Answering a question of Bang-Jensen and Thomassen [4], we prove that the minimum feedback arc set problem is NP-hard for tournaments. A feedback arc set (fas) in a digraph D = (V , A) is a set F of arcs such that D \ F is acyclic. The size of a minimum feedback arc set of D is denoted by mfas(D). A classical result of Lawler and Karp [5] asserts that finding a minimum feedback arc set in a digraph is NP-hard. Bang-Jensen and Thomassen [4] conjectured that finding a minimum fas in a tournament is also NP-hard. A very close answer was given by Ailon, Charikar and Newman in [1], where they prove that the problem is NP-hard under randomized reductions. Our approach is similar, but the reduction we use is simpler and therefore easily derandomized via parity-check matrices (see Alon and Spencer [3, p. 255]). Finally we prove that the minimum fas for tournaments is polynomially equivalent to the minimum fas for digraphs, and thus NP-hard. The following lemma is just Chebyshev inequality applied to the parity matrix of subset-intersection. Lemma 1. Let z be an integer. We denote by A the 2 z × 2 z matrix whose rows and columns are indexed by the subsets F i of {1,. .. , z} (in any order) and whose entries are a ij = (−1) |(F i ∩F j)|. For any subset J of r columns, we have 2 z i=1 j∈J a ij 2 z √ r.

We provide a general method to prove the existence and compute efficiently elimination orderings in graphs. Our method relies on several tools that were known before, but that were not put together so far: the algorithm LexBFS due to Rose, Tarjan and Lueker, one of its properties discovered by Berry and Bordat, and a local decomposition property of graphs discovered by Maffray, Trotignon and Vu\vskovi\'c. We use this method to prove the existence of elimination orderings in several classes of graphs, and to compute them in linear time. Some of the classes have already been studied, namely even-hole-free graphs, square-theta-free Berge graphs, universally signable graphs and wheel-free graphs. Some other classes are new. It turns out that all the classes that we study in this paper can be defined by excluding some of the so-called Truemper configurations. For several classes of graphs, we obtain directly bounds on the chromatic number, or fast algorithms for the maximum clique problem or the coloring problem

We propose a polynomial-time algorithm which takes as input a finite set of points of R 3 and computes, up to arbitrary precision, a maximum subset with diameter at most 1. More precisely, we give the first randomized EPTAS and deterministic PTAS for Maximum Clique in unit ball graphs. Our approximation algorithm also works on disk graphs with arbitrary radii, in the plane.Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Clique on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. Recently, it was shown that the disjoint union of two odd cycles is never the complement of a disk graph [Bonnet, Giannopoulos, Kim, Rzążewski, Sikora; SoCG '18]. This enabled the authors to derive a QPTAS and a subexponential algorithm for Max Clique on disk graphs. In this paper, we improve the approximability to a randomized EPTAS (and a deterministic PTAS). More precisely, we obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VCdimension, and linear independence number. We then address the question of computing Max Clique for disks in higher dimensions. We show that intersection graphs of unit balls, like disk graphs, do not admit the complement of two odd cycles as an induced subgraph. This, in combination with the first result, straightforwardly yields a randomized EPTAS for Max Clique on unit ball graphs. In stark contrast, we show that on ball graphs and unit 4-dimensional disk graphs, Max Clique is NP-hard and does not admit an approximation scheme even in subexponential-time, unless the Exponential Time Hypothesis fails.