arXiv subjects

Pawel Pralat

Publications and source records attributed to Pawel Pralat.

At least 19 recordsLinked to original sources

arXiv2026

Going in Circles: Collaborative Multi-Robot Treasure Hunting

This paper investigates a multi-robot search-and-visit problem involving $n$ robots starting at the origin and $k$ unknown treasures hidden on the unit circle $\mathcal{C}$. The robots move independently anywhere in the plane with a maximum speed of 1 and instantly share discovery information. The goal is to determine an algorithm that minimizes the total time needed for all robots to individually visit every treasure. To establish the foundational limits of this task, we first solve an auxiliary optimal routing problem for a single robot on $\mathcal{C}$ that possesses complete prior knowledge of the treasure locations. The exact analysis of this auxiliary problem yields explicit upper and lower bounds for the original search problem.

cs.DS↗

arXiv2026

Triangle-Saturated Graphs in the Semi-Random Graph Process

The semi-random graph process is an adaptive random graph process in which an online algorithm is initially given an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the algorithm independently and uniformly at random. The algorithm then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. We also consider the offline version of the process in which the algorithm is given the entire sequence of random vertex choices before the selection takes place. For a given graph property, the objective of the algorithm is to force the graph to satisfy this property asymptotically almost surely in as few rounds as possible. In this paper, we focus on the property of being triangle-saturated and establish upper and lower bounds on the number of rounds required to construct a triangle-saturated graph in both the online and offline versions of the process.

math.CO↗

arXiv2026

The stochastic block model has the overlap graph property for modularity

The overlap gap property (OGP) is a statement about the geometry of near-optimal solutions. Exhibiting OGP implies failure of a class of local algorithms; and has been observed to coincide with conjectured algorithmic limits in problems with statistical computational gap. We consider the Stochastic Block Model (SBM), where the graph has a planted partition with $k$ equal-size blocks which form the `communities', and where, for parameters $p>q$, vertices within the same community connect with probability $p$, while vertices in different communities connect with probability $q$, independently across pairs of vertices. Modularity--based clustering algorithms have become ubiquitous in applications. This article studies theoretical limits of local algorithms based on the modularity score on the SBM. We establish that modularity exhibits OGP on the SBM. This rules out a class of local algorithms based on modularity for recovery in the SBM, and shows slow mixing time for a related Markov Chain. Theoretically this is one of the few instances where OGP has been established for a `planted' model, as most such analyses to date consider the `null' model. As part of our analysis, we extend a result by Bickel and Chen 2009, who established that with high probability, the modularity optimal partition of SBM is $o(n)$ local moves away from the planted partition, where $n$ is the graph size. We show that, with high probability, any partition with modularity score sufficiently near the optimal value is close to the planted partition.

math.PR↗

arXiv2026

Canonical labelling of random regular graphs

We prove that whenever $d=d(n)\to\infty$ and $n-d\to\infty$ as $n\to\infty$, then with high probability for any non-trivial initial colouring, the colour refinement algorithm distinguishes all vertices of the random regular graph $\mathcal{G}_{n,d}$. This, in particular, implies that with high probability $\mathcal{G}_{n,d}$ admits a canonical labelling computable in time $O(\min\{n^ω,nd^2+nd\log n\})$, where $ω<2.372$ is the matrix multiplication exponent.

math.CO↗

arXiv2026

Achievable Burning Densities of Growing Grids

Graph burning is a discrete-time process on graphs where vertices are sequentially activated and burning vertices cause their neighbours to burn over time. In this work, we focus on a dynamic setting in which the graph grows over time, and at each step we burn vertices in the growing grid $G_n = [-f(n),f(n)]^2$. We investigate the set of achievable burning densities for functions of the form $f(n)=\lceil cn^α\rceil$, where $α\ge 1$ and $c>0$. We show that for $α=1$, the set of achievable densities is $[1/(2c^2),1]$, for $1<α<3/2$, every density in $[0,1]$ is achievable, and for $α=3/2$, the set of achievable densities is $[0,(1+\sqrt{6}c)^{-2}]$.

math.CO↗

arXiv2025

Creating Subgraphs in Semi-Random Hypergraph Games

The semi-random hypergraph process is a natural generalisation of the semi-random graph process, which can be thought of as a one player game. For fixed $r < s$, starting with an empty hypergraph on $n$ vertices, in each round a set of $r$ vertices $U$ is presented to the player independently and uniformly at random. The player then selects a set of $s-r$ vertices $V$ and adds the hyperedge $U \cup V$ to the $s$-uniform hypergraph. For a fixed (monotone) increasing graph property, the player's objective is to force the graph to satisfy this property with high probability in as few rounds as possible. We focus on the case where the player's objective is to construct a subgraph isomorphic to an arbitrary, fixed hypergraph $H$. In the case $r=1$ the threshold for the number of rounds required was already known in terms of the degeneracy of $H$. In the case $2 \le r < s$, we give upper and lower bounds on this threshold for general $H$, and find further improved upper bounds for cliques in particular. We identify cases where the upper and lower bounds match. We also demonstrate that the lower bounds are not always tight by finding exact thresholds for various paths and cycles.

math.CO↗

arXiv2025

The Fagnano Triangle Patrolling Problem

We investigate a combinatorial optimization problem that involves patrolling the edges of an acute triangle using a unit-speed agent. The goal is to minimize the maximum (1-gap) idle time of any edge, which is defined as the time gap between consecutive visits to that edge. This problem has roots in a centuries-old optimization problem posed by Fagnano in 1775, who sought to determine the inscribed triangle of an acute triangle with the minimum perimeter. It is well-known that the orthic triangle, giving rise to a periodic and cyclic trajectory obeying the laws of geometric optics, is the optimal solution to Fagnano's problem. Such trajectories are known as Fagnano orbits, or more generally as billiard trajectories. We demonstrate that the orthic triangle is also an optimal solution to the patrolling problem. Our main contributions pertain to new connections between billiard trajectories and optimal patrolling schedules in combinatorial optimization. In particular, as an artifact of our arguments, we introduce a novel 2-gap patrolling problem that seeks to minimize the visitation time of objects every three visits. We prove that there exist infinitely many well-structured billiard-type optimal trajectories for this problem, including the orthic trajectory, which has the special property of minimizing the visitation time gap between any two consecutively visited edges. Complementary to that, we also examine the cost of dynamic, sub-optimal trajectories to the 1-gap patrolling optimization problem. These trajectories result from a greedy algorithm and can be implemented by a computationally primitive mobile agent.

cs.DM↗

arXiv2025

Asynchronous Majority Dynamics on Binomial Random Graphs

We study information aggregation in networks when agents interact to learn a binary state of the world. Initially each agent privately observes an independent signal which is "correct" with probability $\frac{1}{2}+δ$ for some $δ> 0$. At each round, a node is selected uniformly at random to update their public opinion to match the majority of their neighbours (breaking ties in favour of their initial private signal). Our main result shows that for sparse and connected binomial random graphs $\mathcal G(n,p)$ the process stabilizes in a "correct" consensus in $\mathcal O(n\log^2 n/\log\log n)$ steps with high probability. In fact, when $\log n/n \ll p = o(1)$ the process terminates at time $\hat T = (1+o(1))n\log n$, where $\hat T$ is the first time when all nodes have been selected at least once. However, in dense binomial random graphs with $p=Ω(1)$, there is an information cascade where the process terminates in the "incorrect" consensus with probability bounded away from zero.

cs.GT↗

arXiv2025

A Direct Proof of the Short-Side Advantage in Random Matching Markets

We study the stable matching problem under the random matching model where the preferences of the doctors and hospitals are sampled uniformly and independently at random. In a balanced market with $n$ doctors and $n$ hospitals, the doctor-proposal deferred-acceptance algorithm gives doctors an expected rank of order $\log n$ for their partners and hospitals an expected rank of order $\frac{n}{\log n}$ for their partners. This situation is reversed in an unbalanced market with $n+1$ doctors and $n$ hospitals, a phenomenon known as the short-side advantage. The current proofs of this fact are indirect, counter-intuitively being based upon analyzing the hospital-proposal deferred-acceptance algorithm. In this paper we provide a direct proof of the short-side advantage, explicitly analyzing the doctor-proposal deferred-acceptance algorithm. Our proof sheds light on how and why the phenomenon arises.

cs.GT↗

arXiv2025

Multiset Metric Dimension of Binomial Random Graphs

For a graph $G = (V,E)$ and a subset $R \subseteq V$, we say that $R$ is \textit{multiset resolving} for $G$ if for every pair of vertices $v,w$, the \textit{multisets} $\{d(v,r): r \in R\}$ and $\{d(w,r):r \in R\}$ are distinct, where $d(x,y)$ is the graph distance between vertices $x$ and $y$. The \textit{multiset metric dimension} of $G$ is the size of a smallest set $R \subseteq V$ that is multiset resolving (or $\infty$ if no such set exists). This graph parameter was introduced by Simanjuntak, Siagian, and Vitr\'{i}k in 2017~\cite{simanjuntak2017multiset}, and has since been studied for a variety of graph families. We prove bounds which hold with high probability for the multiset metric dimension of the binomial random graph $G(n,p)$ in the regime $d = (n-1)p = \Theta(n^{x})$ for fixed $x \in (0,1)$.

math.CO↗

arXiv2025

Almost all 9-regular graphs have a modulo-5 orientation

In 1972 Tutte famously conjectured that every 4-edge-connected graph has a nowhere zero 3-flow; this is known to be equivalent to every 5-regular, 4-edge-connected graph having an edge orientation in which every in-degree is either 1 or 4. Jaeger conjectured a generalization of Tutte's conjecture, namely, that every $4p+1$-regular, $4p$-edge-connected graph has an edge orientation in which every in-degree is either $p$ or $3p+1$. Inspired by the work of Pralat and Wormald investigating $p=1$, for $p=2$ we show this holds asymptotically almost surely for random 9-regular graphs. It follows that the conjecture holds for almost all 9-regular, 8-edge-connected graphs. These results make use of the technical small subgraph conditioning method.

math.CO↗

arXiv2025

Playing Sudoku on random 3-regular graphs

The Sudoku number $s(G)$ of graph $G$ with chromatic number $χ(G)$ is the smallest partial $χ(G)$-colouring of $G$ that determines a unique $χ(G)$-colouring of the entire graph. We show that the Sudoku number of the random $3$-regular graph $\mathcal{G}_{n,3}$ satisfies $s(\mathcal{G}_{n,3}) \leq (1+o(1))\frac{n}{3}$ asymptotically almost surely. We prove this by analyzing an algorithm which $3$-colours $\mathcal{G}_{n,3}$ in a way that produces many locally forced vertices, i.e., vertices which see two distinct colours among their neighbours. The intricacies of the algorithm present some challenges for the analysis, and to overcome these we use a non-standard application of Wormald's differential equations method that incorporates tools from finite Markov chains.

math.CO↗

arXiv2025

Burning Random Trees

Let $\mathcal{T}$ be a Galton-Watson tree with a given offspring distribution $ξ$, where $ξ$ is a $Z_{\geq 0}$-valued random variable with $E[ξ] = 1$ and $0 < σ^{2}:=Var[ξ] < \infty$. For $n \geq 1$, let $T_{n}$ be the tree $\mathcal{T}$ conditioned to have $n$ vertices. In this paper we investigate $b(T_n)$, the burning number of $T_n$. Our main result shows that asymptotically almost surely $b(T_n)$ is of the order of $n^{1/3}$.

math.CO↗

arXiv2024

Makespan Trade-offs for Visiting Triangle Edges

We study a primitive vehicle routing-type problem in which a fleet of $n$unit speed robots start from a point within a non-obtuse triangle $Δ$, where $n \in \{1,2,3\}$. The goal is to design robots' trajectories so as to visit all edges of the triangle with the smallest visitation time makespan. We begin our study by introducing a framework for subdividing $Δ$into regions with respect to the type of optimal trajectory that each point $P$ admits, pertaining to the order that edges are visited and to how the cost of the minimum makespan $R_n(P)$ is determined, for $n\in \{1,2,3\}$. These subdivisions are the starting points for our main result, which is to study makespan trade-offs with respect to the size of the fleet. In particular, we define $ R_{n,m} (Δ)= \max_{P \in Δ} R_n(P)/R_m(P)$, and we prove that, over all non-obtuse triangles $Δ$: (i) $R_{1,3}(Δ)$ ranges from $\sqrt{10}$ to $4$, (ii) $R_{2,3}(Δ)$ ranges from $\sqrt{2}$ to $2$, and (iii) $R_{1,2}(Δ)$ ranges from $5/2$ to $3$. In every case, we pinpoint the starting points within every triangle $Δ$ that maximize $R_{n,m} (Δ)$, as well as we identify the triangles that determine all $\inf_ΔR_{n,m}(Δ)$ and $\sup_ΔR_{n,m}(Δ)$ over the set of non-obtuse triangles.

cs.DM↗

arXiv2024

Hopping Forcing Number in Random $d$-regular Graphs

Hopping forcing is a single player combinatorial game in which the player is presented a graph on $n$ vertices, some of which are initially blue with the remaining vertices being white. In each round $t$, a blue vertex $v$ with all neighbours blue may hop and colour a white vertex blue in the second neighbourhood, provided that $v$ has not performed a hop in the previous $t-1$ rounds. The objective of the game is to eventually colour every vertex blue by repeatedly applying the hopping forcing rule. Subsequently, for a given graph $G$, the hopping forcing number is the minimum number of initial blue vertices that are required to achieve the objective. In this paper, we study the hopping forcing number for random $d$-regular graphs. Specifically, we aim to derive asymptotic upper and lower bounds for the hopping forcing number for various values of $d \geq 2$.

math.CO↗

arXiv2024

Perfect matchings and loose Hamilton cycles in the semirandom hypergraph model

We study the 2-offer semirandom 3-uniform hypergraph model on $n$ vertices. At each step, we are presented with 2 uniformly random vertices. We choose any other vertex, thus creating a hyperedge of size 3. We show a strategy that constructs a perfect matching, and another that constructs a loose Hamilton cycle, both succeeding asymptotically almost surely within $Θ(n)$ steps. Both results extend to $s$-uniform hypergraphs. The challenges with hypergraphs, and our methods, are qualitatively different from what has been seen for semirandom graphs. Much of our analysis is done on an auxiliary graph that is a uniform $k$-out subgraph of a random bipartite graph, and this tool may be useful in other contexts.

math.CO↗

arXiv2024

Cliques, Chromatic Number, and Independent Sets in the Semi-random Process

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. In this paper, we investigate the following three properties: containing a complete graph of order $k$, having the chromatic number at least $k$, and not having an independent set of size at least $k$.

math.CO↗

arXiv2024

Subgraph Games in the Semi-Random Graph Process and Its Generalization to Hypergraphs

The semi-random graph process is a single-player game that begins with an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$ and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. We focus on the problem of constructing a subgraph isomorphic to an arbitrary, fixed graph $G$. Let $ω= ω(n)$ be any function tending to infinity as $n \to \infty$. In (Omri Ben-Eliezer et al. "Semi-random graph process". In: Random Structures & Algorithms 56.3 (2020), pp. 648-675) it was proved that asymptotically almost surely one can construct $G$ in less than $n^{(d-1)/d} ω$ rounds where $d \ge 2$ is the degeneracy of $G$. It was also proved that the result is sharp for $G = K_{d+1}$, that is, asymptotically almost surely it takes at least $n^{(d-1)/d} / ω$ rounds to create $K_{d+1}$. Moreover, the authors conjectured that their general upper bound is sharp for all graphs $G$. We prove this conjecture here. We also consider a natural generalization of the process to $s$-uniform hypergraphs, the semi-random hypergraph process in which $r \ge 1$ vertices are presented at random, and the player then selects $s-r \ge 1$ vertices to form an edge of size~$s$. Our results for graphs easily generalize to hypergraphs when $r=1$; the threshold for constructing a fixed $s$-uniform hypergraph $G$ is, again, determined by the degeneracy of $G$. However, new challenges are mounting when $r \ge 2$; thresholds are not even known for complete hypergraphs. We provide bounds for this family and determine thresholds for some sparser hypergraphs.

math.CO↗