Incremental complexity of a bi-objective hypergraph transversal problem

TL;DR

This paper introduces and analyzes the incremental complexity of a bi-objective hypergraph transversal problem, which generalizes the classical problem by considering pairs of hypergraphs and aims to find minimal hitting sets with additional constraints.

Contribution

It formalizes the bi-objective transversal problem, connects it to monotone boolean formulas, and studies its incremental complexity, extending existing theoretical understanding.

Findings

Incremental complexity is linked to monotone boolean formulas of depth 3.

The problem generalizes classical hypergraph transversals to a bi-objective setting.

Theoretical results on complexity bounds for the new problem.

Abstract

The hypergraph transversal problem has been intensively studied, from both a theoretical and a practical point of view. In particular , its incremental complexity is known to be quasi-polynomial in general and polynomial for bounded hypergraphs. Recent applications in computational biology however require to solve a generalization of this problem, that we call bi-objective transversal problem. The instance is in this case composed of a pair of hypergraphs (A, B), and the aim is to find minimal sets which hit all the hyperedges of A while intersecting a minimal set of hyperedges of B. In this paper, we formalize this problem, link it to a problem on monotone boolean $\land$ -- $\lor$ formulae of depth 3 and study its incremental complexity.