A Combinatorial Decomposition of Knapsack Cones

TL;DR

This paper introduces a new combinatorial decomposition method for knapsack cones, potentially leading to a polynomial-time algorithm for Barvinok's cone decomposition in fixed dimensions, and enhances partition analysis algorithms with LLL techniques.

Contribution

The paper presents exttt{DecDenu}, a novel combinatorial decomposition for knapsack cones, and integrates it into exttt{CTEuclid} to create an improved exttt{LLLCTEuclid} algorithm.

Findings

exttt{DecDenu} aligns with Barvinok's decomposition within algebraic combinatorics.

Computer experiments suggest exttt{DecDenu} may be polynomial for fixed n.

Enhanced exttt{LLLCTEuclid} algorithm effectively combines combinatorial and LLL techniques.

Abstract

In this paper, we focus on knapsack cones, a specific type of simplicial cones that arise naturally in the context of the knapsack problem $x_1 a_1 + \cdots + x_n a_n = a_0$. We present a novel combinatorial decomposition for these cones, named \texttt{DecDenu}, which aligns with Barvinok's unimodular cone decomposition within the broader framework of Algebraic Combinatorics. Computer experiments support us to conjecture that our \texttt{DecDenu} algorithm is polynomial when the number of variables $n$ is fixed. If true, \texttt{DecDenu} will provide the first alternative polynomial algorithm for Barvinok's unimodular cone decomposition, at least for denumerant cones. The \texttt{CTEuclid} algorithm is designed for MacMahon's partition analysis, and is notable for being the first algorithm to solve the counting problem for Magic squares of order 6. We have enhanced the \texttt{CTEuclid} algorithm by incorporating \texttt{DecDenu}, resulting in the \texttt{LLLCTEuclid} algorithm. This enhanced algorithm makes significant use of LLL's algorithm and stands out as an effective elimination-based approach.