The Parameterized Complexity of Packing Arc-Disjoint Cycles in Tournaments

R. Krithika; Abhishek Sahu; Saket Saurabh; Meirav Zehavi

doi:10.48550/arXiv.1802.07090

The Parameterized Complexity of Packing Arc-Disjoint Cycles in Tournaments

R. Krithika
, Abhishek Sahu
, Saket Saurabh
, Meirav Zehavi

Research output: Working paper/Preprint › Preprint

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Abstract

Given a directed graph D on n vertices and a positive integer k, the Arc-Disjoint
Cycle Packing problem is to determine whether D has k arc-disjoint cycles. This
problem is known to be W[1]-hard in general directed graphs. In this paper, we initiate a systematic study on the parameterized complexity of the problem restricted to tournaments. We show that the problem is fixed-parameter tractable and admits a polynomial kernel when parameterized by the solution size k. In particular, we show that it can be solved in 2 ^{O(k log k)}n^O(1) time and has a kernel with O(k) vertices. The primary ingredient in both these results is a min-max theorem that states that every tournament either contains k arc-disjoint triangles or has a feedback arc set of size at most 6k. Our belief is that this combinatorial result is of independent interest and
could be useful in other problems related to cycles in tournaments.

Original language	English
DOIs	https://doi.org/10.48550/arXiv.1802.07090
State	Published - Feb 2018

Keywords

Computer Science - Data Structures and Algorithms

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title = "The Parameterized Complexity of Packing Arc-Disjoint Cycles in Tournaments",

abstract = "Given a directed graph D on n vertices and a positive integer k, the Arc-Disjoint Cycle Packing problem is to determine whether D has k arc-disjoint cycles. This problem is known to be W[1]-hard in general directed graphs. In this paper, we initiate a systematic study on the parameterized complexity of the problem restricted to tournaments. We show that the problem is fixed-parameter tractable and admits a polynomial kernel when parameterized by the solution size k. In particular, we show that it can be solved in 2 O(k log k)nO(1) time and has a kernel with O(k) vertices. The primary ingredient in both these results is a min-max theorem that states that every tournament either contains k arc-disjoint triangles or has a feedback arc set of size at most 6k. Our belief is that this combinatorial result is of independent interest and could be useful in other problems related to cycles in tournaments.",

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author = "R. Krithika and Abhishek Sahu and Saket Saurabh and Meirav Zehavi",

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AU - Krithika, R.

AU - Sahu, Abhishek

AU - Saurabh, Saket

AU - Zehavi, Meirav

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Y1 - 2018/2

N2 - Given a directed graph D on n vertices and a positive integer k, the Arc-Disjoint Cycle Packing problem is to determine whether D has k arc-disjoint cycles. This problem is known to be W[1]-hard in general directed graphs. In this paper, we initiate a systematic study on the parameterized complexity of the problem restricted to tournaments. We show that the problem is fixed-parameter tractable and admits a polynomial kernel when parameterized by the solution size k. In particular, we show that it can be solved in 2 O(k log k)nO(1) time and has a kernel with O(k) vertices. The primary ingredient in both these results is a min-max theorem that states that every tournament either contains k arc-disjoint triangles or has a feedback arc set of size at most 6k. Our belief is that this combinatorial result is of independent interest and could be useful in other problems related to cycles in tournaments.

AB - Given a directed graph D on n vertices and a positive integer k, the Arc-Disjoint Cycle Packing problem is to determine whether D has k arc-disjoint cycles. This problem is known to be W[1]-hard in general directed graphs. In this paper, we initiate a systematic study on the parameterized complexity of the problem restricted to tournaments. We show that the problem is fixed-parameter tractable and admits a polynomial kernel when parameterized by the solution size k. In particular, we show that it can be solved in 2 O(k log k)nO(1) time and has a kernel with O(k) vertices. The primary ingredient in both these results is a min-max theorem that states that every tournament either contains k arc-disjoint triangles or has a feedback arc set of size at most 6k. Our belief is that this combinatorial result is of independent interest and could be useful in other problems related to cycles in tournaments.

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M3 - Preprint

BT - The Parameterized Complexity of Packing Arc-Disjoint Cycles in Tournaments

ER -

The Parameterized Complexity of Packing Arc-Disjoint Cycles in Tournaments

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