Maximizing a Nonnegative, Monotone, Submodular Function Constrained to Matchings

TL;DR

This paper introduces the CSM-Matching problem, a generalization of maximizing submodular functions on matchings, and provides approximation algorithms with proven bounds for this NP-hard problem.

Contribution

It formulates the CSM-Matching problem, proves its NP-hardness, and develops a simple greedy 3-approximation algorithm along with a reduction to matroid-based optimization with a (4+epsilon)-approximation.

Findings

NP-hardness of CSM-Matching within a factor of e/(e-1)

A simple greedy 3-approximation algorithm for CSM-Matching

Reduction to CSM-2-Matroids with a (4+epsilon)-approximation

Abstract

Submodular functions have many applications. Matchings have many applications. The bitext word alignment problem can be modeled as the problem of maximizing a nonnegative, monotone, submodular function constrained to matchings in a complete bipartite graph where each vertex corresponds to a word in the two input sentences and each edge represents a potential word-to-word translation. We propose a more general problem of maximizing a nonnegative, monotone, submodular function defined on the edge set of a complete graph constrained to matchings; we call this problem the CSM-Matching problem. CSM-Matching also generalizes the maximum-weight matching problem, which has a polynomial-time algorithm; however, we show that it is NP-hard to approximate CSM-Matching within a factor of e/(e-1) by reducing the max k-cover problem to it. Our main result is a simple, greedy, 3-approximation algorithm for CSM-Matching. Then we reduce CSM-Matching to maximizing a nonnegative, monotone, submodular function over two matroids, i.e., CSM-2-Matroids. CSM-2-Matroids has a (2+epsilon)-approximation algorithm - called LSV2. We show that we can find a (4+epsilon)-approximate solution to CSM-Matching using LSV2. We extend this approach to similar problems.