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35
Linear vs. Semidefinite Extended Formulations: Exponential Separation and Strong Lower Bounds
, 2012
"... We solve a 20year old problem posed by Yannakakis and prove that there exists no polynomialsize linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope an ..."
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Cited by 51 (11 self)
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We solve a 20year old problem posed by Yannakakis and prove that there exists no polynomialsize linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope and the stable set polytope. These results were discovered through a new connection that we make between oneway quantum communication protocols and semidefinite programming reformulations of LPs.
Graph Expansion and the Unique Games Conjecture
, 2010
"... The edge expansion of a subset of vertices S ⊆ V in a graph G measures the fraction of edges that leave S. In a dregular graph, the edge expansion/conductance Φ(S) of a subset E(S,V \S) dS S ⊆ V is defined as Φ(S) =. Approximating the conductance of small linear sized sets (size δn) is a natur ..."
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Cited by 41 (5 self)
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The edge expansion of a subset of vertices S ⊆ V in a graph G measures the fraction of edges that leave S. In a dregular graph, the edge expansion/conductance Φ(S) of a subset E(S,V \S) dS S ⊆ V is defined as Φ(S) =. Approximating the conductance of small linear sized sets (size δn) is a natural optimization question that is a variant of the wellstudied Sparsest Cut problem. However, there are no known algorithms to even distinguish between almost complete edge expansion (Φ(S) = 1 − ε), and close to 0 expansion. In this work, we investigate the connection between Graph Expansion and the Unique Games Conjecture. Specifically, we show the following: –We show that a simple decision version of the problem of approximating small set expansion reduces to Unique Games. Thus if approximating edge expansion of small sets is hard, then Unique Games is hard. Alternatively, a refutation of the UGC will yield better algorithms to approximate edge expansion in graphs. This is the first nontrivial “reverse ” reduction from a natural optimization problem to Unique Games. –Under a slightly stronger UGC that assumes mild expansion of small sets, we show that it is UGhard to approximate small set expansion. –On instances with sufficiently good expansion of small sets, we show that Unique Games is easy by extending the techniques of [4].
Improved approximation guarantees through higher levels of SDP hierarchies
 In Approximation, randomization and combinatorial optimization
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Integrality Gaps of Linear and Semidefinite Programming Relaxations for Knapsack
"... Recent years have seen an explosion of interest in lift and project methods, such as those proposed by Lovász and Schrijver [40], Sherali and Adams [49], Balas, Ceria and Cornuejols [6], Lasserre [36, 37] and others. These methods are systematic procedures for constructing a sequence of increasingly ..."
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Cited by 20 (0 self)
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Recent years have seen an explosion of interest in lift and project methods, such as those proposed by Lovász and Schrijver [40], Sherali and Adams [49], Balas, Ceria and Cornuejols [6], Lasserre [36, 37] and others. These methods are systematic procedures for constructing a sequence of increasingly tight mathematical programming relaxations for 01 optimization problems. One major line of research in this area has focused on understanding the strengths and limitations of these procedures. Of particular interest to our community is the question of how the integrality gaps for interesting combinatorial optimization problems evolve through a series of rounds of one of these procedures. On the one hand, if the integrality gap of successive relaxations drops sufficiently fast, there is the potential for an improved approximation algorithm. On the other hand, if the integrality gap for a problem persists, this can be viewed as a lower bound in a certain restricted model of computation. In this paper, we study the integrality gap in these hierarchies for the knapsack problem. We have two main results. First, we show that an integrality gap of 2 − ɛ persists up to a linear number of rounds of SheraliAdams. This is interesting, since it is well known that knapsack has a fully polynomial time approximation scheme [30, 39]. Second, we show that Lasserre’s hierarchy closes the gap quickly. Specifically, after t 2 rounds of Lasserre, the integrality gap decreases to t/(t − 1). Thus, we provide a second example of an integrality gap separation between Lasserre and Sherali Adams. The only other such gap we are aware of is in the recent work of Fernandez de la Vega and Mathieu [19] (respectively of Charikar, Makarychev and Makarychev [12]) showing that the integrality gap for MAXCUT remains 2 − ɛ even after ω(1) (respectively n γ) rounds of SheraliAdams. On the other hand, it is known that 2 rounds of Lasserre yields a relaxation as least as strong as the GoemansWilliamson SDP, which has an integrality gap of 0.878.
Approximation Limits of Linear Programs (Beyond Hierarchies)
, 2013
"... We develop a framework for proving approximation limits of polynomialsize linear programs from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any linear program as opposed to only programs generate ..."
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Cited by 17 (6 self)
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We develop a framework for proving approximation limits of polynomialsize linear programs from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any linear program as opposed to only programs generated by hierarchies. Using our framework, we prove that O(n1/2−ɛ)approximations for CLIQUE require linear programs of size 2nΩ(ɛ). This lower bound applies to linear programs using a certain encoding of CLIQUE as a linear optimization problem. Moreover, we establish a similar result for approximations of semidefinite programs by linear programs. Our main technical ingredient is a quantitative improvement of Razborov’s rectangle corruption lemma (1992) for the high error regime, which gives strong lower bounds on the nonnegative rank of shifts of the unique disjointness matrix.
SheraliAdams relaxations of the matching polytope
 In STOC’2009
, 2009
"... We study the SheraliAdams liftandproject hierarchy of linear programming relaxations of the matching polytope. Our main result is an asymptotically tight expression 1 + 1/k for the integrality gap after k rounds of this hierarchy. The result is derived by a detailed analysis of the LP after k rou ..."
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Cited by 16 (0 self)
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We study the SheraliAdams liftandproject hierarchy of linear programming relaxations of the matching polytope. Our main result is an asymptotically tight expression 1 + 1/k for the integrality gap after k rounds of this hierarchy. The result is derived by a detailed analysis of the LP after k rounds applied to the complete graph K2d+1. We give an explicit recurrence for the value of this LP, and hence show that its gap exhibits a “phase transition, ” dropping from close to its maximum value 1 + 1 2d to close to 1 around the threshold k = 2d − √ d. We also show that the rank of the matching polytope (i.e., the number of SheraliAdams rounds until the integer polytope is reached) is exactly 2d − 1.
Approximate Constraint Satisfaction Requires Large LP Relaxations
"... We prove superpolynomial lower bounds on the size of linear programming relaxations for approximation versions of constraint satisfaction problems. We show that for these problems, polynomialsized linear programs are exactly as powerful as programs arising from a constant number of rounds of the S ..."
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Cited by 16 (2 self)
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We prove superpolynomial lower bounds on the size of linear programming relaxations for approximation versions of constraint satisfaction problems. We show that for these problems, polynomialsized linear programs are exactly as powerful as programs arising from a constant number of rounds of the SheraliAdams hierarchy. In particular, any polynomialsized linear program for Max Cut has an integrality gap of 7/8.
On Linear and Semidefinite Programming Relaxations for Hypergraph Matching
"... The hypergraph matching problem is to find a largest collection of disjoint hyperedges in a hypergraph. This is a wellstudied problem in combinatorial optimization and graph theory with various applications. The best known approximation algorithms for this problem are all local search algorithms. I ..."
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Cited by 16 (0 self)
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The hypergraph matching problem is to find a largest collection of disjoint hyperedges in a hypergraph. This is a wellstudied problem in combinatorial optimization and graph theory with various applications. The best known approximation algorithms for this problem are all local search algorithms. In this paper we analyze different linear and semidefinite programming relaxations for the hypergraph matching problem, and study their connections to the local search method. Our main results are the following: • We consider the standard linear programming relaxation of the problem. We provide an algorithmic proof of a result of Füredi, Kahn and Seymour, showing that the integrality gap is exactly k − 1 + 1/k for kuniform hypergraphs, and is exactly k − 1 for kpartite hypergraphs. This yields an improved approximation algorithm for the weighted 3dimensional matching problem. Our algorithm combines the use of the iterative rounding method and the fractional local ratio method, showing a new way to round linear programming solutions for packing problems. • We study the strengthening of the standard LP relaxation by local constraints. We show that, even after linear number of rounds of the SheraliAdams liftandproject procedure on the standard LP relaxation, there are kuniform hypergraphs with integrality gap at least k − 2. On the other hand, we prove that for every constant k, there is a strengthening of the standard LP relaxation by only a polynomial number of constraints, with integrality gap at most (k+1)/2 for kuniform hypergraphs. The construction uses a result in extremal combinatorics. • We consider the standard semidefinite programming relaxation of the problem. We prove that the Lovász ϑfunction provides an SDP relaxation with integrality gap at most (k + 1)/2. The proof gives an indirect way (not by a rounding algorithm) to bound the ratio between any local optimal solution and any optimal SDP solution. This shows a new connection between local search and linear and semidefinite programming relaxations. 1
Polynomial integrality gaps for strong SDP relaxations of Densest ksubgraph
"... The Densest ksubgraph problem (i.e. find a size k subgraph with maximum number of edges), is one of the notorious problems in approximation algorithms. There is a significant gap between known upper and lower bounds for Densest ksubgraph: the current best algorithm gives an ≈ O(n 1/4) approximatio ..."
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Cited by 14 (4 self)
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The Densest ksubgraph problem (i.e. find a size k subgraph with maximum number of edges), is one of the notorious problems in approximation algorithms. There is a significant gap between known upper and lower bounds for Densest ksubgraph: the current best algorithm gives an ≈ O(n 1/4) approximation, while even showing a small constant factor hardness requires significantly stronger assumptions than P ̸ = NP. In addition to interest in designing better algorithms, a number of recent results have exploited the conjectured hardness of Densest ksubgraph and its variants. Thus, understanding the approximability of Densest ksubgraph is an important challenge. In this work, we give evidence for the hardness of approximating Densest ksubgraph within polynomial factors. Specifically, we expose the limitations of strong semidefinite programs from SDP hierarchies in solving Densest ksubgraph. Our results include: • A lower bound of Ω ( n 1/4 / log 3 n) on the integrality gap for Ω(log n / log log n) rounds of the SheraliAdams relaxation for Densest ksubgraph. This also holds for the relaxation obtained from SheraliAdams with an added SDP constraint. Our gap instances are in
Unsplittable flow in paths and trees and columnrestricted packing integer programs
 IN PROCEEDINGS, INTERNATIONAL WORKSHOP ON APPROXIMATION ALGORITHMS FOR COMBINATORIAL OPTIMIZATION PROBLEMS
, 2009
"... We consider the unsplittable flow problem (UFP) and the closely related columnrestricted packing integer programs (CPIPs). In UFP we are given an edgecapacitated graph G = (V, E) and k request pairs R1,..., Rk, where each Ri consists of a sourcedestination pair (si, ti), a demand di and a weigh ..."
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Cited by 11 (0 self)
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We consider the unsplittable flow problem (UFP) and the closely related columnrestricted packing integer programs (CPIPs). In UFP we are given an edgecapacitated graph G = (V, E) and k request pairs R1,..., Rk, where each Ri consists of a sourcedestination pair (si, ti), a demand di and a weight wi. The goal is to find a maximum weight subset of requests that can be routed unsplittably in G. Most previous work on UFP has focused on the nobottleneck case in which the maximum demand of the requests is at most the smallest edge capacity. Inspired by the recent work of Bansal et al. [3] on UFP on a path without the above assumption, we consider UFP on paths as well as trees. We give a simple O(log n) approximation for UFP on trees when all weights are identical; this yields an O(log 2 n) approximation for the weighted case. These are the first nontrivial approximations for UFP on trees. We develop an LP relaxation for UFP on paths that has an integrality gap of O(log 2 n); previously there was no relaxation with o(n) gap. We also consider UFP in general graphs and CPIPs without the nobottleneck assumption and obtain new and useful results.