Results 1  10
of
27
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 ..."
Abstract

Cited by 144 (13 self)
 Add to MetaCart
(Show Context)
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 polynomialtime algorithm for all CSPs. Assuming the Unique Games Conjecture, this algorithm achieves the optimal approximation ratio for every CSP. Unconditionally, for all 2CSPs 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 ..."
Abstract

Cited by 34 (4 self)
 Add to MetaCart
(Show Context)
Abstract: We give a brief introduction to the basic notions of Fourier analysis on the
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 ..."
Abstract

Cited by 32 (1 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 32 (5 self)
 Add to MetaCart
(Show Context)
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 kCSPq problem is UGhard 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
How to Round Any CSP
"... A large number of interesting combinatorial optimization ..."
Abstract

Cited by 26 (3 self)
 Add to MetaCart
(Show Context)
A large number of interesting combinatorial optimization
Towards computing the grothendieck constant
 In SODA ’09: Proceedings of the 20th Annual ACMSIAM 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 ..."
Abstract

Cited by 16 (2 self)
 Add to MetaCart
(Show Context)
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,NQuadratic 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,NQuadratic Programming problem. Using some standard but nontrivial modifications, the reduction in [22] yields the following hardness result: Assuming the Unique Games Conjecture [9], it is NPhard to approximate the KM,NQuadratic 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,NQuadratic 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 ηclose 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 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 ..."
Abstract

Cited by 15 (1 self)
 Add to MetaCart
(Show Context)
This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1
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 ..."
Abstract

Cited by 13 (5 self)
 Add to MetaCart
(Show Context)
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