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Convex Relaxations and Integrality Gaps

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Handbook on Semidefinite, Conic and Polynomial Optimization

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 166))

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Abstract

We discuss the effectiveness of linear and semidefinite relaxations in approximating the optimum for combinatorial optimization problems. Various hierarchies of these relaxations, such as the ones defined by Lovasz and Schrijver, Sherali and Adams, and Lasserre generate increasingly strong linear and semidefinite programming relaxations starting from a basic one. We survey some positive applications of these hierarchies, where their use yields improved approximation algorithms. We also discuss known lower bounds on the integrality gaps of relaxations arising from these hierarchies, demonstrating limits on the applicability of such hierarchies for certain optimization problems.

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Notes

  1. 1.

    Some of these hierarchies can be defined in a more general context. However, we shall limit our discussion to relaxations of 0/1 (or  − 1 ∕ 1) integer programs.

  2. 2.

    The Cholesky decomposition of an n ×n PSD matrix X is a collection of vectors u1, …, un satisfying for all i, j, Xij = ⟨ui, uj⟩. 

  3. 3.

    More recently, 1. 6281…-approximations were obtained using spectral techniques [56, 58] and combinatorially using random walks [32].

  4. 4.

    The Lovász \({\vartheta}\) function itself is actually equivalent to a variant of the above relaxation, called the strict vector chromatic number, in which we have equality in (6.10).

  5. 5.

    Resolution is the proof system where one uses two clauses of the form (ψ1 ∨ x) and (ψ2 ∨  ¬x) to derive (ψ1 ∨ ψ2). Unsatisfiability is proved by deriving the empty clause.

  6. 6.

    The results in [38] were actually for a generalized version of the Sparsest Cut problem, where the denominator is not the total number of pairs \(\vert S\vert \vert \overline{S}\vert \) with one vertex in S, but rather each pair has a different cost associated with it. This is known as the non-uniform version of the problem. The result for the uniform version was proven later by Devanur et al. [22].

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Author information

Authors and Affiliations

  1. Tel Aviv University, Tel Aviv, Israel

    Eden Chlamtac

  2. Toyota Technological Institute at Chicago, Chicago, IL, USA

    Madhur Tulsiani

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  1. Eden Chlamtac
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Correspondence to Eden Chlamtac .

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Editors and Affiliations

  1. Department of Mathematics and Industrial, Ecole Polytechnique Montreal, Édouard-Montpetit Boulevard 2350, Montreal, QC, Canada, H3C 3A7

    Miguel F. Anjos

  2. LAAS-CNRS, Avenue du Colonel Roche 7, 31077, Toulouse Cedex 4, France

    Jean B. Lasserre

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Chlamtac, E., Tulsiani, M. (2012). Convex Relaxations and Integrality Gaps. In: Anjos, M.F., Lasserre, J.B. (eds) Handbook on Semidefinite, Conic and Polynomial Optimization. International Series in Operations Research & Management Science, vol 166. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0769-0_6

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