Efficient deterministic algorithms for maximizing symmetric submodular functions

Zongqi Wan, Jialin Zhang, Xiaoming Sun, Zhijie Zhang

Abstract: Symmetric submodular maximization is an important class of combinatorial optimization problems, including MAX-CUT on graphs and hyper-graphs. The state-of-the-art algorithm for the problem over general constraints has an approximation ratio of 0.432 [16]. The algorithm applies the canonical continuous greedy technique that involves a sampling process. It, therefore, suffers from high query complexity and is inherently randomized. In this paper, we present several efficient deterministic algorithms for maximizing a symmetric submodular function under various constraints. Specifically, for the cardinality constraint, we design a deterministic algorithm that attains a 0.432 ratio and uses O(kn)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">k</mi><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></math> queries. Previously, the best deterministic algorithm attains a 0.385−ϵ<math><mn is="true">0.385</mn><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mi is="true">ϵ</mi></math> ratio and uses O(kn(109ϵ)209ϵ−1)<math><mi is="true">O</mi><mrow is="true"><mo stretchy="true" is="true">(</mo><mi is="true">k</mi><mi is="true">n</mi><msup is="true"><mrow is="true"><mo stretchy="false" is="true">(</mo><mfrac is="true"><mrow is="true"><mn is="true">10</mn></mrow><mrow is="true"><mn is="true">9</mn><mi is="true">ϵ</mi></mrow></mfrac><mo stretchy="false" is="true">)</mo></mrow><mrow is="true"><mfrac is="true"><mrow is="true"><mn is="true">20</mn></mrow><mrow is="true"><mn is="true">9</mn><mi is="true">ϵ</mi></mrow></mfrac><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn></mrow></msup><mo stretchy="true" is="true">)</mo></mrow></math> queries [12]. For the matroid constraint, we design a deterministic algorithm that attains a 1/3−ϵ<math><mn is="true">1</mn><mo stretchy="false" is="true">/</mo><mn is="true">3</mn><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mi is="true">ϵ</mi></math> ratio and uses O(knlog⁡ϵ−1)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">k</mi><mi is="true">n</mi><mi mathvariant="normal" is="true">log</mi><mo is="true">⁡</mo><msup is="true"><mrow is="true"><mi is="true">ϵ</mi></mrow><mrow is="true"><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn></mrow></msup><mo stretchy="false" is="true">)</mo></math> queries. Previously, the best deterministic algorithm can also attain 1/3−ϵ<math><mn is="true">1</mn><mo stretchy="false" is="true">/</mo><mn is="true">3</mn><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mi is="true">ϵ</mi></math> ratio but it uses much larger O(ϵ−1n4)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">ϵ</mi></mrow><mrow is="true"><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn></mrow></msup><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">4</mn></mrow></msup><mo stretchy="false" is="true">)</mo></math> queries [24]. For the packing constraints with a large width, we design a deterministic algorithm that attains a 0.432−ϵ<math><mn is="true">0.432</mn><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mi is="true">ϵ</mi></math> ratio and uses O(n2)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo stretchy="false" is="true">)</mo></math> queries. To the best of our knowledge, there is no deterministic algorithm for the constraint previously. The last algorithm can be adapted to attain a 0.432 ratio for single knapsack constraint using O(n4)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">4</mn></mrow></msup><mo stretchy="false" is="true">)</mo></math> queries. Previously, the best deterministic algorithm attains a 0.316−ϵ<math><mn is="true">0.316</mn><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mi is="true">ϵ</mi></math> ratio and uses O˜(n3)<math><mover accent="true" is="true"><mrow is="true"><mi is="true">O</mi></mrow><mrow is="true"><mo is="true">˜</mo></mrow></mover><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">3</mn></mrow></msup><mo stretchy="false" is="true">)</mo></math> queries [2].