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Consider a model of fire spreading through a graph; initially some vertices are burning, and at every given time-step fire spreads from burning vertices to their neighbours. The firefighter problem is a solitaire game in which a player is allowed, at every time-step, to protect some non-burning vertices (by effectively deleting them) in order to contain the fire growth. How many vertices per turn, on average, must be protected in order to stop the fire from spreading infinitely? Here we consider the problem on $\mathbb{Z}^2\times [h]$ for both nearest neighbour adjacency and strong adjacency. We determine the critical protection rates for these graphs to be $1.5h$ and $3h$, respectively. This establishes the fact that using an optimal two-dimensional strategy for all layers in parallel is asymptotically optimal., Comment: 22 pages, 5 figures

In the $\left(1:b\right)$ component game played on a graph $G$, two players, Maker and Breaker, alternately claim~$1$ and~$b$ previously unclaimed edges of $G$, respectively. Maker's aim is to maximise the size of a largest connected component in her graph, while Breaker is trying to minimise it. We show that the outcome of the game on the binomial random graph is strongly correlated with the appearance of a nonempty $(b+2)$-core in the graph. For any integer $k$, the $k$-core of a graph is its largest subgraph of minimum degree at least $k$. Pittel, Spencer and Wormald showed in 1996 that for any $k\ge3$ there exists an explicitly defined constant $c_{k}$ such that $p=c_{k}/n$ is the threshold function for the appearance of the $k$-core in $G(n,p)$. More precisely, $G(n,c/n)$ has WHP a linear-size $k$-core when the constant $c>c_{k}$, and an empty $k$-core when $cc_{b+2}$, while Breaker can WHP prevent Maker from building larger than polylogarithmic-size components if $c

Every Boolean function can be uniquely represented as a multilinear polynomial. The entropy and the total influence are two ways to measure the concentration of its Fourier coefficients, namely the monomial coefficients in this representation: the entropy roughly measures their spread, while the total influence measures their average level. The Fourier Entropy/Influence conjecture of Friedgut and Kalai from 1996 states that the entropy to influence ratio is bounded by a universal constant $C$. Using lexicographic Boolean functions, we present three explicit asymptotic constructions that improve upon the previously best known lower bound $C>6.278944$ by O'Donnell and Tan, obtained via recursive composition. The first uses their construction with the lexicographic function $\ell\left\langle 2/3\right\rangle $ of measure $2/3$ to demonstrate that $C\ge4+3\log_{4}3>6.377444$. The second generalizes their construction to biased functions and obtains $C>6.413846$ using $\ell\left\langle \Phi\right\rangle $, where $\Phi$ is the inverse golden ratio. The third, independent, construction gives $C>6.454784$, even for monotone functions. Beyond modest improvements to the value of $C$, our constructions shed some new light on the properties sought in potential counterexamples to the conjecture. Additionally, we prove a Lipschitz-type condition on the total influence and spectral entropy, which may be of independent interest., Comment: 20+7 pages

The problem of online checkpointing is a classical problem with numerous applications which had been studied in various forms for almost 50 years. In the simplest version of this problem, a user has to maintain $k$ memorized checkpoints during a long computation, where the only allowed operation is to move one of the checkpoints from its old time to the current time, and his goal is to keep the checkpoints as evenly spread out as possible at all times. Bringmann et al. studied this problem as a special case of an online/offline optimization problem in which the deviation from uniformity is measured by the natural discrepancy metric of the worst case ratio between real and ideal segment lengths. They showed this discrepancy is smaller than $1.59-o(1)$ for all $k$, and smaller than $\ln4-o(1)\approx1.39$ for the sparse subset of $k$'s which are powers of 2. In addition, they obtained upper bounds on the achievable discrepancy for some small values of $k$. In this paper we solve the main problems left open in the above-mentioned paper by proving that $\ln4$ is a tight upper and lower bound on the asymptotic discrepancy for all large $k$, and by providing tight upper and lower bounds (in the form of provably optimal checkpointing algorithms, some of which are in fact better than those of Bringmann et al.) for all the small values of $k \leq 10$. In the last part of the paper we describe some new applications of this online checkpointing problem., Comment: Appeared at ICALP 2018

We study novel variations of Voronoi games and associated random processes that we call Voronoi choice games. These games provide a rich framework for studying questions regarding the power of small numbers of choices in multi-player, competitive scenarios, and they further lead to many interesting, non-trivial random processes that appear worthy of study. As an example of the type of problem we study, suppose a group of $n$ miners are staking land claims through the following process: each miner has $m$ associated points independently and uniformly distributed on an underlying space, so the $k$th miner will have associated points $p_{k1},p_{k2},\ldots,p_{km}$. Each miner chooses one of these points as the base point for their claim. Each miner obtains mining rights for the area of the square that is closest to their chosen base, that is, they obtain the Voronoi cell corresponding to their chosen point in the Voronoi diagram of the $n$ chosen points. Each player's goal is simply to maximize the amount of land under their control. What can we say about the players' strategy and the equilibria of such games? In our main result, we derive bounds on the expected number of pure Nash equilibria for a variation of the 1-dimensional game on the circle where a player owns the arc starting from their point and moving clockwise to the next point. This result uses interesting properties of random arc lengths on circles, and demonstrates the challenges in analyzing these kinds of problems. We also provide several other related results. In particular, for the 1-dimensional game on the circle, we show that a pure Nash equilibrium always exists when each player owns the part of the circle nearest to their point, but it is NP-hard to determine whether a pure Nash equilibrium exists in the variant when each player owns the arc starting from their point clockwise to the next point.

Given a property of Boolean functions, what is the minimum number of queries required to determine with high probability if an input function satisfies this property or is "far" from satisfying it? This is a fundamental question in Property Testing, where traditionally the testing algorithm is allowed to pick its queries among the entire set of inputs. Balcan, Blais, Blum and Yang have recently suggested to restrict the tester to take its queries from a smaller random subset of polynomial size of the inputs. This model is called active testing, and in the extreme case when the size of the set we can query from is exactly the number of queries performed it is known as passive testing. We prove that passive or active testing of k-linear functions (that is, sums of k variables among n over Z_2) requires Theta(k*log n) queries, assuming k is not too large. This extends the case k=1, (that is, dictator functions), analyzed by Balcan et. al. We also consider other classes of functions including low degree polynomials, juntas, and partially symmetric functions. Our methods combine algebraic, combinatorial, and probabilistic techniques, including the Talagrand concentration inequality and the Erdos--Rado theorem on Delta-systems., Comment: 16 pages

Let $n$, $k$, and $t$ be integers satisfying $n>k>t\ge2$. A Steiner system with parameters $t$, $k$, and $n$ is a $k$-uniform hypergraph on $n$ vertices in which every set of $t$ distinct vertices is contained in exactly one edge. An outstanding problem in Design Theory is to determine whether a nontrivial Steiner system exists for $t\geq6$. In this note we prove that for every $k>t\ge2$ and sufficiently large $n$, there exists an almost Steiner system with parameters $t$, $k$, and $n$; that is, there exists a $k$-uniform hypergraph on $n$ vertices such that every set of $t$ distinct vertices is covered by either one or two edges.

We study the (1:b) Maker-Breaker component game, played on the edge set of a d-regular graph. Maker's aim in this game is to build a large connected component, while Breaker's aim is to not let him do so. For all values of Breaker's bias b, we determine whether Breaker wins (on any d-regular graph) or Maker wins (on almost every d-regular graph) and provide explicit winning strategies for both players. To this end, we prove an extension of a theorem by Gallai-Hasse-Roy-Vitaver about graph orientations without long directed simple paths., Comment: 10 pages

A team of players plays the following game. After a strategy session, each player is randomly fitted with a blue or red hat. Then, without further communication, everybody can try to guess simultaneously his or her own hat color by looking at the hat colors of other players. Visibility is defined by a directed graph; that is, vertices correspond to players, and a player can see each player to whom she or he is connected by an arc. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The team aims to maximize the probability of a win, and this maximum is called the hat number of the graph. Previous works focused on the problem on complete graphs and on undirected graphs. Some cases were solved, e.g., complete graphs of certain orders, trees, cycles, bipartite graphs. These led Uriel Feige to conjecture that the hat number of any graph is equal to the hat number of its maximum clique. We show that the conjecture does not hold for directed graphs.Moreover, for every value of the maximum clique size, we provide a tight characterization of the range of possible values of the hat number. We construct families of directed graphs with a fixed clique number the hat number of which is asymptotically optimal. We also determine the hat number of tournaments to be one half., Comment: 9 pages. v2: updated title and abstract to match journal version

The firefighter problem is a monotone dynamic process in graphs that can be viewed as modeling the use of a limited supply of vaccinations to stop the spread of an epidemic. In more detail, a fire spreads through a graph, from burning vertices to their unprotected neighbors. In every round, a small amount of unburnt vertices can be protected by firefighters. How many firefighters per turn, on average, are needed to stop the fire from advancing? We prove tight lower and upper bounds on the amount of firefighters needed to control a fire in the Cartesian planar grid and in the strong planar grid, resolving two conjectures of Ng and Raff., Comment: 8 pages