7,897 research outputs found
The Minimal Covering Set in Large Tournaments
Abstract We prove that in almost all large tournaments, the minimal covering set is the entire set of alternatives. That is, as the number of alternatives gets large, the probability that the minimal covering set of a uniformly chosen random tournament is the entire set of alternatives goes to one. By contrast, it follows from a result of Fisher and Reeves (1995) that the bipartisan set contains about half of the alternatives in almost all large tournaments
Minimal Stable Sets in Tournaments
We propose a systematic methodology for defining tournament solutions as
extensions of maximality. The central concepts of this methodology are maximal
qualified subsets and minimal stable sets. We thus obtain an infinite hierarchy
of tournament solutions, which encompasses the top cycle, the uncovered set,
the Banks set, the minimal covering set, the tournament equilibrium set, the
Copeland set, and the bipartisan set. Moreover, the hierarchy includes a new
tournament solution, the minimal extending set, which is conjectured to refine
both the minimal covering set and the Banks set.Comment: 29 pages, 4 figures, changed conten
Hitting minors, subdivisions, and immersions in tournaments
The Erd\H{o}s-P\'osa property relates parameters of covering and packing of
combinatorial structures and has been mostly studied in the setting of
undirected graphs. In this note, we use results of Chudnovsky, Fradkin, Kim,
and Seymour to show that, for every directed graph (resp.
strongly-connected directed graph ), the class of directed graphs that
contain as a strong minor (resp. butterfly minor, topological minor) has
the vertex-Erd\H{o}s-P\'osa property in the class of tournaments. We also prove
that if is a strongly-connected directed graph, the class of directed
graphs containing as an immersion has the edge-Erd\H{o}s-P\'osa property in
the class of tournaments.Comment: Accepted to Discrete Mathematics & Theoretical Computer Science.
Difference with the previous version: use of the DMTCS article class. For a
version with hyperlinks see the previous versio
Recognizing Members of the Tournament Equilibrium Set is NP-hard
A recurring theme in the mathematical social sciences is how to select the
"most desirable" elements given a binary dominance relation on a set of
alternatives. Schwartz's tournament equilibrium set (TEQ) ranks among the most
intriguing, but also among the most enigmatic, tournament solutions that have
been proposed so far in this context. Due to its unwieldy recursive definition,
little is known about TEQ. In particular, its monotonicity remains an open
problem up to date. Yet, if TEQ were to satisfy monotonicity, it would be a
very attractive tournament solution concept refining both the Banks set and
Dutta's minimal covering set. We show that the problem of deciding whether a
given alternative is contained in TEQ is NP-hard.Comment: 9 pages, 3 figure
Uncovered Sets
This paper covers the theory of the uncovered set used in the literatures on tournaments and spatial voting. I discern three main extant definitions, and I introduce two new concepts that bound exist- ing sets from above and below: the deep uncovered set and the shallow uncovered set. In a general topological setting, I provide relationships to other solutions and give results on existence and external stability for all of the covering concepts, and I establish continuity properties of the two new uncovered sets. Of note, I characterize each of the uncovered sets in terms of a decomposition into choices from externally stable sets; I define the minimal generalized covering solution, a nonempty refinement of the deep uncovered set that employs both of the new relations; and I define the acyclic Banks set, a nonempty generalization of the Banks set.
Computing Tournament Solutions using Relation Algebra and REL VIEW
We describe a simple computing technique for the tournament choice problem. It rests upon a relational modeling and uses the BDD-based computer system RelView for the evaluation of the relation-algebraic expressions that specify the solutions and for the visualization of the computed results. The Copeland set can immediately be identified using RelView's labeling feature. Relation-algebraic specifications of the Condorcet non-losers, the Schwartz set, the top cycle, the uncovered set, the minimal covering set, the Banks set, and the tournament equilibrium set are delivered. We present an example of a tournament on a small set of alternatives, for which the above choice sets are computed and visualized via RelView. The technique described in this paper is very flexible and especially appropriate for prototyping and experimentation, and as such very instructive for educational purposes. It can easily be applied to other problems of social choice and game theory.Tournament, relational algebra, RelView, Copeland set, Condorcet non-losers, Schwartz set, top cycle, uncovered set, minimal covering set, Banks set, tournament equilibrium set.
Density version of the Ramsey problem and the directed Ramsey problem
We discuss a variant of the Ramsey and the directed Ramsey problem. First,
consider a complete graph on vertices and a two-coloring of the edges such
that every edge is colored with at least one color and the number of bicolored
edges is given. The aim is to find the maximal size of a
monochromatic clique which is guaranteed by such a coloring. Analogously, in
the second problem we consider semicomplete digraph on vertices such that
the number of bi-oriented edges is given. The aim is to bound the
size of the maximal transitive subtournament that is guaranteed by such a
digraph.
Applying probabilistic and analytic tools and constructive methods we show
that if , (), then where only depend on , while if then . The latter case is
strongly connected to Tur\'an-type extremal graph theory.Comment: 17 pages. Further lower bound added in case $|E_{RB}|=|E_{bi}| =
p{n\choose 2}
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