The paper introduces a new algebraic approach to solving the Longest Path Problem (LPP), which is a well-known challenge in combinatorial optimization. The authors focus on identifying algebraic conditions that can exactly determine the length of the longest path in polynomial time for certain graph classes.
The key highlights and insights are:
The authors propose a "booleanize" mapping on the adjacency matrix of a graph, which they prove can identify the solution for trees, uniform block graphs, block graphs, and directed acyclic graphs (DAGs) under certain conditions.
For trees, the authors show that the length of the longest path can be found by identifying the minimum number n such that the booleanized matrix power β(A(Γ)^n+1) is equal to β(A(Γ)^n-1).
For uniform block graphs, the authors prove that the length of the longest path is n*(ω(Γ)-1), where n is the minimum number such that the booleanized matrix power β(A(Γ)^n) is a matrix of all 1's.
For block graphs, the authors provide a formula to compute the length of the longest path based on the booleanized matrix powers.
For directed acyclic graphs (DAGs), the authors show that the length of the longest path is the minimum number n such that the booleanized matrix power β(A(D)^n) is a matrix of all 0's.
The authors also provide algorithms to generate all the longest paths for each of the considered graph classes.
The paper includes a detailed complexity analysis of the proposed algorithms, showing that they have polynomial-time complexity.
The authors conclude that the algebraic approach presents a promising method for efficiently solving the LPP on graph classes for which polynomial solutions do not yet exist.
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arxiv.org
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by Omar Al - Kh... о arxiv.org 04-02-2024
https://arxiv.org/pdf/2312.11469.pdfГлибші Запити