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 one hand, we obtain the first intractability results for pseudo-Boolean optimization problems on signed hypergraphs with bounded rank, in terms of the treewidth of the intersection graph. On the other hand, we introduce the nest-set gap, a new hypergraph-theoretic notion that enables us to move beyond hypergraph acyclicity, and obtain a polynomial-size extended formulation for the pseudo-Boolean polytope of a class of signed hypergraphs whose underlying hypergraphs contain beta-cycles.

翻译：本文研究在所有二进制点上最小化含文字多项式函数的问题，通常称为伪布尔优化。我们通过提供新的必要条件和充分条件来探究该问题的基本计算极限。一方面，我们首次基于交集图的树宽，获得了有界秩符号超图上伪布尔优化问题的不可处理性结果。另一方面，我们引入了巢集间隙这一新的超图理论概念，使我们能够超越超图无环性，并为底层超图包含β循环的一类符号超图的伪布尔多胞形，获得多项式规模的扩展表述。