Efficient and Reliable Hitting-Set Computations for the Implicit Hitting Set Approach

TL;DR

This paper investigates alternative algorithms for hitting set computations within the implicit hitting set framework, comparing their efficiency and reliability, especially focusing on pseudo-Boolean reasoning and stochastic local search methods.

Contribution

It introduces and evaluates new algorithmic techniques for hitting set optimization, demonstrating that PB reasoning can be competitive and provide correctness certificates.

Findings

Commercial IP solvers are highly effective but may face correctness issues due to numerical instability.

PB reasoning-based hitting set computations can be made competitive with exact IP solvers.

PB reasoning allows for correctness certificates applicable across various IHS instantiations.

Abstract

The implicit hitting set (IHS) approach offers a general framework for solving computationally hard combinatorial optimization problems declaratively. IHS iterates between a decision oracle used for extracting sources of inconsistency and an optimizer for computing so-called hitting sets (HSs) over the accumulated sources of inconsistency. While the decision oracle is language-specific, the optimizers is usually instantiated through integer programming. We explore alternative algorithmic techniques for hitting set optimization based on different ways of employing pseudo-Boolean (PB) reasoning as well as stochastic local search. We extensively evaluate the practical feasibility of the alternatives in particular in the context of pseudo-Boolean (0-1 IP) optimization as one of the most recent instantiations of IHS. Highlighting a trade-off between efficiency and reliability, while a commercial IP solver turns out to remain the most effective way to instantiate HS computations, it can cause correctness issues due to numerical instability; in fact, we show that exact HS computations instantiated via PB reasoning can be made competitive with a numerically exact IP solver. Furthermore, the use of PB reasoning as a basis for HS computations allows for obtaining certificates for the correctness of IHS computations, generally applicable to any IHS instantiation in which reasoning in the declarative language at hand can be captured in the PB-based proof format we employ.