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On Connectedness of Solutions to Integer Linear Systems

Takasugu Shigenobu, Naoyuki Kamiyama

TL;DR

This paper investigates when the solution graph of an integer linear system is connected for every right-hand side vector, focusing on elimination orderings (EO) of the coefficient matrix. It proves that EO is not necessary for connectivity in general by constructing a counterexample with $m\ge4$ and $n\ge3$, while showing EO is necessary in the narrow cases $m\le3$ or $n\le2$ via a contrapositive argument. The authors introduce the Expansion Lemma to structurally separate columns and analyze connectivity through case analyses for two rows, two columns, and three rows. The work connects algebraic properties of $A$ to reconfiguration problems and SAT-like complexity, and it notes that EO can be determined in polynomial time. This advances understanding of when feasible solutions of ILSs can be reconfigured without breaking connectivity across all right-hand sides.

Abstract

An integer linear system (ILS) is a linear system with integer constraints. The solution graph of an ILS is defined as an undirected graph defined on the set of feasible solutions to the ILS. A pair of feasible solutions is connected by an edge in the solution graph if the Hamming distance between them is 1. We consider a property of the coefficient matrix of an ILS such that the solution graph is connected for any right-hand side vector. Especially, we focus on the existence of an elimination ordering (EO) of a coefficient matrix, which is known as the sufficient condition for the connectedness of the solution graph for any right-hand side vector. That is, we consider the question whether the existence of an EO of the coefficient matrix is a necessary condition for the connectedness of the solution graph for any right-hand side vector. We first prove that if a coefficient matrix has at least four rows and at least three columns, then the existence of an EO may not be a necessary condition. Next, we prove that if a coefficient matrix has at most three rows or at most two columns, then the existence of an EO is a necessary condition.

On Connectedness of Solutions to Integer Linear Systems

TL;DR

This paper investigates when the solution graph of an integer linear system is connected for every right-hand side vector, focusing on elimination orderings (EO) of the coefficient matrix. It proves that EO is not necessary for connectivity in general by constructing a counterexample with and , while showing EO is necessary in the narrow cases or via a contrapositive argument. The authors introduce the Expansion Lemma to structurally separate columns and analyze connectivity through case analyses for two rows, two columns, and three rows. The work connects algebraic properties of to reconfiguration problems and SAT-like complexity, and it notes that EO can be determined in polynomial time. This advances understanding of when feasible solutions of ILSs can be reconfigured without breaking connectivity across all right-hand sides.

Abstract

An integer linear system (ILS) is a linear system with integer constraints. The solution graph of an ILS is defined as an undirected graph defined on the set of feasible solutions to the ILS. A pair of feasible solutions is connected by an edge in the solution graph if the Hamming distance between them is 1. We consider a property of the coefficient matrix of an ILS such that the solution graph is connected for any right-hand side vector. Especially, we focus on the existence of an elimination ordering (EO) of a coefficient matrix, which is known as the sufficient condition for the connectedness of the solution graph for any right-hand side vector. That is, we consider the question whether the existence of an EO of the coefficient matrix is a necessary condition for the connectedness of the solution graph for any right-hand side vector. We first prove that if a coefficient matrix has at least four rows and at least three columns, then the existence of an EO may not be a necessary condition. Next, we prove that if a coefficient matrix has at most three rows or at most two columns, then the existence of an EO is a necessary condition.

Paper Structure

This paper contains 22 sections, 23 theorems, 51 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Our contribution
  4. Proof of Theorem \ref{['thm:main1']}
  5. Proof of Theorem \ref{['thm:main2']}
  6. Expansion Lemma
  7. Two rows
  8. Two columns
  9. Three rows
  10. Omitted Figures
  11. Branch diagram for the proof of Lemma \ref{['lemma:alldm=2']}
  12. Omitted Proofs
  13. Proof of Proposition \ref{['prop:sgna_1_eq_minus_sgna_2']}
  14. Proof of Proposition \ref{['prop:pq_in_R(I)_ver2row']}
  15. Proof of Proposition \ref{['prop:a(1-2s)=|a|_2row']}
  16. ...and 7 more sections

Key Result

Theorem 1

Suppose that $m \ge 4$ and $n \ge 3$. Then there exists a matrix $A \in \mathbb{R}^{[m]\times [n]}$ satisfying the following conditions. (i) $A$ does not have an EO. (ii) For all vectors $b \in \mathbb{R}^{[m]}$, the solution graph $G(R(A,b))$ is connected.

Figures (1)

  • Figure 1: The branch diagram for the proof of Lemma \ref{['lemma:alldm=2']}.

Theorems & Definitions (53)

  • Definition 1: Integer linear system
  • Definition 2: Hamming distance
  • Definition 3: Solution graph
  • Definition 4: Elimination
  • Definition 5: Eliminated matrix
  • Definition 6: Elimination ordering
  • Definition 7: Sign function
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • ...and 43 more