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Tight Approximation for Unconstrained XOS Maximization

Author

Listed:
  • Yuval Filmus

    (The Henry and Marilyn Taub Faculty of Computer Science, Technion–Israel Institute of Technology, Haifa 3200003, Israel)

  • Yasushi Kawase

    (Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan)

  • Yusuke Kobayashi

    (Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan)

  • Yutaro Yamaguchi

    (Graduate School and Faculty of Information Science and Electrical Engineering, Kyushu University, Fukuoka 819-0395, Japan)

Abstract

A set function is called XOS if it can be represented by the maximum of additive functions. When such a representation is fixed, the number of additive functions required to define the XOS function is called the width. In this paper, we study the problem of maximizing XOS functions in the value oracle model. The problem is trivial for the XOS functions of width 1 because they are just additive, but it is already nontrivial even when the width is restricted to 2. We show two types of tight bounds on the polynomial-time approximability for this problem. First, in general, the approximation bound is between O ( n ) and Ω ( n / l o g n ) , and exactly θ ( n / l o g n ) if randomization is allowed, where n is the ground set size. Second, when the width of the input XOS functions is bounded by a constant k ≥ 2, the approximation bound is between k − 1 and k − 1 − ɛ for any ɛ > 0. In particular, we give a linear-time algorithm to find an exact maximizer of a given XOS function of width 2, whereas we show that any exact algorithm requires an exponential number of value oracle calls even when the width is restricted to 3.

Suggested Citation

  • Yuval Filmus & Yasushi Kawase & Yusuke Kobayashi & Yutaro Yamaguchi, 2021. "Tight Approximation for Unconstrained XOS Maximization," Mathematics of Operations Research, INFORMS, vol. 46(4), pages 1599-1610, November.
    • Handle: RePEc:inm:ormoor:v:46:y:2021:i:4:p:1599-1610
    DOI: 10.1287/moor.2020.1088
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