Perfect Graphs and Graphical Modeling

Tractability is the study of computational tasks with the goal of identifying which problem classes are tractable or, in other words, efficiently solvable. The class of tractable problems is traditionally assumed to be solvable in polynomial time by a deterministic Turing machine and is denoted by P. The class contains many natural tasks such as sorting a set of numbers, linear programming (the decision version), determining if a number is prime, and finding a maximum weight matching. Many interesting problems, however, lie in another class that generalizes P and is known as NP: the class of languages decidable in polynomial time on a non-deterministic Turing machine. We trivially have that P is a subset of NP (many researchers also believe that it is a strict subset). It is believed that many problems in the class NP are, in the worst case, intractable and do not admit efficient inference. Problems such as maximum stable set, the traveling salesman problem and graph coloring are known to be NP-hard (at least as hard as the hardest problems in NP). It is, therefore , widely suspected that there are no polynomial-time algorithms for NP-hard problems. Rather than stop after labeling a problem class as NP-hard by identifying its worst-case instances, this chapter explores the question: what instances of otherwise NP-hard problems still admit efficient inference? The study of perfect graphs (Berge, 1963) helps shed light on this question. It turns out that a variety of hard problems such as graph coloring, maximum clique and maximum stable set are all solv-able efficiently in polynomial time when the input graph is restricted to be a perfect graph (Grötschel et al., 1988). Can we extend this promising result to other important problems in applied fields such as data mining and machine learning? This chapter will focus on a central NP-hard problem in statistics and machine learning known as the maximum a posteriori (MAP) problem, in other