We introduce a new hierarchy over monotone set functions, that we refer to as
MPH (Maximum over Positive Hypergraphs). Levels of the hierarchy
correspond to the degree of complementarity in a given function. The highest
level of the hierarchy, MPH-m (where m is the total number of
items) captures all monotone functions. The lowest level, MPH-1,
captures all monotone submodular functions, and more generally, the class of
functions known as XOS. Every monotone function that has a positive
hypergraph representation of rank k (in the sense defined by Abraham,
Babaioff, Dughmi and Roughgarden [EC 2012]) is in MPH-k. Every
monotone function that has supermodular degree k (in the sense defined by
Feige and Izsak [ITCS 2013]) is in MPH-(k+1). In both cases, the
converse direction does not hold, even in an approximate sense. We present
additional results that demonstrate the expressiveness power of
MPH-k.
One can obtain good approximation ratios for some natural optimization
problems, provided that functions are required to lie in low levels of the
MPH hierarchy. We present two such applications. One shows that the
maximum welfare problem can be approximated within a ratio of k+1 if all
players hold valuation functions in MPH-k. The other is an upper
bound of 2k on the price of anarchy of simultaneous first price auctions.
Being in MPH-k can be shown to involve two requirements -- one
is monotonicity and the other is a certain requirement that we refer to as
PLE (Positive Lower Envelope). Removing the monotonicity
requirement, one obtains the PLE hierarchy over all non-negative
set functions (whether monotone or not), which can be fertile ground for
further research