Introduction

In this chapter, we consider a combinatorial resource allocation problem, called the submodular partition or welfare problem, where the objective is to divide or partition a given set of resources among multiple agents (with a possibly different valuation for each subset of resources), such that the sum of the agents’ valuation (for resources assigned to them) is maximized.

When the agents’ valuations of the subsets of resources is arbitrary, this problem is not only NP-hard, but also APX hard, i.e., it is hard to find even a good approximate solution. Thus, a natural, submodularity assumption is made on the agent valuations, that essentially captures the diminishing returns property, i.e., the incremental increase in any agents’ valuation decreases as more and more resources are assigned to it. Important examples of this problem include combinatorial auctions, e.g., spectrum allocation among various cellular telephone service providers, advertisement-display slot assignments on web platforms, public utility allocations, etc.

Under the submodularity assumption, the partitioning problem becomes approximable. Early research in this direction considered an offline setting, but surprisingly the same ideas are applicable in the online setting as well, but with a slightly weaker guarantee.

In this chapter, for the online submodular partitioning problem, we present a simple greedy algorithm, and derive its competitive ratio, as a function of the curvature of the submodular valuation functions and a new metric called the discriminant. Curvature measures the ‘distance’ as to how far the valuation function is from being linear, while the discriminant counts the amount of improvement made by the greedy algorithm in each iteration. We also discuss some important applications of the submodular partition problem.

Submodular Partition Problem

We begin with a formal definition of a submodular function.

Submodularity captures the diminishing returns property exhibited by or expected to hold for natural utility metrics, i.e., the rate of increase of utility function decreases with an increase in the size of the allocated set.

Alternate and equivalent definitions of submodularity are as follows.

Important examples of submodular functions include set union, entropy, mutual information [347], number of edges crossing a graph cut [348], etc. Showing these quantities are submodular can be easy or hard depending on the choice of the three definitions one chooses.