A 7/3-Approximation for Feedback Vertex Sets in Tournaments

We consider the minimum-weight feedback vertex set problem in tournaments: given a tournament with non-negative vertex weights, remove a minimum-weight set of vertices that intersects all cycles. This problem is NP-hard to solve exactly, and Unique Games-hard to approximate by a factor better than 2. We present the first 7/3 approximation algorithm for this problem, improving on the previously best known ratio 5/2 given by Cai et al. [FOCS 1998, SICOMP 2001]

Feedback Vertex Sets in Tournaments

We study combinatorial and algorithmic questions around minimal feedback vertex sets in tournament graphs. On the combinatorial side, we derive strong upper and lower bounds on the maximum number of minimal feedback vertex sets in an n-vertex tournament. We prove that every tournament on n vertices has at most 1.6740^n minimal feedback vertex sets, and that there is an infinite family of tournaments, all having at least 1.5448^n minimal feedback vertex sets. This improves and extends the bounds of Moon (1971). On the algorithmic side, we design the first polynomial space algorithm that enumerates the minimal feedback vertex sets of a tournament with polynomial delay. The combination of our results yields the fastest known algorithm for finding a minimum size feedback vertex set in a tournament

A 7/3-approximation for feedback vertex sets in tournaments

We consider the minimum-weight feedback vertex set problem in tournaments: given a tournament with non-negative vertex weights, remove a minimum-weight set of vertices that intersects all cycles. This problem is NP-hard to solve exactly, and Unique Games-hard to approximate by a factor better than 2. We present the first 7/3 approximation algorithm for this problem, improving on the previously best known ratio 5/2 given by Cai et al. [FOCS 1998, SICOMP 2001]

Hereditary properties of combinatorial structures: posets and oriented graphs

A hereditary property of combinatorial structures is a collection of structures (e.g. graphs, posets) which is closed under isomorphism, closed under taking induced substructures (e.g. induced subgraphs), and contains arbitrarily large structures. Given a property P, we write P_n for the collection of distinct (i.e., non-isomorphic) structures in a property P with n vertices, and call the function n -> |P_n| the speed (or unlabelled speed) of P. Also, we write P^n for the collection of distinct labelled structures in P with vertices labelled 1,...,n, and call the function n -> |P^n| the labelled speed of P. The possible labelled speeds of a hereditary property of graphs have been extensively studied, and the aim of this paper is to investigate the possible speeds of other combinatorial structures, namely posets and oriented graphs. More precisely, we show that (for sufficiently large n), the labelled speed of a hereditary property of posets is either 1, or exactly a polynomial, or at least 2^n - 1. We also show that there is an initial jump in the possible unlabelled speeds of hereditary properties of posets, tournaments and directed graphs, from bounded to linear speed, and give a sharp lower bound on the possible linear speeds in each case.Comment: 26 pgs, no figure

Coloring Tournaments with Few Colors: Algorithms and Complexity

A k-coloring of a tournament is a partition of its vertices into k acyclic sets. Deciding if a tournament is 2-colorable is NP-hard. A natural problem, akin to that of coloring a 3-colorable graph with few colors, is to color a 2-colorable tournament with few colors. This problem does not seem to have been addressed before, although it is a special case of coloring a 2-colorable 3-uniform hypergraph with few colors, which is a well-studied problem with super-constant lower bounds. We present an efficient decomposition lemma for tournaments and show that it can be used to design polynomial-time algorithms to color various classes of tournaments with few colors, including an algorithm to color a 2-colorable tournament with ten colors. For the classes of tournaments considered, we complement our upper bounds with strengthened lower bounds, painting a comprehensive picture of the algorithmic and complexity aspects of coloring tournaments

Splitting a tournament into two subtournaments with given minimum outdegree

A {\it $(k_1,k_2)$-outdegree-splitting} of a digraph $D$ is a partition $(V_1,V_2)$ of its vertex set such that $D[V_1]$ and $D[V_2]$ have minimum outdegree at least $k_1$ and $k_2$, respectively. We show that there exists a minimum function $f_T$ such that every tournament of minimum outdegree at least $f_T(k_1,k_2)$ has a $(k_1,k_2)$-outdegree-splitting, and $f_T(k_1,k_2) \leq k_1^2/2+3k_1/2 +k_2+1$. We also show a polynomial-time algorithm that finds a $(k_1,k_2)$-outdegree-splitting of a tournament if one exists, and returns 'no' otherwise. We give better bound on $f_T$ and faster algorithms when $k_1=1$.Un {\it $(k_1,k_2)$-partage} d'un digraphe $D$ est une partition $(V_1,V_2)$ de son ensemble de sommets telle que $D[V_1]$ et $D[V_2]$ soient de degréß sortant minimum au moins $k_1$ et $k_2$, respectivement. Nous établissons l'existence d'une fonction (minimum) $f_T$ telle que tout tournoi de degré sortant minimum au moins $f_T(k_1,k_2)$ a un $(k_1,k_2)$-partage, et que $f_T(k_1,k_2) \leq k_1^2/2+3k_1/2 +k_2+1$. Nous donnons également un algorithme en temps polynomial qui trouve un $(k_1,k_2)$-partage d'un tournoi s'il en existe un et renvoie 'non' sinon. Nous donnons de meilleures bornes sur $f_T$ et des algorithmes plus rapides pour $k_1=1$

Edge disjoint Hamiltonian cycles in highly connected tournaments

Thomassen conjectured that there is a function f(k) such that every strongly f(k)-connected tournament contains k edge-disjoint Hamiltonian cycles. This conjecture was recently proved by Kühn, Lapinskas, Osthus, and Patel who showed that f(k) ≤ O(k 2 (logk) 2 ) and conjectured that there is a constant C such that f(k) ≤ Ck 2 . We prove this conjecture. As a second application of our methods we answer a question of Thomassen about spanning linkages in highly connected tournaments