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25
Optimal algorithms and inapproximability results for every CSP
- In Proc. 40 th ACM STOC
, 2008
"... Semidefinite Programming(SDP) is one of the strongest algorithmic techniques used in the design of approximation algorithms. In recent years, Unique Games Conjecture(UGC) has proved to be intimately connected to the limitations of Semidefinite Programming. Making this connection precise, we show the ..."
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Cited by 137 (13 self)
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Semidefinite Programming(SDP) is one of the strongest algorithmic techniques used in the design of approximation algorithms. In recent years, Unique Games Conjecture(UGC) has proved to be intimately connected to the limitations of Semidefinite Programming. Making this connection precise, we show the following result: If UGC is true, then for every constraint satisfaction problem(CSP) the best approximation ratio is given by a certain simple SDP. Specifically, we show a generic conversion from SDP integrality gaps to UGC hardness results for every CSP. This result holds both for maximization and minimization problems over arbitrary finite domains. Using this connection between integrality gaps and hardness results we obtain a generic polynomial-time algorithm for all CSPs. Assuming the Unique Games Conjecture, this algorithm achieves the optimal approximation ratio for every CSP. Unconditionally, for all 2-CSPs the algorithm achieves an approximation ratio equal to the integrality gap of a natural SDP used in literature. Further the algorithm achieves at least as good an approximation ratio as the best known algorithms for several problems like MaxCut, Max2Sat, MaxDiCut
A brief introduction to Fourier analysis on the Boolean cube
- Theory of Computing Library– Graduate Surveys
, 2008
"... Abstract: We give a brief introduction to the basic notions of Fourier analysis on the ..."
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Cited by 34 (4 self)
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Abstract: We give a brief introduction to the basic notions of Fourier analysis on the
Approximation Resistant Predicates From Pairwise Independence
, 2008
"... We study the approximability of predicates on k variables from a domain [q], and give a new sufficient condition for such predicates to be approximation resistant under the Unique Games Conjecture. Specifically, we show that a predicate P is approximation resistant if there exists a balanced pairwis ..."
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Cited by 32 (6 self)
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We study the approximability of predicates on k variables from a domain [q], and give a new sufficient condition for such predicates to be approximation resistant under the Unique Games Conjecture. Specifically, we show that a predicate P is approximation resistant if there exists a balanced pairwise independent distribution over [q] k whose support is contained in the set of satisfying assignments to P. Using constructions of pairwise independent distributions this result implies that • For general k ≥ 3 and q ≥ 2, theMAX k-CSPq problem is UG-hard to approximate within O(kq 2)/q k + ɛ. • For the special case of q =2, i.e., boolean variables, we can sharpen this bound to (k + O(k 0.525))/2 k + ɛ, improving upon the best previous bound of 2k/2 k +ɛ (Samorodnitsky and Trevisan, STOC’06) by essentially a factor 2. • Finally, again for q =2, assuming that the famous Hadamard Conjecture is true, this can be improved even further, and the O(k 0.525) term can be replaced by the constant 4. 1
Towards Sharp Inapproximability For Any 2-CSP
"... We continue the recent line of work on the connection between semidefinite programming-based approximation algorithms and the Unique Games Conjecture. Given any boolean 2-CSP (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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Cited by 29 (1 self)
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We continue the recent line of work on the connection between semidefinite programming-based approximation algorithms and the Unique Games Conjecture. Given any boolean 2-CSP (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 clause-by-clause 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 Games-based 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 2-CSPs. As an application, we show that MAX 2-AND 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 2-AND, 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.
Understanding Parallel Repetition Requires Understanding Foams
, 2007
"... Motivated by the study of Parallel Repetition and also by the Unique Games Conjecture, we investigate the value of the “Odd Cycle Games ” under parallel repetition. Using tools from discrete harmonic analysis, we show that after d rounds on the cycle of length m, the value of the game is at most 1 − ..."
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Cited by 28 (3 self)
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Motivated by the study of Parallel Repetition and also by the Unique Games Conjecture, we investigate the value of the “Odd Cycle Games ” under parallel repetition. Using tools from discrete harmonic analysis, we show that after d rounds on the cycle of length m, the value of the game is at most 1 − (1/m) · Ω̃( d) (for d ≤ m2, say). This beats the natural barrier of 1−Θ(1/m)2 ·d for Raz-style proofs [Raz98, Hol06] (see [Fei95]) and also the SDP bound of Feige-Lovász [FL92, GW95]; however, it just barely fails to have implications for Unique Games. On the other hand, we also show that improving our bound would require proving nontrivial lower bounds on the surface area of high-dimensional foams. Specifically, one would need to answer: What is the least surface area of a cell that tiles Rd by the lattice Z d?
How to Round Any CSP
"... A large number of interesting combinatorial optimization ..."
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Cited by 27 (3 self)
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A large number of interesting combinatorial optimization
Towards computing the grothendieck constant
- In SODA ’09: Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms
, 2009
"... The Grothendieck constant KG is the smallest constant such that for every d ∈ N and every matrix A = (aij), sup u i,v j ∈B (d) X aij〈ui, vj 〉 � KG · ij sup x i,y j ∈[−1,1] X ij aijxiyj, where B (d) is the unit ball in R d. Despite several efforts [15, 23], the value of the constant KG remains unkno ..."
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Cited by 16 (2 self)
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The Grothendieck constant KG is the smallest constant such that for every d ∈ N and every matrix A = (aij), sup u i,v j ∈B (d) X aij〈ui, vj 〉 � KG · ij sup x i,y j ∈[−1,1] X ij aijxiyj, where B (d) is the unit ball in R d. Despite several efforts [15, 23], the value of the constant KG remains unknown. The Grothendieck constant KG is precisely the integrality gap of a natural SDP relaxation for the KM,N-Quadratic Programming problem. The input to this problem is a matrix A = (aij) and the objective is to maximize the quadratic form P ij aijxiyj over xi, yj ∈ [−1, 1]. In this work, we apply techniques from [22] to the KM,N-Quadratic Programming problem. Using some standard but non-trivial modifications, the reduction in [22] yields the following hardness result: Assuming the Unique Games Conjecture [9], it is NP-hard to approximate the KM,N-Quadratic Programming problem to any factor better than the Grothendieck constant KG. By adapting a “bootstrapping ” argument used in a proof of Grothendieck inequality [5], we are able to perform a tighter analysis than [22]. Through this careful analysis, we obtain the following new results: ◦ An approximation algorithm for KM,N-Quadratic Programming that is guaranteed to achieve an approximation ratio arbitrarily close to the Grothendieck constant KG (optimal approximation ratio assuming the Unique Games Conjecture). ◦ We show that the Grothendieck constant KG can be computed within an error η, in time depending only on η. Specifically, for each η, we formulate an explicit finite linear program, whose optimum is η-close to the Grothendieck constant. We also exhibit a simple family of operators on the Gaussian Hilbert space that is guaranteed to contain tight examples for the Grothendieck inequality.
On the Advantage over Random for Maximum Acyclic Subgraph
"... In this paper we present a new approximation algorithm for the MAX ACYCLIC SUBGRAPH problem. Given an instance where the maximum acyclic subgraph contains 1/2+δ fraction of all edges, our algorithm finds an acyclic subgraph with 1/2 + Ω(δ / logn) fraction of all edges. 1 ..."
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Cited by 13 (5 self)
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In this paper we present a new approximation algorithm for the MAX ACYCLIC SUBGRAPH problem. Given an instance where the maximum acyclic subgraph contains 1/2+δ fraction of all edges, our algorithm finds an acyclic subgraph with 1/2 + Ω(δ / logn) fraction of all edges. 1
