The positive semidefinite Grothendieck problem with rank constraint

Authors:
Jop Briët;Fernando Mário De Oliveira Filho;Frank Vallentin
Affiliations:
Centrum Wiskunde & Informatica, Amsterdam, The Netherlands;Department of Econometrics ad OR, Tilburg University, Tilburg, The Netherlands;Delft Institute of Applied Mathematics, Technical University of Delft, Delft, The Netherlands
Venue:
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Year:
2010

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Abstract

Given a positive integer n and a positive semidefinite matrix A = (Aij) ∈ Rm × m the positive semidefinite Grothendieck problem with rank-n-constraint (SDPn) is maximize Σi=1m Σj=1m Aij xi ċ xj, where x1, ..., xm ∈ Sn-1. In this paper we design a randomized polynomial-time approximation algorithm for SDPn achieving an approximation ratio of γ(n) = 2/n(Γ((n + 1)/2)/Γ(n/2))2 = 1 - Θ(1/n). We show that under the assumption of the unique games conjecture the achieved approximation ratio is optimal: There is no polynomial-time algorithm which approximates SDPn with a ratio greater than γ(n). We improve the approximation ratio of the best known polynomial-time algorithm for SDP1 from 2/π to 2/(πγ(m)) = 2/π + Θ(1/m), and we show a tighter approximation ratio for SDPn when A is the Laplacian matrix of a weighted graph with nonnegative edge weights.