An independent set in a graph \(G\) is a set of pairwise non-adjacent vertices. A graph \(G\) is bipartite if its vertex set can be partitioned into two independent sets. In the Odd Cycle Transversal problem, the input is a graph \(G\) along with a weight function w associating a rational weight with each vertex, and the task is to find a minimum weight vertex subset \(S\) in \(G\) such that \(G-S\) is bipartite; the weight of \(S\) , \(\text{w}(S)=\sum_{v\in S}\text{w}(v)\) . We show that Odd Cycle Transversal is polynomial-time solvable on graphs excluding \(P_{5}\) (a path on five vertices) as an induced subgraph. The problem was previously known to be polynomial-time solvable on \(P_{4}\) -free graphs and NP -hard on \(P_{6}\) -free graphs [Dabrowski, Feghali, Johnson, Paesani, Paulusma and Rzążewski, Algorithmica 2020]. Bonamy, Dabrowski, Feghali, Johnson and Paulusma [Algorithmica 2019] posed the existence of a polynomial-time algorithm on \(P_{5}\) -free graphs as an open problem. This was later re-stated by Rzążewski [Dagstuhl Reports, 9(6): 2019], by Chudnovsky, King, Pilipczuk, Rzążewski, and Spirkl [SIDMA 2021] who gave an algorithm with running time \(n^{O(\sqrt{n})}\) for the problem, and by Agrawal, Lima, Lokshtanov, Saurabh, and Sharma [SODA 2024] who gave a quasi-polynomial time algorithm.

Akanksha Agrawal

Paloma T. Lima

Daniel Lokshtanov

Paweł Rzążewski

Saket Saurabh

Roohani Sharma

ACM Transactions on Algorithms

ACM TRANSACTIONS ON ALGORITHMS 欢迎提交最高质量的原始研究，处理本质上离散和有限的算法，并且在目标或分析中以自然的方式具有数学内容。最受欢迎的是新算法和数据结构、新的和改进的分析以及复杂性结果。该杂志涵盖的具体计算领域包括组合搜索和对象；计数；离散优化和近似；随机化和量子计算；并行和分布式计算；图形算法、几何、算术、数论、字符串；在线分析；密码学；编码;数据压缩;学习算法;算法分析方法;离散算法应用于生物学、经济学、博弈论、通信、计算机系统与体系结构、硬件设计、科学计算等应用领域

ACM Transactions on Algorithms welcomes submissions of high-quality original research that deals with discrete and finite algorithms, naturally incorporating mathematical content in its objectives or analyses. We particularly encourage new algorithms and data structures, novel and improved analyses, and complexity results. The specific computational areas covered by the journal include combinatorial search and objects, counting, discrete optimization and approximation, randomized and quantum computing, parallel and distributed computing, graph algorithms, geometry, arithmetic, number theory, string processing, online analysis, cryptography, coding, data compression, learning algorithms, algorithmic analysis methods, and the application of discrete algorithms in fields such as biology, economics, game theory, communications, computer systems and architecture, hardware design, and scientific computing.

《ACM Transactions on Algorithms》欢迎提交高质量的原创研究，主要涉及离散和有限算法，且在目标或分析中自然包含数学内容。我们特别欢迎新算法和数据结构的提出、新的和改进的分析方法，以及复杂性结果的研究。该期刊涵盖的具体计算领域包括组合搜索与对象、计数、离散优化与近似、随机化与量子计算、并行与分布式计算、图算法、几何、算术、数论、字符串处理、在线分析、密码学、编码、数据压缩、学习算法、算法分析方法，以及离散算法在生物学、经济学、博弈论、通信、计算机系统与体系结构、硬件设计、科学计算等领域的应用。

Category	Quartile
计算机科学	3
计算机科学, 计算机理论方法	2
计算机科学, 应用数学	2

Odd Cycle Transversal on \(P_{5}\) -free Graphs in Polynomial Time

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Journal Info

Category	Quartile
MATHEMATICS, APPLIED	3