Simple Approximation Algorithms for Balanced MAX 2SAT

Abstract

We study simple algorithms for the balanced MAX 2SAT problem, where we are given weighted clauses of length one and two with the property that for each variable x the total weight of clauses that x appears in equals the total weight of clauses for \(\overline{x}\). We show that such instances have a simple structural property in that any optimal solution can satisfy at most the total weight of the clauses minus half the total weight of the unit clauses. Using this property and a novel analysis of the computation tree, we are able to show that a large class of greedy algorithms, including Johnson’s algorithm, gives a \(\frac{3}{4}\)-approximation algorithm for balanced MAX 2SAT; a similar statement is false for general MAX 2SAT instances. We further give a spectral 0.81-approximation algorithm for balanced MAX E2SAT instances (in which each clause has exactly 2 literals) by a reduction to a spectral algorithm of Trevisan for the maximum colored cut problem. We provide experimental results showing that this spectral algorithm performs well and is slightly better than Johnson’s algorithm and the Goemans–Williamson semidefinite programming algorithm on balanced MAX E2SAT instances.

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Fig. 1

Notes

  1. 1.

    Although Khot et al. state their guarantee for balanced MAX 2SAT, their argument (in the proof of Theorem 3 in [10, p. 340]) implicitly assumes that no clauses of length one are present. Only then the observation that “linear terms cancel out” in the arithmetization is valid, a fact that we also utilize (see Sect. 4).

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Correspondence to Matthias Poloczek.

Additional information

Alice Paul’s work was supported by an NDSEG fellowship. Matthias Poloczek’s work was supported by the Alexander von Humboldt Foundation within the Feodor Lynen program, and in part by NSF Grant CCF-1115256 and DFG Grant SCHN 503/5-1. David P. Williamson’s work was supported in part by NSF Grant CCF-1115256.

Appendix: Soto’s Bound for MAX CC

Appendix: Soto’s Bound for MAX CC

Recall from Sect. 4 that \(\mathrm {LB}_{G}(\varepsilon )\) is a lower bound on the fraction of weight achieved by Trevisan’s spectral algorithm on G, where G is the MAX CC instance that was created by our reduction on the balanced set of 2-clauses C.

Lemma 3

(Section 3.1 in [19]) Let \(\varepsilon _0\) be the unique solution of the equation \(\frac{1}{1 + 2 \sqrt{\varepsilon (1- \varepsilon )}} = \frac{ -1 + \sqrt{4 \varepsilon ^2 - 8 \varepsilon + 5}}{2 (1-\varepsilon )} .\) Then,

If \(\varepsilon \ge \frac{1}{3}\),

$$\begin{aligned} \mathrm {LB}_G(\varepsilon ) := \frac{1}{2} . \end{aligned}$$

If \(\varepsilon _0 \le \varepsilon \le \frac{1}{3}\),

$$\begin{aligned} \mathrm {LB}_G(\varepsilon ) :=&\frac{1}{2} \cdot \bigg ( \varepsilon - 1 + \sqrt{4 \varepsilon ^2 - 8 \varepsilon + 5} - \varepsilon \ln \left( \frac{1+ \sqrt{4 \varepsilon ^2 - 8 \varepsilon + 5}}{8 \varepsilon } \right) \\&+ \,\frac{ \sqrt{5}}{5} \varepsilon \ln \left( \frac{5 - 4 \varepsilon + \sqrt{5 (4 \varepsilon ^2 - 8 \varepsilon +5)}}{(11 + 5 \sqrt{5}) \varepsilon } \right) \bigg ). \end{aligned}$$

If \(\varepsilon \le \varepsilon _0\),

$$\begin{aligned} \mathrm {LB}_G(\varepsilon ) :=&\, \frac{1}{2} \cdot \bigg ( \varepsilon \left( 1 - \frac{3}{\varepsilon _0} \right) + 2 + \frac{\varepsilon }{\varepsilon _0} \sqrt{4 \varepsilon _0^2 - 8 \varepsilon _0 +5} \\&- \varepsilon \ln \left( \frac{1 + \sqrt{4 \varepsilon _0^2 - 8 \varepsilon _0 + 5}}{8 \varepsilon _0} \right) \\&+\, \frac{ \sqrt{5}}{5} \varepsilon \ln \left( \frac{5 - 4 \varepsilon _0 + \sqrt{5 (4 \varepsilon _0^2 - 8 \varepsilon _0 +5)}}{(11 + 5 \sqrt{5}) \varepsilon _0} \right) \\&+\, 16 \varepsilon \ln \left( \frac{ \sqrt{\varepsilon } + \sqrt{1- \varepsilon }}{\sqrt{\varepsilon } + \sqrt{ \frac{\varepsilon }{\varepsilon _0} - \varepsilon }} \right) + 8 \varepsilon \frac{ \sqrt{ \varepsilon _0 (1- \varepsilon _0)} +1 - 2 \varepsilon _0}{ \varepsilon _0 + \sqrt{\varepsilon _0 (1- \varepsilon _0)}}\\&-\, 8 \sqrt{\varepsilon } \frac{ \sqrt{ \varepsilon (1- \varepsilon )} +1 - 2 \varepsilon }{ \sqrt{\varepsilon } + \sqrt{\varepsilon (1- \varepsilon )}} \bigg ) . \end{aligned}$$

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Paul, A., Poloczek, M. & Williamson, D.P. Simple Approximation Algorithms for Balanced MAX 2SAT. Algorithmica 80, 995–1012 (2018). https://doi.org/10.1007/s00453-017-0312-6

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Keywords

  • Maximum satisfiability
  • Approximation algorithm
  • Greedy algorithm
  • Spectral algorithm
  • Balanced instances
  • Priority algorithm