An Algebraic Approach to the Longest Path Problem
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916340039680000 |
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| author | Khazali, Omar Al - |
| author_facet | Khazali, Omar Al - |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_11469 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An Algebraic Approach to the Longest Path Problem Khazali, Omar Al - Data Structures and Algorithms Discrete Mathematics Combinatorics 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. |
| title | An Algebraic Approach to the Longest Path Problem |
| topic | Data Structures and Algorithms Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2312.11469 |