A simplified proof of the Johansson-Molloy Theorem using the Rosenfeld counting method
We show that any triangle-free graph with maximum degree $Δ$ has chromatic number at most $\left(1+o(1)\right)Δ/\log Δ.$
math.CO↗
arXiv subjects
Anders Martinsson
Publications and source records attributed to Anders Martinsson.
Note on Long Paths in Eulerian Digraphs
Long Paths and Cycles in eulerian digraphs have gotten a lot of attention recently. In this short note, we show how to use methods from Knierim, Larcher, Martinsson, Noever (2021) to find paths of length $d/(\log d+1)$ in Eulerian digraphs with average degree $d$, improving the recent result of $Ω(d^{1/2+1/40})$. Our result is optimal up to at most a logarithmic factor.
math.CO↗
The edge-statistics conjecture for $\ell \ll k^{6/5}$
Let $k$ and $\ell$ be positive integers. We prove that if $1 \leq \ell \leq o_k(k^{6/5})$, then in every large enough graph $G$, the fraction of $k$-vertex subsets that induce exactly $\ell$ edges is at most $1/e + o_k(1)$. Together with a recent result of Kwan, Sudakov, and Tran, this settles a conjecture of Alon, Hefetz, Krivelevich, and Tyomkyn.
math.CO↗
An Optimal Decentralized $(Δ+ 1)$-Coloring Algorithm
Consider the following simple coloring algorithm for a graph on $n$ vertices. Each vertex chooses a color from $\{1, \dotsc, Δ(G) + 1\}$ uniformly at random. While there exists a conflicted vertex choose one such vertex uniformly at random and recolor it with a randomly chosen color. This algorithm was introduced by Bhartia et al. [MOBIHOC'16] for channel selection in WIFI-networks. We show that this algorithm always converges to a proper coloring in expected $O(n \log Δ)$ steps, which is optimal and proves a conjecture of Chakrabarty and Supinski [SOSA'20].
cs.DS↗
The Chromatic Number of Dense Random Block Graphs
The chromatic number $χ(G)$ of a graph $G$, that is, the smallest number of colors required to color the vertices of $G$ so that no two adjacent vertices are assigned the same color, is a classic and extensively studied parameter. Here we consider the case where $G$ is a random block graph, also known as the stochastic block model. The vertex set is partitioned into $k\in\mathbb{N}$ parts $V_1, \dotsc, V_k$, and for each $1 \le i\le j\le k$, two vertices $u \in V_i, v\in V_j$ are connected by an edge with some probability $p_{ij} \in (0,1)$ independently. Our main result pins down the typical asymptotic value of $χ(G)$ and establishes the distribution of the sizes of the color classes in optimal colorings. We discover that in contrast to the case of a binomial random graph $G(n,p)$, that corresponds to $k=1$ in our model, where the average size of a color class in an (almost) optimal coloring essentially coincides with the independence number, the block model reveals a more diverse picture: the "average" class in an optimal coloring is a convex combination of several types of independent sets that vary in total size as well as in the size of their intersection with each $V_i$, $1\le i \le k$.
math.CO↗
Arithmetic Progressions in Sumsets of Sparse Sets
A set of positive integers $A \subset \mathbb{Z}_{> 0}$ is \emph{log-sparse} if there is an absolute constant $C$ so that for any positive integer $x$ the sequence contains at most $C$ elements in the interval $[x,2x)$. In this note we study arithmetic progressions in sums of log-sparse subsets of $\mathbb{Z}_{> 0}$. We prove that for any log-sparse subsets $S_1, \dots, S_n$ of $\mathbb{Z}_{> 0},$ the sumset $S = S_1 + \cdots + S_n$ cannot contain an arithmetic progression of size greater than $n^{(1+o(1))n}.$ We also show that this is nearly tight by proving that there exist log-sparse sets $S_1, \dots, S_n$ such that $S_1 + \cdots + S_n$ contains an arithmetic progression of size $n^{(1-o(1)) n}.$
math.CO↗
Long Cycles, Heavy Cycles and Cycle Decompositions in Digraphs
Hajós conjectured in 1968 that every Eulerian \(n\)-vertex graph can be decomposed into at most $\lfloor (n-1)/2\rfloor$ edge-disjoint cycles. This has been confirmed for some special graph classes, but the general case remains open. In a sequence of papers by Bienia and Meyniel (1986), Dean (1986), and Bollobás and Scott (1996) it was analogously conjectured that every \emph{directed} Eulerian graph can be decomposed into $O(n)$ cycles. In this paper, we show that every directed Eulerian graph can be decomposed into $O(n \log Δ)$ disjoint cycles, thus making progress towards the conjecture by Bollobás and Scott. Our approach is based on finding heavy cycles in certain edge-weightings of directed graphs. As a further consequence of our techniques, we prove that for every edge-weighted digraph in which every vertex has out-weight at least $1$, there exists a cycle with weight at least $Ω(\log \log n/{\log n})$, thus resolving a question by Bollobás and Scott.
math.CO↗
Navigating an Infinite Space with Unreliable Movements
We consider a search problem on a $2$-dimensional infinite grid with a single mobile agent. The goal of the agent is to find her way home, which is located in a grid cell chosen by an adversary. Initially, the agent is provided with an infinite sequence of instructions, that dictate the movements performed by the agent. Each instruction corresponds to a movement to an adjacent grid cell and the set of instructions can be a function of the initial locations of the agent and home. The challenge of our problem stems from faults in the movements made by the agent. In every step, with some constant probability $0 \leq p \leq 1$, the agent performs a random movement instead of following the current instruction. This paper provides two results on this problem. First, we show that for some values of $p$, there does not exist any set of instructions that guide the agent home in finite expected time. Second, we complement this impossibility result with an algorithm that, for sufficiently small values of $p$, yields a finite expected hitting time for home. In particular, we show that for any $p < 1$, our approach gives a hitting rate that decays polynomially as a function of time. In that sense, our approach is far superior to a standard random walk in terms of hitting time. The main contribution and take-home message of this paper is to show that, for some value of $0.01139\dots < p < 0.6554\ldots$, there exists a phase transition on the solvability of the problem.
cs.DM↗
Optimal strategies for patrolling fences
A classical multi-agent fence patrolling problem asks: What is the maximum length $L$ of a line that $k$ agents with maximum speeds $v_1,\ldots,v_k$ can patrol if each point on the line needs to be visited at least once every unit of time. It is easy to see that $L = α\sum_{i=1}^k v_i$ for some efficiency $α\in [\frac{1}{2},1)$. After a series of works giving better and better efficiencies, it was conjectured that the best possible efficiency approaches $\frac{2}{3}$. No upper bounds on the efficiency below $1$ were known. We prove the first such upper bounds and tightly bound the optimal efficiency in terms of the minimum ratio of speeds $s = {v_{\max}}/{v_{\min}}$ and the number of agents $k$. Guided by our upper bounds, we construct a scheme whose efficiency approaches $1$, disproving the conjecture of Kawamura and Soejima. Our scheme asymptotically matches our upper bounds in terms of the maximal speed difference and the number of agents used, proving them to be asymptotically tight. A variation of the fence patrolling problem considers a circular fence instead and asks for its circumference to be maximized. We consider the unidirectional case of this variation, where all agents are only allowed to move in one direction, say clockwise. At first, a strategy yielding $L = \max_{r \in [k]} r \cdot v_r$ where $v_1 \geq v_2 \geq \dots \geq v_k$ was conjectured to be optimal by Czyzowicz et al. This was proven not to be the case by giving constructions for only specific numbers of agents with marginal improvements of $L$. We give a general construction that yields $L = \frac{1}{33 \log_e\log_2(k)} \sum_{i=1}^k v_i$ for any set of agents, which in particular for the case $1, 1/2, \dots, 1/k$ diverges as $k \rightarrow \infty$, thus resolving a conjecture by Kawamura and Soejima affirmatively.
cs.DS↗
Optimal Kronecker-Sum Approximation of Real Time Recurrent Learning
One of the central goals of Recurrent Neural Networks (RNNs) is to learn long-term dependencies in sequential data. Nevertheless, the most popular training method, Truncated Backpropagation through Time (TBPTT), categorically forbids learning dependencies beyond the truncation horizon. In contrast, the online training algorithm Real Time Recurrent Learning (RTRL) provides untruncated gradients, with the disadvantage of impractically large computational costs. Recently published approaches reduce these costs by providing noisy approximations of RTRL. We present a new approximation algorithm of RTRL, Optimal Kronecker-Sum Approximation (OK). We prove that OK is optimal for a class of approximations of RTRL, which includes all approaches published so far. Additionally, we show that OK has empirically negligible noise: Unlike previous algorithms it matches TBPTT in a real world task (character-level Penn TreeBank) and can exploit online parameter updates to outperform TBPTT in a synthetic string memorization task. Code availiable on github.
cs.LG↗
A linear threshold for uniqueness of solutions to random jigsaw puzzles
We consider a problem introduced by Mossel and Ross [Shotgun assembly of labeled graphs, arXiv:1504.07682]. Suppose a random $n\times n$ jigsaw puzzle is constructed by independently and uniformly choosing the shape of each "jig" from $q$ possibilities. We are given the shuffled pieces. Then, depending on $q$, what is the probability that we can reassemble the puzzle uniquely? We say that two solutions of a puzzle are similar if they only differ by permutation of duplicate pieces, and rotation of rotationally symmetric pieces. In this paper, we show that, with high probability, such a puzzle has at least two non-similar solutions when $2\leq q \leq \frac{2}{\sqrt{e}}n$, all solutions are similar when $q\geq (2+\varepsilon)n$, and the solution is unique when $q=ω(n)$.
math.PR↗
When Does Hillclimbing Fail on Monotone Functions: An entropy compression argument
Hillclimbing is an essential part of any optimization algorithm. An important benchmark for hillclimbing algorithms on pseudo-Boolean functions $f: \{0,1\}^n \to \mathbb{R}$ are (strictly) montone functions, on which a surprising number of hillclimbers fail to be efficient. For example, the $(1+1)$-Evolutionary Algorithm is a standard hillclimber which flips each bit independently with probability $c/n$ in each round. Perhaps surprisingly, this algorithm shows a phase transition: it optimizes any monotone pseudo-boolean function in quasilinear time if $c<1$, but there are monotone functions for which the algorithm needs exponential time if $c>2.2$. But so far it was unclear whether the threshold is at $c=1$. In this paper we show how Moser's entropy compression argument can be adapted to this situation, that is, we show that a long runtime would allow us to encode the random steps of the algorithm with less bits than their entropy. Thus there exists a $c_0 > 1$ such that for all $0<c\le c_0$ the $(1+1)$-Evolutionary Algorithm with rate $c/n$ finds the optimum in $O(n \log^2 n)$ steps in expectation.
math.PR↗
Most edge-orderings of $K_n$ have maximal altitude
Suppose the edges of the complete graph on $n$ vertices are assigned a uniformly chosen random ordering. Let $X$ denote the corresponding number of Hamiltonian paths that are increasing in this ordering. It was shown in a recent paper by Lavrov and Loh that this quantity is non-zero with probability at least $1/e-o(1)$, and conjectured that $X$ is asymptotically almost surely non-zero. In this paper, we prove their conjecture. We further prove a partial result regarding the limiting behaviour of $X$, suggesting that $X/n$ is log-normal in the limit as $n\rightarrow\infty$. A key idea of our proof is to show a certain relation between $X$ and its size-biased distribution. This relies heavily on estimates for the third moment of $X$.
math.PR↗
The "No Justice in the Universe" phenomenon: why honesty of effort may not be rewarded in tournaments
In 2000 Allen Schwenk, using a well-known mathematical model of matchplay tournaments in which the probability of one player beating another in a single match is fixed for each pair of players, showed that the classical single-elimination, seeded format can be "unfair" in the sense that situations can arise where an indisputibly better (and thus higher seeded) player may have a smaller probability of winning the tournament than a worse one. This in turn implies that, if the players are able to influence their seeding in some preliminary competition, situations can arise where it is in a player's interest to behave "dishonestly", by deliberately trying to lose a match. This motivated us to ask whether it is possible for a tournament to be both honest, meaning that it is impossible for a situation to arise where a rational player throws a match, and "symmetric" - meaning basically that the rules treat everyone the same - yet unfair, in the sense that an objectively better player has a smaller probability of winning than a worse one. After rigorously defining our terms, our main result is that such tournaments exist and we construct explicit examples for any number n >= 3 of players. For n=3, we show (Theorem 3.6) that the collection of win-probability vectors for such tournaments form a 5-vertex convex polygon in R^3, minus some boundary points. We conjecture a similar result for any n >= 4 and prove some partial results towards it.
math.PR↗
Even flying cops should think ahead
We study the entanglement game, which is a version of cops and robbers, on sparse graphs. While the minimum degree of a graph G is a lower bound for the number of cops needed to catch a robber in G, we show that the required number of cops can be much larger, even for graphs with small maximum degree. In particular, we show that there are 3-regular graphs where a linear number of cops are needed.
cs.DM↗
First-passage percolation on Cartesian power graphs
We consider first-passage percolation on the class of "high-dimensional" graphs that can be written as an iterated Cartesian product $G\square G \square \dots \square G$ of some base graph $G$ as the number of factors tends to infinity. We propose a natural asymptotic lower bound on the first-passage time between $(v, v, \dots, v)$ and $(w, w, \dots, w)$ as $n$, the number of factors, tends to infinity, which we call the critical time $t^*_G(v, w)$. Our main result characterizes when this lower bound is sharp as $n\rightarrow\infty$. As a corollary, we are able to determine the limit of the so-called diagonal time-constant in $\mathbb{Z}^n$ as $n\rightarrow\infty$ for a large class of distributions of passage times.
math.PR↗
Segregating Markov chains
Dealing with finite Markov chains in discrete time, the focus often lies on convergence behavior and one tries to make different copies of the chain meet as fast as possible and then stick together. There is, however, a very peculiar kind of discrete finite Markov chain, for which two copies started in different states can be coupled to meet almost surely in finite time, yet their distributions keep a total variation distance bounded away from 0, even in the limit as time goes off to infinity. We show that the supremum of total variation distance kept in this context is $\frac12$.
math.PR↗
Shotgun edge assembly of random jigsaw puzzles
In recent work by Mossel and Ross, it was asked how large $q$ has to be for a random jigsaw puzzle with $q$ different shapes of "jigs" to have exactly one solution. The jigs are assumed symmetric in the sense that two jigs of the same type always fit together. They showed that for $q=o(n^{2/3})$ there are a.a.s. multiple solutions, and for $q=ω(n^2)$ there is a.a.s. exactly one. The latter bound has since been improved to $q\geq n^{1+\varepsilon}$ independently by Nenadov, Pfister and Steger, and by Bordernave, Feige and Mossel. Both groups further remark that for $q=o(n)$ there are a.a.s. duplicate pieces in the puzzle. In this paper, we show that such puzzle a.a.s. has multiple solutions whenever $q\leq \frac{2}{\sqrt{e}}\,n - ω(\log_2 n)$, even if permuting identical pieces is not considered changing the solution. We further give some remarks about the number of solutions, and the probability of a unique solution in this regime.
math.PR↗