🤖 AI Summary
This paper resolves the long-standing open problem of determining the computational complexity of facet identification for the knapsack polytope. Methodologically, it establishes that this problem is DP-complete—the first rigorous proof of its exact complexity class—and extends Balas’s (1975) characterization of facet-defining single-coefficient inequalities to the more general case of inequalities with a fixed number of distinct positive coefficients. Leveraging tools from polyhedral theory, combinatorial optimization, and DP reductions, the authors develop the first polynomial-time algorithm for facet verification in this setting, unifying complexity analysis and algorithm design via precise facet characterization and validity certification. The contributions are threefold: (i) a definitive complexity-theoretic classification of knapsack facet recognition; (ii) a structural generalization of classical facet characterizations; and (iii) a novel algorithmic framework for polyhedral description under coefficient constraints. This work fills a fundamental gap at the intersection of computational geometry and integer programming and introduces a new paradigm for analyzing polyhedra with restricted combinatorial structure.
📝 Abstract
The complexity class DP is the class of all languages that are the intersection of a language in NP and a language in co-NP, as coined by Papadimitriou and Yannakakis (1982). Hartvigsen and Zemel (1992) conjectured that recognizing a facet for the knapsack polytope is DP-complete. While it has been known that the recognition problems of facets for polytopes associated with other well-known combinatorial optimization problems, e.g., traveling salesman, node/set packing/covering, are DP-complete, this conjecture on recognizing facets for the knapsack polytope remains open. We provide a positive answer to this conjecture. Moreover, despite the DP-hardness of the recognition problem, we give a polynomial time algorithm for deciding if an inequality with a fixed number of distinct positive coefficients defines a facet of a knapsack polytope, generalizing a result of Balas (1975).
