Robust bounds on choosing from large tournaments

Journal of Graph Theory, 1998

We survey results concerning various generalizations of tournaments. The reader will see that tournaments are by no means the only class of directed graphs with a very rich structure. We describe, among numerous other topics mostly related to paths and cycles, results on hamiltonian paths and cycles. The reader will see that although these problems are polynomially solvable for all of the classes described, they can be highly non-trivial, even for these "tournament-like" digraphs.

2012

Let G be an undirected graph and let T be a tournament on the same vertex set as G. Define the cost of G relative to T to be ∑ u,v∈E(G) (T (u, v) + T (v, u)) + |E|, where T (u, v) denotes the number of two-step paths from u v, in T . In this paper, we determine for several classes of graphs which tournaments minimize the cost. Pelsmajer, et al. [5] conjecture that for each graph there is a transitive tournament that minimizes the graph’s cost. We prove that a transitive tournament minimizes the cost for complete graphs, nearly complete graphs, paths, star graphs, and cycles.

Discrete & Computational Geometry, 2013

We study a high-dimensional analog for the notion of an acyclic (aka transitive) tournament. We give upper and lower bounds on the number of d-dimensional n-vertex acyclic tournaments. In addition, we prove that every n-vertex d-dimensional tournament contains an acyclic subtournament of Ω(log 1/d n) vertices and the bound is tight. This statement for tournaments (i.e., the case d = 1) is a well-known fact. We indicate a connection between acyclic high-dimensional tournaments and Ramsey numbers of hypergraphs. We investigate as well the interrelations among various other notions of acyclicity in high-dimensional tournaments. These include combinatorial, geometric and topological concepts.

Journal of Combinatorial Theory, Series B, 1991

An n-partite tournament, n 3 2, or multipartite tournament is an oriented graph obtained by orienting each edge of a complete n-partite graph. The cycle structure of multipartite tournaments is investigated and properties of vertices with maximum score are studied.

The Electronic Journal of Combinatorics, 2000

This note extends a recent result of Kannan, Tetali and Vempala to completely solve, via a simple proof, the problem of random generation of a labeled tournament chain on the set of labeled tournaments with the same score vector.

The Electronic Journal of Combinatorics, 1995

We examine the size $s(n)$ of a smallest tournament having the arcs of an acyclic tournament on $n$ vertices as a minimum feedback arc set. Using an integer linear programming formulation we obtain lower bounds $s(n) \geq 3n - 2 - \lfloor \log_2 n \rfloor$ or $s(n) \geq 3n - 1 - \lfloor \log_2 n \rfloor$, depending on the binary expansion of $n$. When $n = 2^k - 2^t$ we show that the bounds are tight with $s(n) = 3n - 2 - \lfloor \log_2 n \rfloor$. One view of this problem is that if the 'teams' in a tournament are ranked to minimize inconsistencies there is some tournament with $s(n)$ 'teams' in which $n$ are ranked wrong. We will also pose some questions about conditions on feedback arc sets, motivated by our proofs, which ensure equality between the maximum number of arc disjoint cycles and the minimum size of a feedback arc set in a tournament.

We study the Maker-Breaker tournament game played on the edge set of a given graph $G$. Two players, Maker and Breaker claim unclaimed edges of $G$ in turns, and Maker wins if by the end of the game she claims all the edges of a pre-defined goal tournament. Given a tournament $T_k$ on $k$ vertices, we determine the threshold bias for the $(1:b)$ $T_k$-tournament game on $K_n$. We also look at the $(1:1)$ $T_k$-tournament game played on the edge set of a random graph ${\mathcal{G}_{n,p}}$ and determine the threshold probability for Maker's win. We compare these games with the clique game and discuss whether a random graph intuition is satisfied.

2011

A $k$-majority digraph is a directed graph created by combining $k$ individual rankings on the same ground set to form a consensus where edges point in the direction indicated by a strict majority of the rankings. The $k$-majority digraph is used to model voting scenarios, where the vertices correspond to options ranked by $k$ voters. When $k$ is odd, the resulting digraph is always a tournament, called $k$-majority tournament. Let $f_k(n)$ be the minimum, over all $k$-majority tournaments with $n$ vertices, of the maximum order of an induced transitive sub-tournament. Recently, Milans, Schreiber, and West proved that $\sqrt n \le f_3(n) \le 2 \sqrt n +1 $. In this paper, we improve the upper bound of $f_3(n)$ by showing that $f_3(n) < \sqrt {2n} +\frac 12 $.

Journal of Experimental Algorithmics (JEA), 2009

Ranking data is a fundamental organizational activity. Given advice, we may wish to rank a set of items to satisfy as much of that advice as possible. In the Feedback Arc Set (FAS) problem, advice takes the form of pairwise ordering statements, 'a should be ranked before b'. Instances in which there is advice about every pair of items is known as a tournament. This task is equivalent to ordering the nodes of a given directed graph to minimize the number of arcs pointing in one direction.

Graphs and Combinatorics, 2006

An n-partite tournament is an orientation of a complete n-partite graph. An npartite tournament is a tournament, if it contains exactly one vertex in each partite set. Douglas, Proc. London Math. Soc. 21 (1970) 716-730, obtained a characterization of strongly connected tournaments with exactly one Hamilton cycle (i.e., n-cycle). For n ≥ 3, we characterize strongly connected n-partite tournaments that are not tournaments with exactly one n-cycle. For n ≥ 5, we enumerate such non-isomorphic n-partite tournaments.

International Journal of Game Theory, 2005

Consider the problem of ranking social alternatives based on the number of voters that prefer one alternative to the other. Or consider the problem of ranking chess players by their past performance. A wide variety of ranking methods have been proposed to deal with these problems. Using six independent axioms, we characterize the fair-bets ranking method proposed by Daniels [4] and Moon and Pullman [14].

Lecture Notes in Computer Science, 2011

We design the first polynomial time approximation schemes (PTASs) for the Minimum Betweenness problem in tournaments and some related higher arity ranking problems. This settles the approximation status of the Betweenness problem in tournaments along with other ranking problems which were open for some time now. The results depend on a new technique of dealing with fragile ranking constraints and could be of independent interest.

SIAM Journal on Computing, 2001

We obtain a necessary and sufficient condition in terms of forbidden structures for tournaments to possess the min-max relation on packing and covering directed cycles, together with strongly polynomial time algorithms for the feedback vertex set problem and the cycle packing problem in this class of tournaments. Applying the local ratio technique of Bar-Yehuda and Even to the forbidden structures, we find a 2.5-approximation polynomial time algorithm for the feedback vertex set problem in any tournament.

Discrete Mathematics, 1989

The notions of rotational tournament and the associated symbol set are generalized to r-tournaments. It is shown that a necessary and sufficient condition for the existence of a rotational r-tournament on n vertices is (n, r) = 1. A scheme to generate rotational r'tournaments is given, along with some examples.

arXiv (Cornell University), 2018

A vertex x in a tournament T is called a king if for every vertex y of T there is a directed path from x to y of length at most 2. It is not hard to show that every vertex of maximum out-degree in a tournament is a king. However, tournaments may have kings which are not vertices of maximum out-degree. A binary inquiry asks for the orientation of the edge between a pair of vertices and receives the answer. The cost of finding a king in an unknown tournament is the number of binary inquiries required to detect a king. For the cost of finding a king in a tournament, in the worst case, Shen, Sheng and Wu (SIAM J. Comput., 2003) proved a lower and upper bounds of Ω(n 4/3) and O(n 3/2), respectively. In contrast to their result, we prove that the cost of finding a vertex of maximum out-degree is n 2 − O(n) in the worst case.