Convex Relaxations and Integrality Gaps

Chlamtáč, Eden; Tulsiani, Madhur

doi:10.1007/978-1-4614-0769-0_6

Cited by 76 publications

(64 citation statements)

References 51 publications

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“…See Laurent [15] for a survey of these systems; several other Lift-and-Project systems are known, see [9,3]. The index of each relaxation in the sequence of tightened relaxations is known as the level in the hierarchy; the level of the original relaxation is defined to be zero.…”

Section: Hierarchies Of Convex Relaxationsmentioning

confidence: 99%

“…Starting with the work of Arora et al [1], substantial research efforts have been devoted to showing that tightened relaxations (for many levels) fail to reduce the integrality ratio for many combinatorial optimization problems (see [9] for a list of negative results). This task seems especially difficult for the SA system because it strengthens relaxations in a "global manner;" this enhances its algorithmic leverage for deriving positive results, but makes it more challenging to design instances with bad integrality ratios.…”

Section: Hierarchies Of Convex Relaxationsmentioning

confidence: 99%

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On integrality ratios for asymmetric TSP in the Sherali–Adams hierarchy

et al. 2015

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We study the ATSP (Asymmetric Traveling Salesman Problem), and our focus is on negative results in the framework of the Sherali-Adams (SA) Lift and Project method.Our main result pertains to the standard LP (linear programming) relaxation of ATSP, due to Dantzig, Fulkerson, and Johnson. For any fixed integer t ≥ 0 and small ǫ, 0 < ǫ ≪ 1, there exists a digraph G on ν = ν(t, ǫ) = O(t/ǫ) vertices such that the integrality ratio for level t of the SA system starting with the standard LP on G is ≥ 1 + 1−ǫ 2t+3 ≈ 4 3 , 6 5 , 8 7 , . . . . Thus, in terms of the input size, the result holds for any t = 0, 1, . . . , Θ(ν) levels. Our key contribution is to identify a structural property of digraphs that allows us to construct fractional feasible solutions for any level t of the SA system starting from the standard LP. Our hard instances are simple and satisfy the structural property.There is a further relaxation of the standard LP called the balanced LP, and our methods simplify considerably when the starting LP for the SA system is the balanced LP; in particular, the relevant structural property (of digraphs) simplifies such that it is satisfied by the digraphs given by the wellknown construction of Charikar, Goemans and Karloff (CGK). Consequently, the CGK digraphs serve as hard instances, and we obtain an integrality ratio of 1 + 1−ǫ t+1 for any level t of the SA system, where 0 < ǫ ≪ 1 and the number of vertices is ν(t, ǫ) = O((t/ǫ) (t/ǫ) ).Also, our results for the standard LP extend to the PATH ATSP (find a min cost Hamiltonian dipath from a given source vertex to a given sink vertex).

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Section: Hierarchies Of Convex Relaxationsmentioning

confidence: 99%

Section: Hierarchies Of Convex Relaxationsmentioning

confidence: 99%

On integrality ratios for asymmetric TSP in the Sherali–Adams hierarchy

et al. 2015

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“…An excellent introduction to hierarchies and their algorithmic uses can be found in [9,19] (in fact both these surveys discuss the independent set problem as an example). Here, we only describe here the most basic facts that we need.…”

Section: Preliminariesmentioning

confidence: 99%

Approximating independent sets in sparse graphs

Bansal

2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…, N }, |S| ≤ q, and then writing a "locally integral" relaxation for this extended 0-1 program to guarantee that x S = i∈S x i (by convention x ∅ = 1). For more details, see the survey [15].…”

Section: Lp Hierarchies and Faithful Roundingmentioning

confidence: 99%

Everywhere-Sparse Spanners via Dense Subgraphs

E¹,

Dinitz

Krauthgamer

2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

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Abstract-The significant progressg in constructing graph spanners that are sparse (small number of edges) or light (low total weight) has skipped spanners that are everywhere-sparse (small maximum degree). This disparity is in line with other network design problems, where the maximum-degree objective has been a notorious technical challenge. Our main result is for the LOWEST DEGREE 2-SPANNER (LD2S) problem, where the goal is to compute a 2-spanner of an input graph so as to minimize the maximum degree. We design a polynomial-time algorithm achieving approximation factorÕ(∆where ∆ is the maximum degree of the input graph. The previous O(∆ 1/4 )-approximation was proved nearly two decades ago by Kortsarz and Peleg [SODA 1994, SICOMP 1998].Our main conceptual contribution is to establish a formal connection between LD2S and a variant of the DENSEST k-SUBGRAPH (DkS) problem. Specifically, we design for both problems strong relaxations based on the Sherali-Adams linear programming (LP) hierarchy, and show that "faithful" randomized rounding of the DkS-variant can be used to round LD2S solutions. Our notion of faithfulness intuitively means that all vertices and edges are chosen with probability proportional to their LP value, but the precise formulation is more subtle.Unfortunately, the best algorithms known for DkS use the Lovász-Schrijver LP hierarchy in a non-faithful way [Bhaskara, Charikar, Chlamtac, Feige, and Vijayaraghavan, STOC 2010]. Our main technical contribution is to overcome this shortcoming, while still matching the gap that arises in random graphs by planting a subgraph with same log-density.

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Convex Relaxations and Integrality Gaps

Cited by 76 publications

References 51 publications

On integrality ratios for asymmetric TSP in the Sherali–Adams hierarchy

On integrality ratios for asymmetric TSP in the Sherali–Adams hierarchy

Approximating independent sets in sparse graphs

Everywhere-Sparse Spanners via Dense Subgraphs

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