All Feedback Arc Sets of a Random Turán Tournament Have $\lfloor {n}/{k}\rfloor-{k}+1$ Disjoint ${k}$-Cliques (and This Is Tight)

Arc-Disjoint Cycles and Feedback Arc Sets

Isaak posed the following problem. Suppose T is a tournament having a minimum feedback arc set which induces an acyclic digraph with a hamiltonian path. Is it true that the maximum number of arc-disjoint cycles in T equals the cardinality of minimum feedback arc set of T? We prove that the answer to the problem is in the negative. Further, we study the number of arc-disjoint cycles through a ve...

Turán numbers of vertex-disjoint cliques in -partite graphs

(Arc-disjoint) cycle packing in tournament: classical and parameterized complexity

Given a tournament T , the problem MaxCT consists of finding a maximum (arc-disjoint) cycle packing of T . In the same way, MaxTT corresponds to the specific case where the collection of cycles are triangles (i.e. directed 3-cycles). Although MaxCT can be seen as the LP dual of minimum feedback arc set in tournaments which have been widely studied, surprisingly no algorithmic results seem to ex...

Efficient Simulation of a Random Knockout Tournament

We consider the problem of using simulation to efficiently estimate the win probabilities for participants in a general random knockout tournament. Both of our proposed estimators, one based on the notion of “observed survivals” and the other based on conditional expectation and post-stratification, are highly effective in terms of variance reduction when compared to the raw simulation estimato...

Disjoint cliques and disjoint maximal independent sets of vertices in graphs

In this paper, we find lower bounds for the maximum and minimum numbers of cliques in maximal sets of pairwise disjoint cliques in a graph . By complementation, these yield lower bounds for the maximum and minimum numbers of independent sets in maximal sets of pairwise disjoint maximal independent sets of vertices in a graph . In the latter context, we show by examples that one of our bounds is...