexity, control theory, and mathematical programming, and has found applications in fields as diverse as quantum information theory, formal verification, game theory and many others. We survey some connections that were recently uncovered between the Unique Games Conjecture and the SumofSquares method. In particular, we discuss new tools to rigorously bound the running time of the SOS method for obtaining approximate solutions to hard optimization problems, and how these tools give the potential for the sumofsquares method to provide new guarantees for many problems of interest, and possibly to even refute the UGC.
Approximability and Mathematical Relaxations
, 2012
"... The thesis ascertains the approximability of classic combinatorial optimization problems using mathematical relaxations. The general flavor of results in the thesis is: a problem P is hard to approximate to a factor better than one obtained from the R relaxation, unless the Unique Games Conjecture i ..."
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The thesis ascertains the approximability of classic combinatorial optimization problems using mathematical relaxations. The general flavor of results in the thesis is: a problem P is hard to approximate to a factor better than one obtained from the R relaxation, unless the Unique Games Conjecture is false. Almost optimal inapproximability is shown for a wide set of problems including Metric Labeling, Max. Acyclic Subgraph, various packing and covering problems. The key new idea in this thesis is in coverting hard instances of relaxations (a.k.a integrality gap instances) into a proof of inapproximability (assuming the UGC). In most cases, the hard instances were discovered prior to this work; our results imply that these hard instances are possibly strong bottlenecks in designing approximation algorithms of better quality for these problems. For ordering problems such as Max. Acyclic Subgraph and Feedback Arc Set, such hard instances were previously unknown. For these problems (see chapter 6), we construct such hard instance by using the reduction designed to prove the inapproximability.
A Generalized Grothendieck Inequality and Nonlocal Correlations that Require High Entanglement
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2011
"... Suppose that Alice and Bob make local twooutcome measurements on a shared entangled quantum state. We show that, for all positive integers d, there exist correlations that can only be reproduced if the local Hilbertspace dimension is at least d. This establishes that the amount of entanglement re ..."
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Suppose that Alice and Bob make local twooutcome measurements on a shared entangled quantum state. We show that, for all positive integers d, there exist correlations that can only be reproduced if the local Hilbertspace dimension is at least d. This establishes that the amount of entanglement required to maximally violate a Bell inequality must depend on the number of measurement settings, not just the number of measurement outcomes. We prove this result by establishing a lower bound on a new generalization of Grothendieck’s constant.