Extremal Probability Bounds in Combinatorial Optimization

TL;DR

This paper derives the tightest probability bounds for the optimal value of combinatorial optimization problems with random objectives, based solely on marginal distributions, and analyzes their computational complexity.

Contribution

It introduces extremal probability bounds for combinatorial optimization, analyzes their computational complexity, and provides algorithms for specific cases like 0/1 V- and H-polytopes.

Findings

Tightest bounds are valid across all joint distributions with given marginals.

Upper bounds are weakly NP-hard to compute for 0/1 V-polytopes.

Lower bounds are strongly NP-hard to compute for 0/1 V-polytopes.

Abstract

In this paper, we compute the tightest possible bounds on the probability that the optimal value of a combinatorial optimization problem in maximization form with a random objective exceeds a given number, assuming only knowledge of the marginal distributions of the objective coefficient vector. The bounds are ``extremal'' since they are valid across all joint distributions with the given marginals. We analyze the complexity of computing the bounds assuming discrete marginals and identify instances when the bounds are computable in polynomial time. For compact 0/1 V-polytopes, we show that the tightest upper bound is weakly NP-hard to compute by providing a pseudopolynomial time algorithm. On the other hand, the tightest lower bound is shown to be strongly NP-hard to compute for compact 0/1 V-polytopes by restricting attention to Bernoulli random variables. For compact 0/1 H-polytopes, for the special case of PERT networks arising in project management, we show that the tightest upper bound is weakly NP-hard to compute by providing a pseudopolynomial time algorithm. The results in the paper complement existing results in the literature for computing the probability with independent random variables.