arXiv subjects

Kostas Lakis

Publications and source records attributed to Kostas Lakis.

7 recordsLinked to original sources

arXiv2026

Distributed Colouring with 4/3 chi Colours for Hyperbolic Random Graphs

We study distributed vertex colouring on Hyperbolic Random Graphs (HRGs), a geometric random graph model capturing key structural features of real-world networks. This provides a natural setting for analysing distributed algorithms beyond worst-case general graphs. We introduce Sequential Radial Colouring, a CONGEST algorithm using only efficient local computation. The algorithm achieves a near-optimal palette, colouring HRGs with $\frac{4}{3}\chi$ colours and running in $O((\log\log n)^2)$ rounds a.a.s. We also give a variant that speeds this up to $O(\log\log n)$ rounds a.a.s., at the price of using $O(\chi\log\log n)$ colours. Finally, for every constant $\varepsilon>0$, it runs in $O(1)$ rounds a.a.s. when $\chi^{1+\varepsilon}$ colours are used. This greatly reduces the number of colours over the previous constant-round algorithm of Maus and Ruff (SODA 2026) by a factor of at least $n^{1/6}$. Our analysis contains a phase in which we consider a classical randomised colouring protocol on a (large) clique of the graph. We also delve deeper into this part of the analysis and improve upon previous results for colouring a clique $C$, bounding the number of rounds required as a function of the additive slack $s = |\Psi| - \chi$, where $\Psi$ is the set of colours used. In particular, constant-round colouring is possible if and only if $s=|C|^{1+\Omega(1)}$, while $s=|C|/\log |C|$ already gives the optimal $\Theta(\log\log |C|)$ round complexity.

cs.DS↗

arXiv2026

Geometric Routing in Geometric Inhomogeneous Random Graphs

We present the first rigorous analysis of decentralized geometric routing in Geometric Inhomogeneous Random Graphs (GIRGs), a weight-agnostic variant of the greedy routing protocol. While greedy routing in GIRGs is known to explain the algorithmic small-world phenomenon by finding ultra-short paths of length $\Theta (\log \log n)$, it assumes additional knowledge of vertex weights beyond geometry, an assumption that is often restrictive or unavailable. We investigate whether the underlying geometry alone is sufficient for efficient navigation. We prove that for power-law weight exponent $\tau \in (2,3)$ and geometric decay parameter $\alpha > \tau - 1$, geometric routing succeeds with constant probability and finds ultra-short paths of length $\Theta (\log \log n)$, matching the optimal asymptotic guarantees for greedy routing. Our analysis further reveals that, upon success, both protocols follow a similar two-phase trajectory, consisting of a rapid ascent to the heavy vertices, followed by efficient navigation to the target. These results demonstrate that, in the appropriate regime, the network's geometry alone implicitly guides the path to the target through its high-weight core.

math.PR↗

arXiv2025

The Diameter of (Threshold) Geometric Inhomogeneous Random Graphs

We prove that the diameter of threshold (zero temperature) Geometric Inhomogeneous Random Graphs (GIRG) is $\Theta(\log n)$. This has strong implications for the runtime of many distributed protocols on those graphs, which often have runtimes bounded as a function of the diameter. The GIRG model exhibits many properties empirically found in real-world networks, and the runtime of various practical algorithms has empirically been found to scale in the same way for GIRG and for real-world networks, in particular related to computing distances, diameter, clustering, cliques and chromatic numbers. Thus the GIRG model is a promising candidate for deriving insight about the performance of algorithms in real-world instances. The diameter was previously only known in the one-dimensional case, and the proof relied very heavily on dimension one. Our proof employs a similar Peierls-type argument alongside a novel renormalization scheme. Moreover, instead of using topological arguments (which become complicated in high dimensions) in establishing the connectivity of certain boundaries, we employ some comparatively recent and clearer graph-theoretic machinery. The lower bound is proven via a simple ad-hoc construction.

math.PR↗

arXiv2025

Exact Matching and Top-k Perfect Matching Parameterized by Neighborhood Diversity or Bandwidth

The Exact Matching (EM) problem asks whether there exists a perfect matching which uses a prescribed number of red edges in a red/blue edge-colored graph. While there exists a randomized polynomial-time algorithm for the problem, only some special cases admit a deterministic one so far, making it a natural candidate for testing the P=RP hypothesis. A polynomial-time equivalent problem, Top-k Perfect Matching (TkPM), asks for a perfect matching maximizing the weight of the $k$ heaviest edges. We study the above problems, mainly the latter, in the scenario where the input is a blown-up graph, meaning a graph which had its vertices replaced by cliques or independent sets. We describe an FPT algorithm for TkPM parameterized by $k$ and the neighborhood diversity of the input graph, which is essentially the size of the graph before the blow-up; this graph is also called the prototype. We extend this algorithm into an approximation scheme with a much softer dependency on the aforementioned parameters, time-complexity wise. Moreover, for prototypes with bounded bandwidth but unbounded size, we develop a recursive algorithm that runs in subexponential time. Utilizing another algorithm for EM on bounded neighborhood diversity graphs, we adapt this recursive subexponential algorithm to EM. Our approach is similar to the use of dynamic programming on e.g. bounded treewidth instances for various problems. The main point is that the existence of many disjoint separators is utilized to avoid including in the separator any of a set of ``bad'' vertices during the split phase.

cs.DS↗

arXiv2024

Rumour Spreading Depends on the Latent Geometry and Degree Distribution in Social Network Models

We study push-pull rumour spreading in ultra-small-world models for social networks where the degrees follow a power-law distribution. In a non-geometric setting, Fountoulakis, Panagiotou and Sauerwald have shown that rumours always spread ultra-fast (SODA 2012), i.e. in doubly logarithmic time. On the other hand, Janssen and Mehrabian have found that rumours spread slowly (polynomial time) in a spatial preferential attachment model (SIDMA 2017). We study the question systematically for the model of Geometric Inhomogeneous Random Graphs (GIRGs). Our results are two-fold: first, with Euclidean geometry slow, fast (polylogarithmic) and ultra-fast rumour spreading may occur, depending on the exponent of the power law and the strength of the geometry in the networks, and we fully characterise the phase boundaries in between. The regimes do not coincide with the graph distance regimes, i.e., polylogarithmic or even polynomial rumour spreading may occur even if graph distances are doubly logarithmic. We expect these results to hold with little effort for related models, e.g. Scale-Free Percolation. Second, we show that rumour spreading is always (at least) fast in a non-metric geometry. The considered non-metric geometry allows to model social connections where resemblance of vertices in a single attribute, such as familial kinship, already strongly indicates the presence of an edge. Euclidean geometry fails to capture such ties. For some regimes in the Euclidean setting, the efficient pathways for spreading rumours differ from previously identified paths. For example, a vertex of degree $d$ can transmit the rumour to a vertex of larger degree by a chain of length $3$, where one of the two intermediaries has constant degree, and the other has degree $d^{c}$ for some constant $c<1$. Similar but longer chains of vertices, all having non-constant degree, turn out to be useful as well.

math.PR↗

arXiv2024

Improved bounds for polylogarithmic graph distances in scale-free percolation and related models

In this paper, we study graph distances in the geometric random graph models scale-free percolation SFP, geometric inhomogeneous random graphs GIRG, and hyperbolic random graphs HRG. Despite the wide success of the models, the parameter regime in which graph distances are polylogarithmic is poorly understood. We provide new and improved lower bounds. In a certain portion of the parameter regime, those match the known upper bounds. Compared to the best previous lower bounds by Hao and Heydenreich, our result has several advantages: it gives matching bounds for a larger range of parameters, thus settling the question for a larger portion of the parameter space. It strictly improves the lower bounds by Hao and Heydenreich for all parameters settings in which those bounds were not tight. It gives tail bounds on the probability of having short paths, which imply shape theorems for the $k$-neighbourhood of a vertex whenever our lower bounds are tight, and tight bounds for the size of this $k$-neighbourhood. And last but not least, our proof is much simpler and not much longer than two pages, and we demonstrate that it generalizes well by showing that the same technique also works for first passage percolation.

math.PR↗

arXiv2023

Learning-Augmented Online TSP on Rings, Trees, Flowers and (almost) Everywhere Else

We study the Online Traveling Salesperson Problem (OLTSP) with predictions. In OLTSP, a sequence of initially unknown requests arrive over time at points (locations) of a metric space. The goal is, starting from a particular point of the metric space (the origin), to serve all these requests while minimizing the total time spent. The server moves with unit speed or is "waiting" (zero speed) at some location. We consider two variants: in the open variant, the goal is achieved when the last request is served. In the closed one, the server additionally has to return to the origin. We adopt a prediction model, introduced for OLTSP on the line, in which the predictions correspond to the locations of the requests and extend it to more general metric spaces. We first propose an oracle-based algorithmic framework, inspired by previous work. This framework allows us to design online algorithms for general metric spaces that provide competitive ratio guarantees which, given perfect predictions, beat the best possible classical guarantee (consistency). Moreover, they degrade gracefully along with the increase in error (smoothness), but always within a constant factor of the best known competitive ratio in the classical case (robustness). Having reduced the problem to designing suitable efficient oracles, we describe how to achieve this for general metric spaces as well as specific metric spaces (rings, trees and flowers), the resulting algorithms being tractable in the latter case. The consistency guarantees of our algorithms are tight in almost all cases, and their smoothness guarantees only suffer a linear dependency on the error, which we show is necessary. Finally, we provide robustness guarantees improving previous results.

cs.DS↗