Standard Quadratic Formulations of Many NP Problems: A Simplex-Based Compilation Framework for Combinatorial Optimization

2026-10-01

AI Summary

This study addresses the absence of a unified continuous modeling and compilation framework for discrete NP combinatorial optimization problems. To this end, it proposes a unified paradigm that reformulates discrete NP problems as continuous standard quadratic programs (StQPs). Building upon a simplex framework, the work integrates graph reductions, regularized Motzkin–Straus formulations, and QUBO mapping techniques to construct a coefficient-bounded finite-domain factor model compiler, requiring at most four simplex coordinates per binary factor. The primary contribution is the exact, standardized StQP compilation of Karp’s 21 NP-complete problems, accompanied by explicit formulas, dimensionality analyses, and separation bounds. This formulation rigorously guarantees that all valid assignments correspond to global or local minima, thereby establishing a complete continuous solution pathway for combinatorial optimization.

Abstract

The standard quadratic program (StQP) minimizes a quadratic form over nonnegative variables that sum to one. We compose classical graph reductions with regularized Motzkin--Straus clique formulations to express discrete optimization problems in this continuous domain. The graph matrix has diagonal entries $τ$, zeros on edges, and ones on nonedges. For $0<τ<1$, its minimum is $τ/ω(G)$, where $ω(G)$ is the clique number. Its strict local minimizers are precisely the uniform distributions on maximal cliques, and its global minimizers encode maximum cliques. At $τ=1/2$, integer scaling gives coefficients in $\{0,1,2\}$ and minimum $1/ω(G)$, yielding an NP-complete StQP threshold problem with a restricted coefficient alphabet. We give explicit formulations for satisfiability, coloring, Hamiltonian cycles, independent set, vertex cover, set packing, three-dimensional matching, and graph isomorphism. A regularized weighted clique formulation combined with local-state compatibility graphs gives an exact compiler for finite-domain factor models specified by complete local tables, including QUBO, with at most four simplex coordinates per binary pair factor. The catalog covers Karp's 21 problems: twelve use direct graph formulations, and nine use factor-state formulations, including six obtained through binary-linear feasibility. For each route we record dimensions, coefficient structure, and recovery rules. We analyze interaction count, coefficient range, objective separation, perturbation tolerance, support recovery, and decoding overhead. The separation bounds quantify the effects of clique size, factor weights, and offsets. In the complete factor-state construction, every assignment, including each suboptimal assignment, is a strict local minimum.

Problem

• Standard Quadratic Program
• Combinatorial Optimization
• NP-hard Problems
• Motzkin-Straus Formulation
• Factor Models

Innovation

• Standard Quadratic Program
• Combinatorial Optimization
• Simplex-Based Compilation
• Motzkin-Straus Formulation
• Factor Models