  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 $3$.
