A bicriteria algorithm for the minimum submodular cost partial set multi-cover problem

Yishuo Shi, Yingli Ran, Zhao Zhang, Ding-Zhu Du

Abstract: This paper presents a bicriteria approximation algorithm for the minimum submodular cost partial set multi-cover problem (SCPSMC), the goal of which is to find a minimum cost sub-collection of sets to fully cover q percentage of total profit of all elements, where the cost on sub-collections is a submodular function, and an element e with covering requirement re<math><msub is="true"><mrow is="true"><mi is="true">r</mi></mrow><mrow is="true"><mi is="true">e</mi></mrow></msub></math> is fully covered if it belongs to at least re<math><msub is="true"><mrow is="true"><mi is="true">r</mi></mrow><mrow is="true"><mi is="true">e</mi></mrow></msub></math> picked sets. Assuming that the maximum covering requirement rmax=maxe∈E⁡re<math><msub is="true"><mrow is="true"><mi is="true">r</mi></mrow><mrow is="true"><mi mathvariant="normal" is="true">max</mi></mrow></msub><mo is="true">=</mo><msub is="true"><mrow is="true"><mi mathvariant="normal" is="true">max</mi></mrow><mrow is="true"><mi is="true">e</mi><mo is="true">∈</mo><mi is="true">E</mi></mrow></msub><mo is="true">⁡</mo><msub is="true"><mrow is="true"><mi is="true">r</mi></mrow><mrow is="true"><mi is="true">e</mi></mrow></msub></math> is a constant and the cost function is nonnegative and submodular, we give a deterministic (b/qε,(1−ε))<math><mo stretchy="false" is="true">(</mo><mi is="true">b</mi><mo stretchy="false" is="true">/</mo><mi is="true">q</mi><mi is="true">ε</mi><mo is="true">,</mo><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo is="true">−</mo><mi is="true">ε</mi><mo stretchy="false" is="true">)</mo><mo stretchy="false" is="true">)</mo></math>-bicriteria algorithm for SCPSMC, the output of which fully covers at least (1−ε)q<math><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo is="true">−</mo><mi is="true">ε</mi><mo stretchy="false" is="true">)</mo><mi is="true">q</mi></math>-percentage of the total profit and the performance ratio is b/qε<math><mi is="true">b</mi><mo stretchy="false" is="true">/</mo><mi is="true">q</mi><mi is="true">ε</mi></math>, where b=maxe⁡(fere)<math><mi is="true">b</mi><mo is="true">=</mo><msub is="true"><mrow is="true"><mi mathvariant="normal" is="true">max</mi></mrow><mrow is="true"><mi is="true">e</mi></mrow></msub><mo is="true">⁡</mo><mrow is="true"><mo is="true">(</mo><mtable is="true"><mtr is="true"><mtd is="true"><msub is="true"><mrow is="true"><mi is="true">f</mi></mrow><mrow is="true"><mi is="true">e</mi></mrow></msub></mtd></mtr><mtr is="true"><mtd is="true"><msub is="true"><mrow is="true"><mi is="true">r</mi></mrow><mrow is="true"><mi is="true">e</mi></mrow></msub></mtd></mtr></mtable><mo is="true">)</mo></mrow></math> and fe<math><msub is="true"><mrow is="true"><mi is="true">f</mi></mrow><mrow is="true"><mi is="true">e</mi></mrow></msub></math> is the number of sets containing element e.