Acyclic Tournaments Research Articles

P.013 The neural signature of confidence abnormalities in compulsive disorders: obsessive-compulsive disorder and gambling disorder

We show that Γ-free fillings and lonesum fillings of Ferrers shapes are equinumerous by applying a previously defined bijection on matrices for this more general case and by constructing a new bijection between Callan sequences and Dumont-like permutations. As an application, we give a new combinatorial interpretation of Genocchi numbers in terms of Callan sequences.Further, we recover some of Hetyei’s results on alternating acyclic tournaments. Finally, we present an interesting result in the case of certain other special shapes.

Lonesum and [formula omitted]-free [formula omitted]-[formula omitted] fillings of Ferrers shapes

We show that Γ-free fillings and lonesum fillings of Ferrers shapes are equinumerous by applying a previously defined bijection on matrices for this more general case and by constructing a new bijection between Callan sequences and Dumont-like permutations. As an application, we give a new combinatorial interpretation of Genocchi numbers in terms of Callan sequences.Further, we recover some of Hetyei’s results on alternating acyclic tournaments. Finally, we present an interesting result in the case of certain other special shapes.

On the graph Laplacian and the rankability of data

Recently, Anderson et al. (2019) proposed the concept of rankability, which refers to a dataset's inherent ability to produce a meaningful ranking of its items. In the same paper, they proposed a rankability measure that is based on an integer program for computing the minimum number of edge changes made to a directed graph in order to obtain a complete dominance graph, i.e., an acyclic tournament graph. In this article, we prove a spectral-degree characterization of complete dominance graphs and apply this characterization to produce a new measure of rankability that is cost-effective and more widely applicable. We support the details of our algorithm with several results regarding the conditioning of the Laplacian spectrum of complete dominance graphs and the Hausdorff distance between their Laplacian spectrum and that of an arbitrary directed graph with weights between zero and one. Finally, we analyze the rankability of datasets from the world of chess and college football.

Alternation acyclic tournaments

We define a tournament to be alternation acyclic if it does not contain a cycle in which descents and ascents alternate. Using a result by Athanasiadis on hyperplane arrangements, we show that these tournaments are counted by the median Genocchi numbers. By establishing a bijection with objects defined by Dumont, we show that alternation acyclic tournaments in which at least one ascent begins at each vertex, except for the largest one, are counted by the Genocchi numbers of the first kind. Unexpected consequences of our results include a pair of ordinary generating function formulas for the Genocchi numbers of both kinds and a simple model for the normalized median Genocchi numbers.

Combinatorial Interpretations of the Kreweras Triangle in Terms of Subset Tuples

We show how the combinatorial interpretation of the normalized median Genocchi numbers in terms of multiset tuples, defined by Hetyei in his study of the alternation acyclic tournaments, is bijectively equivalent to previous models like the normalized Dumont permutations or the Dellac configurations, and we extend the interpretation to the Kreweras triangle.

Diletter circular codes over finite alphabets

The graph approach of circular codes recently developed (Fimmel et al., 2016) allows here a detailed study of diletter circular codes over finite alphabets. A new class of circular codes is identified, strong comma-free codes. New theorems are proved with the diletter circular codes of maximal length in relation to (i) a characterisation of their graphs as acyclic tournaments; (ii) their explicit description; and (iii) the non-existence of other maximal diletter circular codes. The maximal lengths of paths in the graphs of the comma-free and strong comma-free codes are determined. Furthermore, for the first time, diletter circular codes are enumerated over finite alphabets. Biological consequences of dinucleotide circular codes are analysed with respect to their embedding in the trinucleotide circular code X identified in genes and to the periodicity modulo 2 observed in introns. An evolutionary hypothesis of circular codes is also proposed according to their combinatorial properties.

A Note on the Feedback Arc Set Problem and Acyclic Subdigraphs in Bipartite Tournaments

Given a digraph D a feedback arc set is a subset X of the arcs of D such that D − X is acyclic. Let β(D) denote de minimum cardinality of a feedback arc set of D. In this paper we prove that a bipartite tournament T with minimum out-degree at least r satisfies β(T) ≥ r2. A lower bound and an upper bound for β(T) are given in terms of the bipartite dichromatic number. We define the bipartite dichromatic number of a balanced bipartite tournament Tn,n and use this invariant to give an upper bound for the minimum cardinality of a feedback arc set of Tn,n. We also prove that for each positive integer k ≥ 3 there is an integer N(k) such that if n ≥ N(k), then each balanced bipartite tournament contains an acyclic bipartite tournament Tk,k.

Rigging Nearly Acyclic Tournaments Is Fixed-Parameter Tractable

Single-elimination tournaments (or knockout tournaments) are a popular format in sports competitions that is also widely used for decision making and elections. In this paper we study the algorithmic problem of manipulating the outcome of a tournament. More specifically, we study the problem of finding a seeding of the players such that a certain player wins the resulting tournament. The problem is known to be NP-hard in general. In this paper we present an algorithm for this problem that exploits structural restrictions on the tournament. More specifically, we establish that the problem is fixed-parameter tractable when parameterized by the size of a smallest feedback arc set of the tournament (interpreting the tournament as an oriented complete graph). This is a natural parameter because most problems on tournaments (including this one) are either trivial or easily solvable on acyclic tournaments, leading to the question — what about nearly acyclic tournaments or tournaments with a small feedback arc set? Our result significantly improves upon a recent algorithm by Aziz et al. (2014) whose running time is bounded by an exponential function where the size of a smallest feedback arc set appears in the exponent and the base is the number of players.

Great Deluge Algorithm for the Linear Ordering Problem: The Case of Tanzanian Input-Output Table

Given a weighted complete digraph, the Linear Ordering Problem (LOP) consists of finding and acyclic tournament with maximum weight. It is sometimes referred to as triangulation problem or permutation problem depending on the context of its application. This study introduces an algorithm for LOP and applied for triangulation of Tanzanian Input-Output tables. The algorithm development process uses Great Deluge heuristic method. It is implemented using C++ programming language and tested on a personal computer with 2.40GHZ speed processor. The algorithm has been able to triangulate the Tanzanian input-output tables of size 79×79 within a reasonable time (1.17 seconds). It has been able to order the corresponding economic sectors in the linear order, with upper triangle weight increased from 585,481 to 839,842 giving the degree of linearity of 94.3%. Index Terms — Optimization, Linear Ordering, Input-Output Tables, Great Deluge Algorithm

Induced acyclic tournaments in random digraphs: Sharp concentration, thresholds and algorithms

Given a simple directed graph D = (V,A), let the size of the largest induced acyclic tournament be denoted by mat(D). Let D ∈ D(n, p) (with p = p(n)) be a random instance, obtained by randomly orienting each edge of a random graph drawn from G(n, 2p). We show that mat(D) is asymptotically almost surely (a.a.s.) one of only 2 possible values, namely either b∗ or b∗ + 1, where b∗ = ⌊2(log r n) + 0.5⌋ and r = p. It is also shown that if, asymptotically, 2(log r n) + 1 is not within a distance of w(n)/(lnn) (for any sufficiently slow w(n) → ∞) from an integer, then mat(D) is ⌊2(log r n) + 1⌋ a.a.s. As a consequence, it is shown that mat(D) is 1-point concentrated for all n belonging to a subset of positive integers of density 1 if p is independent of n. It is also shown that there are functions p = p(n) for which mat(D) is provably not concentrated in a single value. We also establish thresholds (on p) for the existence of induced acyclic tournaments of size i which are sharp for i = i(n) → ∞. We also analyze a polynomial time heuristic and show that it produces a solution whose size is at least log r n + Θ( √ log r n). Our results are valid as long as p ≥ 1/n. All of these results also carry over (with some slight changes) to a related model which allows 2-cycles.

On High-Dimensional Acyclic Tournaments

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 $\Omega(\log^{1/d}n)$ 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 inter-relations among various other notions of acyclicity in high-dimensional to tournaments. These include combinatorial, geometric and topological concepts.

On the number of update digraphs and its relation with the feedback arc sets and tournaments

An update digraph corresponds to a labeled digraph that indicates a relative order of its nodes introduced to define equivalence classes of deterministic update schedules yielding the same dynamical behavior of a Boolean network. In Aracena et al. [1], the authors exhibited relationships between update digraphs and the feedback arc sets of a given digraph G. In this paper, we delve into the study of these relations. Specifically, we show differences and similarities between both sets through increasing and decreasing monotony properties in terms of their structural characteristics. Besides, we prove that these sets are equivalent if and only if all the digraph circuits are cycles. On the other hand, we characterize the minimal feedback arc sets of a given digraph in terms of their associated update digraphs. In particular, for complete digraphs, this characterization shows a close relation with acyclic tournaments. For the latter, we show that the size of the associated equivalence classes is a power of two. Finally, we determine exactly the number of update digraphs associated to digraphs containing a tournament.

A benchmark library and a comparison of heuristic methods for the linear ordering problem

The linear ordering problem consists of finding an acyclic tournament in a complete weighted digraph of maximum weight. It is one of the classical NP-hard combinatorial optimization problems. This paper surveys a collection of heuristics and metaheuristic algorithms for finding near-optimal solutions and reports about extensive computational experiments with them. We also present the new benchmark library LOLIB which includes all instances previously used for this problem as well as new ones.

Tabu search for the linear ordering problem with cumulative costs

Given a matrix of weights, the Linear Ordering Problem (LOP) consists of finding a permutation of the columns and rows in order to maximize the sum of the weights in the upper triangle. This well known NP-complete problem can also be formulated on a complete weighted graph, where the objective is to find an acyclic tournament that maximizes the sum of arc weights. The variant of the LOP that we target here was recently introduced and adds a cumulative non-linear propagation of the costs to the sum of the arc weights. We first review the previous methods for the LOP and for this variant with cumulative costs (LOPCC) and then propose a heuristic algorithm for the LOPCC, which is based on the Tabu Search (TS) methodology. Our method achieves search intensification and diversification through the implementation of both short and long term memory structures. Our extensive experimentation with 224 instances shows that the proposed procedure outperforms existing methods in terms of solution quality and has reasonable computing-time requirements.

Les graphes critiquement sans duo

Let G = ( V , A ) be a digraph. G is a tournament if for x , y ∈ V , ( x , y ) ∈ A if and only if ( y , x ) ∉ A . The induced subgraph of G by a subset X of V is denoted by G [ X ] . A subset I of V is an interval of G provided that for any a , b ∈ I and x ∈ V \\ I , ( a , x ) ∈ A if and only if ( b , x ) ∈ A , and ( x , a ) ∈ A if and only if ( x , b ) ∈ A . An interval of G on 2 elements is called duo of G. G is called critically without duo if it is without duo and for all x ∈ V , the subgraph G [ V \\ { x } ] has at least one duo. Following the study of acyclic tournaments made by J.F. Culus and B. Jouve in 2005, we give, in this Note, a complete morphologic description of critically without duo graphs. To cite this article: Y. Boudabbous, A. Salhi, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Smallest tournaments not realizable by $${\frac{2}{3}}$$ -majority voting

Define the predictability number α(T) of a tournament T to be the largest supermajority threshold \({\frac{1}{2} < \alpha\leq 1}\) for which T could represent the pairwise voting outcomes from some population of voter preference orders. We establish that the predictability number always exists and is rational. Only acyclic tournaments have predictability 1; the Condorcet voting paradox tournament has predictability \({\frac{2}{3}}\); Gilboa has found a tournament on 54 alternatives (i.e. vertices) that has predictability less than \({\frac{2}{3}}\) , and has asked whether a smaller such tournament exists. We exhibit an 8-vertex tournament that has predictability \({\frac{13}{20}}\) , and prove that it is the smallest tournament with predictability < \({\frac{2}{3}}\) . Our methodology is to formulate the problem as a finite set of two-person zero-sum games, employ the minimax duality and linear programming basic solution theorems, and solve using rational arithmetic.

The Linear Ordering Problem with cumulative costs

The optimization problem of finding a permutation of a given set of items that minimizes a certain cost function is naturally modeled by introducing a complete digraph G whose vertices correspond to the items to be sorted. Depending on the cost function to be used, different optimization problems can be defined on G. The most familiar one is the min-cost Hamiltonian path problem (or its closed-path version, the Traveling Salesman Problem), arising when the cost of a given permutation only depends on consecutive node pairs. A more complex situation arises when a given cost has to be paid whenever an item is ranked before another one in the final permutation. In this case, a feasible solution is associated with an acyclic tournament (the transitive closure of an Hamiltonian path), and the resulting problem is known as the Linear Ordering Problem (LOP). In this paper we introduce and study a relevant case of LOP arising when the overall permutation cost can be expressed as the sum of terms α u associated with each item u, each defined as a linear combination of the values α v of all items v that follow u in the permutation. This setting implies a cumulative (nonlinear) propagation of the value of variables α v along the node permutation, hence the name Linear Ordering Problem with Cumulative Costs. We illustrate the practical application in wireless telecommunication system that motivated the present study. We prove complexity results, and propose a Mixed-Integer Linear Programming model as well as an ad hoc enumerative algorithm for the exact solution of the problem. A dynamic-programming heuristic is also described. Extensive computational results on large sets of instances are presented, showing that the proposed techniques are capable of solving, in reasonable computing times, all the instances coming from our application.

Extension of Arrow's theorem to symmetric sets of tournaments

Arrow's impossibility theorem [K.J. Arrow, Social Choice and Individual Values, Wiley, New York, NY, 1951] shows that the set of acyclic tournaments is not closed to non-dictatorial Boolean aggregation. In this paper we extend the notion of aggregation to general tournaments and we show that for tournaments with four vertices or more any proper symmetric (closed to vertex permutations) subset cannot be closed to non-dictatorial monotone aggregation and to non-neutral aggregation. We also demonstrate a proper subset of tournaments that is closed to parity aggregation for an arbitrarily large number of vertices. This proves a conjecture of Kalai [Social choice without rationality, Reviewed NAJ Economics 3(4)] for the non-neutral and the non-dictatorial and monotone cases and gives a counter example for the general case.

Variable neighborhood search for the linear ordering problem

Given a matrix of weights, the linear ordering problem (LOP) consists of finding a permutation of the columns and rows in order to maximize the sum of the weights in the upper triangle. This NP-complete problem can also be formulated in terms of graphs, as finding an acyclic tournament with a maximal sum of arc weights in a complete weighted graph. In this paper, we first review the previous methods for the LOP and then propose a heuristic algorithm based on the variable neighborhood search (VNS) methodology. The method combines different neighborhoods for an efficient exploration of the search space. We explore different search strategies and propose a hybrid method in which the VNS is coupled with a short-term tabu search for improved outcomes. Our extensive experimentation with both real and random instances shows that the proposed procedure competes with the best-known algorithms in terms of solution quality, and has reasonable computing-time requirements. Variable neighborhood search (VNS) is a metaheuristic method that has recently been shown to yield promising outcomes for solving combinatorial optimization problems. Based on a systematic change of neighborhood in a local search procedure, VNS uses both deterministic and random strategies in search for the global optimum. In this paper, we present a VNS implementation designed to find high quality solutions for the NP-hard LOP, which has a significant number of applications in practice. The LOP, for example, is equivalent to the so-called triangulation problem for input–output tables in economics. Our implementation incorporates innovative mechanisms to include memory structures within the VNS methodology. Moreover we study the hybridization with other methodologies such as tabu search.

$k$-Colour Partitions of Acyclic Tournaments

Let $G_{1}$ be the acyclic tournament with the topological sort $0 &lt; 1 &lt; 2 &lt; \dots &lt; n &lt; n+1$ defined on node set $N\cup \{0,n+1\}$, where $N=\{1,2,\dots,n\}$. For integer $k\geq 2$, let $G_{k}$ be the graph obtained by taking $k$ copies of every arc in $G_{1}$ and colouring every copy with one of $k$ different colours. A $k$-colour partition of $N$ is a set of $k$ paths from 0 to $n+1$ such that all arcs of each path have the same colour, different paths have different colours, and every node of $N$ is included in exactly one path. If there are costs associated with the arcs of $G_{k}$, the cost of a $k$-colour partition is the sum of the costs of its arcs. For determining minimum cost $k$-colour partitions we describe an $O(k^{2}n^{2k})$ algorithm, and show this is an NP-$hard$ problem. We also study the convex hull of the incidence vectors of $k$-colour partitions. We derive the dimension, and establish a minimal equality set. For $k&gt;2$ we identify a class of facet inducing inequalities. For $k=2$ we show that these inequalities turn out to be equations, and that no other facet defining inequalities exists besides the trivial nonnegativity constraints.