arXiv subjects

Mads Vestergaard Jensen

Publications and source records attributed to Mads Vestergaard Jensen.

4 recordsLinked to original sources

arXiv2026

Instance-Optimality of Bidirectional Dijkstra on Simple Graphs

We study the shortest-path problem on graphs with positive real-valued edge weights. Given a source vertex $s$ and a target vertex $t$, the goal is to calculate the length of the shortest path from $s$ to $t$. We are particularly interested in instances that can be solved in sublinear time. Recently, Haeupler, Hlad\'ik, Rozho\v{n}, Tarjan, and T\v{e}tek proved that (a version of) the bidirectional Dijkstra's algorithm is instance-optimal on positively weighted multigraphs, both directed and undirected, considering the number of vertices and edges queried by the algorithm. However, multigraphs are not the canonical setting for the shortest-path problem. The problem is typically formulated on simple graphs without loops and parallel edges. They therefore left as an open problem whether bidirectional Dijkstra remains instance-optimal on simple weighted graphs. We answer this question, but for simple graphs, the answer is more complex, depending on the setting. We show that bidirectional Dijkstra is still instance-optimal on simple undirected weighted graphs under the order-oblivious model, where incident edges are given in a random order. In contrast, under the order-dependent model, where incident edges have a given order, we show that bidirectional Dijkstra is not instance-optimal. For simple directed weighted graphs, we show that bidirectional Dijkstra is not instance-optimal under either the order-oblivious or the order-dependent model, being off by a factor of $\Theta(m/n)$ in both cases. We further show that no algorithm can have instance-optimality ratio $o(m/n)$ under the order-dependent model, or under the order-oblivious model when $m=O(n\sqrt{n})$. On the positive side, the above results imply that bidirectional Dijkstra is instance-optimal up to logarithmic factors on all sparse directed and undirected graphs satisfying $m/n=\log^{O(1)} n$.

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arXiv2026

Personalized PageRank Estimation in Undirected Graphs

Given an undirected graph $G=(V, E)$, the Personalized PageRank (PPR) of $t\in V$ with respect to $s\in V$, denoted $\pi(s,t)$, is the probability that an $\alpha$-discounted random walk starting at $s$ terminates at $t$. We study the time complexity of estimating $\pi(s,t)$ with constant relative error and constant failure probability, whenever $\pi(s,t)$ is above a given threshold parameter $\delta\in(0,1)$. We consider common graph-access models and furthermore study the single source, single target, and single node (PageRank centrality) variants of the problem. We provide a complete characterization of PPR estimation in undirected graphs by giving tight bounds (up to logarithmic factors) for all problems and model variants in both the worst-case and average-case setting. This includes both new upper and lower bounds. Tight bounds were recently obtained by Bertram, Jensen, Thorup, Wang, and Yan for directed graphs. However, their lower bound constructions rely on asymmetry and therefore do not carry over to undirected graphs. At the same time, undirected graphs exhibit additional structure that can be exploited algorithmically. Our results resolve the undirected case by developing new techniques that capture both aspects, yielding tight bounds.

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arXiv2025

Dynamic Meta-Kernelization

Kernelization studies polynomial-time preprocessing algorithms. Over the last 20 years, the most celebrated positive results of the field have been linear kernels for classical NP-hard graph problems on sparse graph classes. In this paper, we lift these results to the dynamic setting. As the canonical example, Alber, Fellows, and Niedermeier [J. ACM 2004] gave a linear kernel for dominating set on planar graphs. We provide the following dynamic version of their kernel: Our data structure is initialized with an $n$-vertex planar graph $G$ in $O(n \log n)$ amortized time, and, at initialization, outputs a planar graph $K$ with $\mathrm{OPT}(K) = \mathrm{OPT}(G)$ and $|K| = O(\mathrm{OPT}(G))$, where $\mathrm{OPT}(\cdot)$ denotes the size of a minimum dominating set. The graph $G$ can be updated by insertions and deletions of edges and isolated vertices in $O(\log n)$ amortized time per update, under the promise that it remains planar. After each update to $G$, the data structure outputs $O(1)$ updates to $K$, maintaining $\mathrm{OPT}(K) = \mathrm{OPT}(G)$, $|K| = O(\mathrm{OPT}(G))$, and planarity of $K$. Furthermore, we obtain similar dynamic kernelization algorithms for all problems satisfying certain conditions on (topological-)minor-free graph classes. Besides kernelization, this directly implies new dynamic constant-approximation algorithms and improvements to dynamic FPT algorithms for such problems. Our main technical contribution is a dynamic data structure for maintaining an approximately optimal protrusion decomposition of a dynamic topological-minor-free graph. Protrusion decompositions were introduced by Bodlaender, Fomin, Lokshtanov, Penninkx, Saurabh, and Thilikos [J. ACM 2016], and have since developed into a part of the core toolbox in kernelization and parameterized algorithms.

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arXiv2025

Estimating Random-Walk Probabilities in Directed Graphs

We study discounted random walks in directed graphs. In each step, the walk either terminates with a constant probability $\alpha$, or proceeds to a random out-neighbor. Our goal is to estimate the probability $\pi(s, t)$ that a discounted random walk starting from $s$ terminates at $t$. This probability is also known as the Personalized PageRank (PPR) score, which measures the relevance of $t$ to $s$, for instance, when $s$ and $t$ are web pages on the Internet. We aim to estimate $\pi(s, t)$ within a constant relative error with constant probability. A variety of algorithms have been developed for several problem variants, such as single-pair, single-source, single-target, and single-node estimation, under both worst-case and average-case settings, and for different combinations of allowed graph queries. However, in many important cases, there remain polynomial gaps between known upper and lower bounds. In this paper, we establish tight upper and lower bounds (up to logarithmic factors of $n$) for all problem variants and query combinations, closing all existing gaps in both the worst-case and average-case settings. Below we give some examples for the worst-case settings. As an upper-bound example, the classic power method estimates $\pi(s,t)$ if it is above a threshold $\delta$ in time $O(m\log(1/\delta))$ but $\pi(s,t)$ can be as small as $1/n^{\Theta(n)}$. For contrast, we propose algorithms that deterministically estimate arbitrarily small $\pi(s,t)$ in $O(m\log n)$ time. As a lower-bound example, we improve the lower bound for the single-pair problem from $\Omega(\min\{n,1/\delta\})$ to $\Omega(\min\{m,1/\delta\})$, which is optimal (up to logarithmic factors) since a simple Monte Carlo estimate takes $O(1/\delta)$ time.

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