Revolutionaries and spies on random graphs

Pursuit-evasion games, such as the game of Revolutionaries and Spies, are a simplified model for network security. In the game we consider in this paper, a team of $r$ revolutionaries tries to hold an unguarded meeting consisting of $m$ revolutionaries. A team of $s$ spies wants to prevent this forever. For given $r$ and $m$, the minimum number of spies required to win on a graph $G$ is the spy number $\sigma(G,r,m)$. We present asymptotic results for the game played on random graphs $G(n,p)$ for a large range of $p = p(n), r=r(n)$, and $m=m(n)$. The behaviour of the spy number is analyzed completely for dense graphs (that is, graphs with average degree at least n^{1/2+\eps} for some \eps > 0). For sparser graphs, some bounds are provided

Approximation bounds on maximum edge 2-coloring of dense graphs

For a graph $G$ and integer $q\geq 2$, an edge $q$-coloring of $G$ is an assignment of colors to edges of $G$, such that edges incident on a vertex span at most $q$ distinct colors. The maximum edge $q$-coloring problem seeks to maximize the number of colors in an edge $q$-coloring of a graph $G$. The problem has been studied in combinatorics in the context of {\em anti-Ramsey} numbers. Algorithmically, the problem is NP-Hard for $q\geq 2$ and assuming the unique games conjecture, it cannot be approximated in polynomial time to a factor less than $1+1/q$. The case $q=2$, is particularly relevant in practice, and has been well studied from the view point of approximation algorithms. A $2$-factor algorithm is known for general graphs, and recently a $5/3$-factor approximation bound was shown for graphs with perfect matching. The algorithm (which we refer to as the matching based algorithm) is as follows: "Find a maximum matching $M$ of $G$. Give distinct colors to the edges of $M$. Let $C_1,C_2,\ldots, C_t$ be the connected components that results when M is removed from G. To all edges of $C_i$ give the $(|M|+i)$th color." In this paper, we first show that the approximation guarantee of the matching based algorithm is $(1 + \frac {2} {\delta})$ for graphs with perfect matching and minimum degree $\delta$. For $\delta \ge 4$, this is better than the $\frac {5} {3}$ approximation guarantee proved in {AAAP}. For triangle free graphs with perfect matching, we prove that the approximation factor is $(1 + \frac {1}{\delta - 1})$, which is better than $5/3$ for $\delta \ge 3$.Comment: 11pages, 3 figure

Size Ramsey Number of Bounded Degree Graphs for Games

We study Maker/Breaker games on the edges of sparse graphs. Maker and Breaker take turns at claiming previously unclaimed edges of a given graph H. Maker aims to occupy a given target graph G and Breaker tries to prevent Maker from achieving his goal. We show that for every d there is a constant c = c(d) with the property that for every graph G on n vertices of maximum degree d there is a graph H on at most cn edges such that Maker has a strategy to occupy a copy of G in the game on H. This is a result about a game-theoretic variant of the size Ramsey number. For a given graph G, $\hat{r}'(G)$ is defined as the smallest number M for which there exists a graph H with M edges such that Maker has a strategy to occupy a copy of G in the game on H. In this language, our result yields that for every connected graph G of constant maximum degree, $\hat{r}'(G) = \Theta(n)$ . Moreover, we can also use our method to settle the corresponding extremal number for universal graphs: for a constant d and for the class ${\cal G}_{n}$ of n-vertex graphs of maximum degree d, $s({\cal G}_{n})$ denotes the minimum number such that there exists a graph H with M edges where, for every G ∈ ${\cal G}_{n}$ , Maker has a strategy to build a copy of G in the game on H. We obtain that $s({\cal G}_{n}) = \Theta(n^{2 - \frac{2}{d}})

Hitting time results for Maker-Breaker games

We study Maker-Breaker games played on the edge set of a random graph. Specifically, we consider the random graph process and analyze the first time in a typical random graph process that Maker starts having a winning strategy for his final graph to admit some property \mP. We focus on three natural properties for Maker's graph, namely being $k$-vertex-connected, admitting a perfect matching, and being Hamiltonian. We prove the following optimal hitting time results: with high probability Maker wins the $k$-vertex connectivity game exactly at the time the random graph process first reaches minimum degree $2k$; with high probability Maker wins the perfect matching game exactly at the time the random graph process first reaches minimum degree $2$; with high probability Maker wins the Hamiltonicity game exactly at the time the random graph process first reaches minimum degree $4$. The latter two statements settle conjectures of Stojakovi\'{c} and Szab\'{o}.Comment: 24 page

Positional Games

Positional games are a branch of combinatorics, researching a variety of two-player games, ranging from popular recreational games such as Tic-Tac-Toe and Hex, to purely abstract games played on graphs and hypergraphs. It is closely connected to many other combinatorial disciplines such as Ramsey theory, extremal graph and set theory, probabilistic combinatorics, and to computer science. We survey the basic notions of the field, its approaches and tools, as well as numerous recent advances, standing open problems and promising research directions.Comment: Submitted to Proceedings of the ICM 201

Evolutionary Games on Networks and Payoff Invariance Under Replicator Dynamics

The commonly used accumulated payoff scheme is not invariant with respect to shifts of payoff values when applied locally in degree-inhomogeneous population structures. We propose a suitably modified payoff scheme and we show both formally and by numerical simulation, that it leaves the replicator dynamics invariant with respect to affine transformations of the game payoff matrix. We then show empirically that, using the modified payoff scheme, an interesting amount of cooperation can be reached in three paradigmatic non-cooperative two-person games in populations that are structured according to graphs that have a marked degree inhomogeneity, similar to actual graphs found in society. The three games are the Prisoner's Dilemma, the Hawks-Doves and the Stag-Hunt. This confirms previous important observations that, under certain conditions, cooperation may emerge in such network-structured populations, even though standard replicator dynamics for mixing populations prescribes equilibria in which cooperation is totally absent in the Prisoner's Dilemma, and it is less widespread in the other two games.Comment: 20 pages, 8 figures; to appear on BioSystem

Path Coupling Using Stopping Times and Counting Independent Sets and Colourings in Hypergraphs

We give a new method for analysing the mixing time of a Markov chain using path coupling with stopping times. We apply this approach to two hypergraph problems. We show that the Glauber dynamics for independent sets in a hypergraph mixes rapidly as long as the maximum degree Delta of a vertex and the minimum size m of an edge satisfy m>= 2Delta+1. We also show that the Glauber dynamics for proper q-colourings of a hypergraph mixes rapidly if m>= 4 and q > Delta, and if m=3 and q>=1.65Delta. We give related results on the hardness of exact and approximate counting for both problems.Comment: Simpler proof of main theorem. Improved bound on mixing time. 19 page