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An Algebraic Approach to the Longest Path Problem

Omar Al - Khazali

TL;DR

This work introduces an algebraic framework for the Longest Path Problem (LPP) based on a booleanization map $\beta$ applied to adjacency matrices, enabling exact polynomial-time solutions for trees, uniform block graphs, block graphs, and directed acyclic graphs (DAGs). By analyzing booleanized matrix powers $\beta(A(\Gamma)^{k})$, the authors derive algebraic criteria for the length of the longest path (e.g., $\mathfrak{D}(\Gamma) = \min \{ n : \beta(A(\Gamma)^{n+1}) = \beta(A(\Gamma)^{n-1}) \}$ for trees) and extend these to block-graph variants. They present binary-search-based algorithms to compute the longest path length and Kruskal-based methods to handle block graphs, along with path-generation algorithms that can enumerate all longest paths, not just one. While polynomial-time results are achieved for several graph classes, enumerating all longest paths in Block Graphs involves combinatorial growth tied to block sizes, highlighting both the power and limits of the algebraic approach. Overall, the paper demonstrates a rigorous algebraic route to LPP with exact correctness proofs and new tools for generating longest paths, offering a promising avenue for extending LPP-solutions to broader graph families.

Abstract

The Longest Path Problem is a question of finding the maximum length between pairs of vertices of a graph. In the general case, the problem is NP-complete. However, there is a small collection of graph classes for which there exists an efficient solution. Current approaches involve either approximation or computational enumeration. For Tree-like classes of graphs, there are approximation and enumeration algorithms which solves the problem efficiently. Despite this, we propose a new method of approaching the longest path problem with exact algebraic solutions that give rise to polynomial-time algorithms. Our method provides algorithms that are proven correct by their underlying algebraic operations unlike existing purely algorithmic solutions to this problem. We introduce a `booleanize' mapping on the adjacency matrix of a graph which we prove identifies the solution for trees, uniform block graphs, block graphs, and directed acyclic graphs with exact conditions and associated polynomial-time algorithms. In addition, we display additional algorithms that can generate every possible longest path of acyclic graphs in efficient time, as well as for block graphs.

An Algebraic Approach to the Longest Path Problem

TL;DR

This work introduces an algebraic framework for the Longest Path Problem (LPP) based on a booleanization map applied to adjacency matrices, enabling exact polynomial-time solutions for trees, uniform block graphs, block graphs, and directed acyclic graphs (DAGs). By analyzing booleanized matrix powers , the authors derive algebraic criteria for the length of the longest path (e.g., for trees) and extend these to block-graph variants. They present binary-search-based algorithms to compute the longest path length and Kruskal-based methods to handle block graphs, along with path-generation algorithms that can enumerate all longest paths, not just one. While polynomial-time results are achieved for several graph classes, enumerating all longest paths in Block Graphs involves combinatorial growth tied to block sizes, highlighting both the power and limits of the algebraic approach. Overall, the paper demonstrates a rigorous algebraic route to LPP with exact correctness proofs and new tools for generating longest paths, offering a promising avenue for extending LPP-solutions to broader graph families.

Abstract

The Longest Path Problem is a question of finding the maximum length between pairs of vertices of a graph. In the general case, the problem is NP-complete. However, there is a small collection of graph classes for which there exists an efficient solution. Current approaches involve either approximation or computational enumeration. For Tree-like classes of graphs, there are approximation and enumeration algorithms which solves the problem efficiently. Despite this, we propose a new method of approaching the longest path problem with exact algebraic solutions that give rise to polynomial-time algorithms. Our method provides algorithms that are proven correct by their underlying algebraic operations unlike existing purely algorithmic solutions to this problem. We introduce a `booleanize' mapping on the adjacency matrix of a graph which we prove identifies the solution for trees, uniform block graphs, block graphs, and directed acyclic graphs with exact conditions and associated polynomial-time algorithms. In addition, we display additional algorithms that can generate every possible longest path of acyclic graphs in efficient time, as well as for block graphs.
Paper Structure (7 sections, 20 theorems, 57 equations, 11 algorithms)

This paper contains 7 sections, 20 theorems, 57 equations, 11 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Longest Path Problem
  4. Longest Path Length Algorithms
  5. Path Generating Algorithms
  6. Complexity Analysis
  7. Concluding Remarks

Key Result

Lemma 1

Let $\mathbb{Z}_2^{n}$ be binary numbers with a logical-and operation $\land_l$ such that $(a \land_l b)_i := a_i \land_l b_i$. Let $f$ be a natural map from a matrix row or column vector (of 0's and 1's) to $\mathbb{Z}_2^{n}$. Then, the booleanized product can be computed equivalently by

Theorems & Definitions (48)

  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2: N. Biggs Biggs
  • Remark
  • Remark
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • ...and 38 more