Integer Points in Knapsack Polytopes and s-covering Radius

TL;DR

This paper studies the structure of integer solutions in knapsack polytopes using the s-covering radius, providing bounds for the s-Frobenius number under certain conditions.

Contribution

It introduces the concept of s-covering radius to analyze the set of integer vectors with multiple solutions in knapsack problems and establishes an optimal lower bound for the s-Frobenius number in a special case.

Findings

Established an optimal lower bound for the s-Frobenius number.

Analyzed the structure of integer points in knapsack polytopes using s-covering radius.

Provided insights into the distribution of solutions in knapsack polytopes.

Abstract

Given an integer matrix A satisfying certain regularity assumptions, we consider for a positive integer s the set F_s(A) of all integer vectors b such that the associated knapsack polytope P(A,b)={x: Ax=b, x non-negative} contains at least s integer points. In this paper we investigate the structure of the set F_s(A) sing the concept of s-covering radius. In particular, in a special case we prove an optimal lower bound for the s-Frobenius number.