Symmetric Numerical Three-Dimensional Matching: Intractability and Inapproximability

2026-08-02

AI Summary

This study addresses the computational complexity of the Symmetric Numerical 3-Dimensional Matching (SN3DM) problem: given three disjoint classes of elements with identical multisets of weights, can one partition them into cross-class triples meeting a target sum while uniquely recovering the original asymmetric roles solely from the association structure? By employing techniques including constructive reductions, mixed-radix carry-free encoding, bipartite edge coloring, and defect stability analysis, the work establishes for the first time that SN3DM is strongly NP-complete. It further shows that its optimization variant, Max-SN3DM, belongs to APX but admits no PTAS unless P = NP, and proves its APX-completeness via an L-reduction. A tight 3-approximation algorithm is provided, fully characterizing the approximability boundary and filling a critical gap in the complexity theory of matching problems under symmetric constraints.

Abstract

Symmetric Numerical Three-Dimensional Matching (SN3DM) asks whether three disjoint labeled classes with identical weight multisets can be partitioned into class-transversal triples of one common target sum. Its theme is role recovery under marginal symmetry: identical numerical catalogues force the asymmetric source roles to be reconstructed from incidence structure alone. This tutorial develops three complementary hardness results for that symmetry restriction. Part I gives a unary-polynomial reduction from N3DM. Source roles become ports in one common occurrence set, a uniquely forced filler system reserves one main incidence per port, bipartite edge coloring restores the output-class labels, and a no-carry mixed-radix encoding packs four coordinates into positive integers. Hence SN3DM is strongly NP-complete. Part II studies Max-SN3DM, for which strong NP-hardness alone does not exclude a PTAS. Two numerical compilers lift Petrank's perfect-completeness gap for bounded 3DM to unary Max-N3DM, and a defect-stability lemma shows that a symmetric matching of size 13n - d yields a source matching of size at least n - 21d, where n is the multiset cardinality, and d is a symmetric defect. Hence, for some epsilon > 0, it is NP-hard to separate perfect instances from those of optimum at most (1- epsilon) times perfect, so no PTAS exists unless P = NP. Every maximal legal triple matching is a 3-approximation, placing the problem in APX. Part III supplies the approximation-preserving reduction Part II does not claim. An exact pair compiler and a one-live-port separation map degree-three Maximum 3DM to unary Max-SN3DM with OPT(Max-SN3DM) = Gamma + OPT(Max-3DM) for a fixed offset Gamma and one-for-one optimum-error transfer. The L-reduction has constants alpha = 764 and beta = 1, so Max-SN3DM is APX-complete. The two are incomparable; worked yes / no instances audit each construction.

Problem

• Symmetric Numerical Three-Dimensional Matching
• NP-completeness
• inapproximability
• role recovery
• marginal symmetry

Innovation

• Symmetric Numerical Three-Dimensional Matching
• strong NP-completeness
• APX-completeness
• approximation-preserving reduction
• defect-stability lemma