Improved approximation algorithms for k-submodular maximization under a knapsack constraint

Dung T. K. Ha, Canh V. Pham, Tan D. Tran

Abstract: Highlights•We investigate the problem of k submodular maximization under a knapsack constraint over the ground set of size n<math><mi is="true">n</mi></math>.•We propose three deterministic approximation algorithms for the studied the problem: FA<math><mi mathvariant="sans-serif" is="true">FA</mi></math>, IFA<math><mi mathvariant="sans-serif" is="true">IFA</mi></math> and IFA+<math><mrow is="true"><mi mathvariant="sans-serif" is="true">IFA</mi><mtext is="true">+</mtext></mrow></math> with approximation ratios of 110,14−ϵ<math><mrow is="true"><mfrac is="true"><mrow is="true"><mn is="true">1</mn></mrow><mrow is="true"><mn is="true">10</mn></mrow></mfrac><mo is="true">,</mo><mfrac is="true"><mrow is="true"><mn is="true">1</mn></mrow><mrow is="true"><mn is="true">4</mn></mrow></mfrac><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mi is="true">ϵ</mi></mrow></math> and 13−ϵ<math><mrow is="true"><mfrac is="true"><mrow is="true"><mn is="true">1</mn></mrow><mrow is="true"><mn is="true">3</mn></mrow></mfrac><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mi is="true">ϵ</mi></mrow></math>, respectively. Both run in O(nk)<math><mrow is="true"><mi is="true">O</mi><mrow is="true"><mo is="true">(</mo><mi is="true">n</mi><mi is="true">k</mi><mo is="true">)</mo></mrow></mrow></math> query complexity.•Our IFA+<math><mrow is="true"><mi mathvariant="sans-serif" is="true">IFA</mi><mtext is="true">+</mtext></mrow></math> algorithm improves both approximation ratio and query complexity of any existing deterministic algorithms