Beyond hypergraph acyclicity: limits of tractability for pseudo-Boolean optimization
Alberto Del Pia, Aida Khajavirad
TL;DR
This work probes the limits of tractability for pseudo-Boolean optimization by studying signed hypergraphs that encode polynomial objectives over binary points. It combines lower-bound results on extension complexity and algorithmic hardness with new structural notions—nest-set gap and nest-set width—to identify tractable regimes beyond traditional hypergraph acyclicity and bounded treewidth. The authors establish exponential extension complexity for certain bounded-rank scenarios, and, crucially, introduce nest-set-based conditions under which polynomial-size extended formulations exist, yielding polynomial-time solvability for wide classes that include $eta$-cycles. Together, these results delineate when pseudo-Boolean optimization can be solved via LP relaxations and when it inherently resists such formulations, with implications for modeling and algorithm design in discrete optimization.
Abstract
In this paper, we study the problem of minimizing a polynomial function with literals over all binary points, often referred to as pseudo-Boolean optimization. We investigate the fundamental limits of computation for this problem by providing new necessary conditions and sufficient conditions for tractability. On the one hand, we obtain the first intractability results, in the best-case sense, for pseudo-Boolean optimization problems on signed hypergraphs with bounded rank, in terms of the treewidth of the intersection graph. Namely, first, under some mild assumptions, we show that for every sequence of hypergraphs indexed by the treewidth and with bounded rank, the complexity of solving the associated pseudo-Boolean optimization problem grows super-polynomially in the treewidth. Second, we show that any hypergraph of bounded rank is the underlying hypergraph of some signed hypergraph for which the corresponding pseudo-Boolean polytope has an exponential extension complexity in the treewidth. On the other hand, we introduce the nest-set gap, a new hypergraph-theoretic notion that enables us to define a notion of "distance" from the hypergaph acyclicity. We prove that if this distance is bounded, the pseudo-Boolean polytope admits a polynomial-size extended formulation. This in turn enables us to obtain a polynomial-time algorithm for a large class of pseudo-Boolean optimization problems whose underlying hypergraphs contain beta-cycles.