Lower bound of computational complexity of knapsack problems

TL;DR

This paper establishes a lower bound on the computational complexity of knapsack problems using quantum statistical mechanics, revealing topological structures and phase diagrams that distinguish NP, P, and intermediate problems.

Contribution

It introduces a novel approach linking quantum statistics and topology to analyze the complexity bounds of knapsack problems, identifying the NP-intermediate region.

Findings

Identified topological structures arising from lattice and transfer matrix contradictions.

Mapped a phase diagram with NP, P, and NP-intermediate regions.

Proposed implications for developing optimal algorithms.

Abstract

The quantum statistics mechanism is very powerful for investigating the equilibrium states and the phase transitions in complex spin disorder systems. The spin disorder systems act as an interdisciplinary platform for solving the optimum processes in computer science. In this work, I determined the lower bound of the computational complexity of knapsack problems. I investigated the origin of nontrivial topological structures in these hard problems. It was uncovered that the nontrivial topological structures arise from the contradictory between the three-dimensional character of the lattice and the two-dimensional character of the transfer matrices used in the quantum statistics mechanism. I illustrated a phase diagram for the non-deterministic polynomial (NP) vs polynomial (P) problems, in which a NP-intermediate (NPI) area exists between the NP-complete problems and the P-problems, while the absolute minimum core model is at the border between the NPI and the NP-complete problems. The absolute minimum core model of the knapsack problem cannot collapse directly into the P-problem. Under the guide of the results, one may develop the best algorithms for solving various optimum problems in the shortest time, being in subexponential and superpolynomial. This work illuminates the road on various fields of science ranging from physics to biology to finances, and to information technologies.