How experience as a Support Scheme volunteer and Secretary has enhanced the author's insight and understanding as a probation officer and increased police sensitivity to victims.
We prove a stability version of a general result that bounds the permanent of a matrix in terms of its operator norm. More specifically, suppose $A$ is an $n \times n$ matrix over $\mathbb{C}$ (resp. $\mathbb{R}$), and let $\mathcal{P}$ denote the set of $n \times n$ matrices over $\mathbb{C}$ (resp. $\mathbb{R}$) that can be written as a permutation matrix times a unitary diagonal matrix. Then it is known that the permanent of $A$ satisfies $|\text{perm}(A)| \leq \Vert A \Vert_{2} ^n$ with equality iff $A/ \Vert A \Vert_{2} \in \mathcal{P}$ (where $\Vert A \Vert_2$ is the operator $2$-norm of $A$). We show a stability version of this result asserting that unless $A$ is very close (in a particular sense) to one of these extremal matrices, its permanent is exponentially smaller (as a function of $n$) than $\Vert A \Vert_2 ^n$. In particular, for any fixed $\alpha, \beta > 0$, we show that $|\text{perm}(A)|$ is exponentially smaller than $\Vert A \Vert_2 ^n$ unless all but at most $\alpha n$ rows contain entries of modulus at least $\Vert A \Vert_2 (1 - \beta)$.
For a graph property $\mathcal{P}$ and a common vertex set $V = \{1, 2, \ldots, n\}$, a family of graphs on $V$ is \emph{$\mathcal{P}$-intersecting} iff $G \cap H$ satisfies $\mathcal{P}$ for all $G,H$ in the family. Addressing a question of Chung, Graham, Frankl, and Shearer, we explore---for various $\mathcal{P}$---the maximum cardinality among all $\mathcal{P}$-intersecting families of graphs. In the connected-intersecting case, we resolve the question completely by a short linear algebraic proof showing this maximum is attained by taking all graphs containing a fixed spanning tree (though we show other extremal constructions as well). We also present a new lower bound for containing unions of a fixed subgraph.
In this note, we investigate combinatorial games where both players move randomly (each turn, independently selecting a legal move uniformly at random). In this model, we provide closed-form expressions for the expected number of turns in a game of Chomp with any starting condition. We also derive and prove formulas for the win probabilities for any game of Chomp with at most two rows. Additionally, we completely analyze the game of nim under random play by finding the expected number of turns and win probabilities from any starting position.
Given a graph $G$ and some initial labelling $\sigma : V(G) \to \{Red, Blue\}$ of its vertices, the \textit{majority dynamics model} is the deterministic process where at each stage, every vertex simultaneously replaces its label with the majority label among its neighbors (remaining unchanged in the case of a tie). We prove---for a wide range of parameters---that if an initial assignment is fixed and we independently sample an Erd\H{o}s--R\'enyi random graph, $G_{n,p}$, then after one step of majority dynamics, the number of vertices of each label follows a central limit law. As a corollary, we provide a strengthening of a theorem of Benjamini, Chan, O'Donnell, Tamuz, and Tan about the number of steps required for the process to reach unanimity when the initial assignment is also chosen randomly. Moreover, suppose there are initially three more red vertices than blue. In this setting, we prove that if we independently sample the graph $G_{n,1/2}$, then with probability at least $51\%$, the majority dynamics process will converge to every vertex being red. This improves a result of Tran and Vu who addressed the case that the initial lead is at least 10.
In this paper, we use the notion of twisted subgroups (i.e., subsets of group elements closed under the binary operation $(a,b) \mapsto aba$) to provide the first structural characterization of optimal play in the Explorer-Director game, introduced as the Magnus-Derek game by Nedev and Muthukrishnan and generalized to finite groups by Gerbner. In particular, we reduce the game to the problem of finding the largest proper twisted subgroup, and as a corollary we resolve the Explorer-Director game completely for all nilpotent groups.
We prove the following conjecture of Leighton and Moitra. Let $T$ be a tournament on $[n]$ and $S_n$ the set of permutations of $[n]$. For an arc $uv$ of $T$, let $A_{uv}=\{\sigma \in S_n \, : \, \sigma(u)<\sigma(v) \}$.
$\textbf{Theorem.}$ For a fixed $\varepsilon>0$, if $\mathbb{P}$ is a probability distribution on $S_n$ such that $\mathbb{P}(A_{uv})>1/2+\varepsilon$ for every arc $uv$ of $T$, then the binary entropy of $\mathbb{P}$ is at most $(1-\vartheta_{\varepsilon})\log_2 n!$ for some (fixed) positive $\vartheta_\varepsilon$.
When $T$ is transitive the theorem is due to Leighton and Moitra; for this case we give a short proof with a better $\vartheta_\varepsilon$.
