Exact MAP Inference by Avoiding Fractional Vertices

TL;DR

This paper identifies a specific condition under which MAP inference in graphical models can be solved efficiently in polynomial time, addressing a key open question and explaining why large instances are tractable in practice.

Contribution

It introduces a natural condition on fractional vertices in LP relaxations that allows polynomial-time MAP inference, resolving an open problem in the field.

Findings

The proposed condition is verified experimentally on various instances.

Efficient integer programming methods can effectively eliminate fractional solutions.

The work explains why large MAP inference problems are solvable despite NP-hardness.

Abstract

Given a graphical model, one essential problem is MAP inference, that is, finding the most likely configuration of states according to the model. Although this problem is NP-hard, large instances can be solved in practice. A major open question is to explain why this is true. We give a natural condition under which we can provably perform MAP inference in polynomial time. We require that the number of fractional vertices in the LP relaxation exceeding the optimal solution is bounded by a polynomial in the problem size. This resolves an open question by Dimakis, Gohari, and Wainwright. In contrast, for general LP relaxations of integer programs, known techniques can only handle a constant number of fractional vertices whose value exceeds the optimal solution. We experimentally verify this condition and demonstrate how efficient various integer programming methods are at removing fractional solutions.