3,573 research outputs found

    Excluding pairs of tournaments

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    The Erd\H{o}s-Hajnal conjecture states that for every given undirected graph HH there exists a constant c(H)>0c(H)>0 such that every graph GG that does not contain HH as an induced subgraph contains a clique or a stable set of size at least V(G)c(H)|V(G)|^{c(H)}. The conjecture is still open. Its equivalent directed version states that for every given tournament HH there exists a constant c(H)>0c(H)>0 such that every HH-free tournament TT contains a transitive subtournament of order at least V(T)c(H)|V(T)|^{c(H)}. We prove in this paper that {H1,H2}\{H_{1},H_{2}\}-free tournaments TT contain transitive subtournaments of size at least V(T)c(H1,H2)|V(T)|^{c(H_{1},H_{2})} for some c(H1,H2)>0c(H_{1},H_{2})>0 and several pairs of tournaments: H1H_{1}, H2H_{2}. In particular we prove that {H,Hc}\{H,H^{c}\}-freeness implies existence of the polynomial-size transitive subtournaments for several tournaments HH for which the conjecture is still open (HcH^{c} stands for the \textit{complement of HH}). To the best of our knowledge these are first nontrivial results of this type

    Hereditary properties of combinatorial structures: posets and oriented graphs

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    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

    Who Can Win a Single-Elimination Tournament?

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    A single-elimination (SE) tournament is a popular way to select a winner in both sports competitions and in elections. A natural and well-studied question is the tournament fixing problem (TFP): given the set of all pairwise match outcomes, can a tournament organizer rig an SE tournament by adjusting the initial seeding so that their favorite player wins? We prove new sufficient conditions on the pairwise match outcome information and the favorite player, under which there is guaranteed to be a seeding where the player wins the tournament. Our results greatly generalize previous results. We also investigate the relationship between the set of players that can win an SE tournament under some seeding (so called SE winners) and other traditional tournament solutions. In addition, we generalize and strengthen prior work on probabilistic models for generating tournaments. For instance, we show that \emph{every} player in an nn player tournament generated by the Condorcet Random Model will be an SE winner even when the noise is as small as possible, p=Θ(lnn/n)p=\Theta(\ln n/n); prior work only had such results for pΩ(lnn/n)p\geq \Omega(\sqrt{\ln n/n}). We also establish new results for significantly more general generative models.Comment: A preliminary version appeared in Proceedings of the 30th AAAI Conference on Artificial Intelligence (AAAI), 201

    Algorithmic Aspects of a General Modular Decomposition Theory

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    A new general decomposition theory inspired from modular graph decomposition is presented. This helps unifying modular decomposition on different structures, including (but not restricted to) graphs. Moreover, even in the case of graphs, the terminology ``module'' not only captures the classical graph modules but also allows to handle 2-connected components, star-cutsets, and other vertex subsets. The main result is that most of the nice algorithmic tools developed for modular decomposition of graphs still apply efficiently on our generalisation of modules. Besides, when an essential axiom is satisfied, almost all the important properties can be retrieved. For this case, an algorithm given by Ehrenfeucht, Gabow, McConnell and Sullivan 1994 is generalised and yields a very efficient solution to the associated decomposition problem

    Incentives and Superstars on the LPGA Tour

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    Following Ehrenberg and Bognanno (1990a, b), this paper explores the role of incentives on the 2000 LPGA Tour. Overall, it finds them to have limited effectiveness. Several possible explanations are considered, including unmeasured differences in both abilities and courses and variations in the distribution of prizes across tournaments. The existence of a “superstar effect” is also considered.
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