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A Necessary Condition for Connectedness of Solutions to Integer Linear Systems

Takasugu Shigenobu, Naoyuki Kamiyama

TL;DR

Defines universal connectedness for the solution graph $G(A,b)$ of an integer linear system and motivates the connectivity question for all RHS vectors. The authors introduce a forbidden-pattern condition on the coefficient matrix and prove it is a necessary condition for universal connectedness, then establish sufficiency for three small-dimension regimes, including the challenging $(m,n)=(4,3)$ case via a uniquely determined sign pattern and constructive connectivity arguments. This work strengthens prior results that used elimination-ordering as a criterion and provides a tighter characterization in these small-dimension regimes. The findings have implications for reconfiguration problems in ILP and inform when any initial feasible solution can be transformed to any target one via single-coordinate moves, though the general sufficiency question remains open.

Abstract

An integer linear system is a set of inequalities with integer constraints. The solution graph of an integer linear system is an undirected graph defined on the set of feasible solutions to the integer linear system. In this graph, a pair of feasible solutions is connected by an edge if the Hamming distance between them is one. In this paper, we consider a condition under which the solution graph is connected for any right-hand side vector. First, we prove that if the solution graph is connected for any right-hand side vector, then the coefficient matrix of the system does not contain some forbidden pattern as a submatrix. Next, we prove that if at least one of (i) the number of rows is at most 3, (ii) the number of columns is at most 2, (iii) the number of rows is 4 and the number of columns is 3 holds, then the condition that the coefficient matrix of the system does not contain the forbidden pattern is a sufficient condition under which the solution graph is connected for any right-hand side vector. This result is stronger than a known necessary and sufficient condition since the set of coefficient matrix dimensions is strictly larger.

A Necessary Condition for Connectedness of Solutions to Integer Linear Systems

TL;DR

Defines universal connectedness for the solution graph of an integer linear system and motivates the connectivity question for all RHS vectors. The authors introduce a forbidden-pattern condition on the coefficient matrix and prove it is a necessary condition for universal connectedness, then establish sufficiency for three small-dimension regimes, including the challenging case via a uniquely determined sign pattern and constructive connectivity arguments. This work strengthens prior results that used elimination-ordering as a criterion and provides a tighter characterization in these small-dimension regimes. The findings have implications for reconfiguration problems in ILP and inform when any initial feasible solution can be transformed to any target one via single-coordinate moves, though the general sufficiency question remains open.

Abstract

An integer linear system is a set of inequalities with integer constraints. The solution graph of an integer linear system is an undirected graph defined on the set of feasible solutions to the integer linear system. In this graph, a pair of feasible solutions is connected by an edge if the Hamming distance between them is one. In this paper, we consider a condition under which the solution graph is connected for any right-hand side vector. First, we prove that if the solution graph is connected for any right-hand side vector, then the coefficient matrix of the system does not contain some forbidden pattern as a submatrix. Next, we prove that if at least one of (i) the number of rows is at most 3, (ii) the number of columns is at most 2, (iii) the number of rows is 4 and the number of columns is 3 holds, then the condition that the coefficient matrix of the system does not contain the forbidden pattern is a sufficient condition under which the solution graph is connected for any right-hand side vector. This result is stronger than a known necessary and sufficient condition since the set of coefficient matrix dimensions is strictly larger.
Paper Structure (6 sections, 12 theorems, 59 equations, 1 algorithm)

This paper contains 6 sections, 12 theorems, 59 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['theorem:necessary']}
  4. Proof of Theorem \ref{['theorem:sufficient']}
  5. Proof of Lemma \ref{['lem:unique_pattern_dicided']}
  6. Conclusion

Key Result

Lemma 1

Let $A$ be a matrix in $\mathbb{R}^{[m]\times [n]}$. Define $A^r$ as a submatrix of $A$ whose index set of columns is $\Delta$ defined by Algorithm alg. If $A^r$ is not universally connected, then $A$ is not universally connected.

Theorems & Definitions (27)

  • Definition 1: Sign function
  • Definition 2: Integer linear system
  • Definition 3: Hamming distance
  • Definition 4: Solution graph
  • Definition 5: Universal connectedness
  • Definition 6: Forbidden pattern
  • Example 1
  • Definition 7: Elimination
  • Definition 8: Eliminated matrix
  • Definition 9: Elimination ordering
  • ...and 17 more