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

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Abstract

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).

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Notes

  1. An odd hole is an induced C k for odd k, and an odd anti-hole is an induced \(\overline{C_{k}}\).

  2. In Alizadeh’s paper [4], it is stated that the interior point method for SDP takes \(O(\sqrt{n})\) iterations; and Nayakkankuppam and Overton [20] explicitly analyzed the running time of each iteration to be O(m 3).

  3. Throughout this paper, we use \(\tilde{O}(\cdot)\) to hide poly-logarithmic factor; formally, \(f(n) = \tilde{O}(g(n))\) if there exists k>0 such that f(n)=O(g(n)logk n).

  4. In Arora and Kale’s description, it is suggested that the Oracle is run with a “candidate” primal solution \(\mathbf {X}:= \frac{\mathbf {W}}{\mathbf {Tr}(\mathbf {W})}\), but this is not entirely necessary, and we incorporate this rescaling operation in the Oracle itself to simplify the description. Moreover, in their description of the Oracle, the primal solution returned is always X itself. However, in their applications, the Oracle can also return a slightly modified version of X, in a manner which is more in accordance with our description.

  5. According to Arora and Kale’s framework, it is sufficient to have z≤β. However, in our case, it is to our advantage for the Oracle to produce a dual solution with z=β.

  6. We remind the reader here that the width ρ β of the Oracle depends on the candidate value β currently being tested.

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Acknowledgement

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

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Authors and Affiliations

  1. Department of Computer Science, University of Hong Kong, Pokfulam, Hong Kong

    T.-H. Hubert Chan

  2. TCS Innovation Labs, TRDDC, Pune, India

    Rajiv Raman

  3. Mountain View, CA, USA

    Kevin L. Chang

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  1. T.-H. Hubert Chan
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  2. Kevin L. Chang
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Correspondence to T.-H. Hubert Chan.

Additional information

A preliminary version of the paper appeared in IEEE International Symposium on Information Theory 2009. This research was done while the authors were at Max-Planck-Institut für Informatik, 66123 Saarbrücken, Germany.

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Chan, TH.H., Chang, K.L. & Raman, R. An SDP Primal-Dual Algorithm for Approximating the Lovász-Theta Function. Algorithmica 69, 605–618 (2014). https://doi.org/10.1007/s00453-013-9756-5

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