Tight Approximation for Unconstrained XOS Maximization

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 $\Omega(n / \log n)$, and exactly $\Theta(n / \log n)$ 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 $k \geq 2$, the approximation bound is between $k - 1$ and $k - 1 - \epsilon$ for any $\epsilon > 0$. In particular, we give a linear-time algorithm to find an exact maximizer of a given XOS function of width $2$, while we show that any exact algorithm requires an exponential number of value oracle calls even when the width is restricted to $3$.