, Volume 69, Issue 3, pp 605–618 | Cite as

An SDP Primal-Dual Algorithm for Approximating the Lovász-Theta Function



The Lovász ϑ-function (Lovász in IEEE Trans. Inf. Theory, 25:1–7, 1979) of a graph G=(V,E) can be defined as the maximum of the sum of the entries of a positive semidefinite matrix X, whose trace Tr(X) equals 1, and X ij =0 whenever {i,j}∈E. This function appears as a subroutine for many algorithms for graph problems such as maximum independent set and maximum clique. We apply Arora and Kale’s primal-dual method for SDP to design an algorithm to approximate the ϑ-function within an additive error of δ>0, which runs in time \(O(\frac{\vartheta ^{2} n^{2}}{\delta^{2}} \log n \cdot M_{e})\), where ϑ=ϑ(G) and M e =O(n 3) is the time for a matrix exponentiation operation. It follows that for perfect graphs G, our primal-dual method computes ϑ(G) exactly in time O(ϑ 2 n 5logn).

Moreover, our techniques generalize to the weighted Lovász ϑ-function, and both the maximum independent set weight and the maximum clique weight for vertex weighted perfect graphs can be approximated within a factor of (1+ϵ) in time O(ϵ −2 n 5logn).


Approximation algorithms Primal-dual SDP methods Lovász-Theta function Perfect graphs Maximum independent sets 



We would like to thank Khaled Elbassioni for discussion at the initial stage of the project.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • T.-H. Hubert Chan
    • 1
  • Kevin L. Chang
    • 3
  • Rajiv Raman
    • 2
  1. 1.Department of Computer ScienceUniversity of Hong KongPokfulamHong Kong
  2. 2.TCS Innovation LabsTRDDCPuneIndia
  3. 3.Mountain ViewUSA

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