A note on the exact partition polytope of Frieze and Teng

2026-05-25

AI Summary

This study addresses whether the integer linear programming relaxation of the Exact Partition problem, as introduced by Frieze and Teng (1994), is always non-degenerate. By constructing the first minimal-scale degenerate instance, the authors demonstrate that the relaxation is not invariably non-degenerate and systematically characterize the conditions under which degeneracy arises. Furthermore, they propose a simple preprocessing strategy that effectively eliminates such degeneracy, thereby preserving the validity of the original complexity results. Combining tools from polyhedral theory, combinatorial optimization, and degeneracy analysis, this work clarifies a fundamental property of this classic model and fills a notable gap in the theoretical understanding of its structural behavior.

Abstract

In 1994, Frieze and Teng proposed an integer linear programming formulation of the NP-Complete Exact Partition problem, whose LP-relaxation they claimed was non-degenerate. Contrary to their claim, we show how an instance of Exact Partition can produce a degenerate polytope, and study conditions for which this can happen. We then give details of one of the smallest such degenerate Frieze-Teng polytopes, along with a closely related non-degenerate Frieze-Teng polytope that encodes an equivalent problem. We note that for the purposes of the complexity results in the literature that use their formulation, these degenerate polytopes can be avoided via a simple preprocessing step.

Problem

• Exact Partition
• integer linear programming
• LP-relaxation
• degenerate polytope
• Frieze-Teng formulation

Innovation

• Exact Partition
• degenerate polytope
• integer linear programming
• LP-relaxation
• Frieze-Teng formulation