arXiv subjects

Tijn de Vos

Publications and source records attributed to Tijn de Vos.

17 recordsLinked to original sources

arXiv2026

Distributed Santa Claus via Global Rounding

In this paper, we initiate the study of a new class of problems in the CONGEST model: Mixed packing and covering linear programs (LP). Previously, the design of optimization algorithms in the distributed setting was heavily focused on solving linear programs with either packing or covering constraints. We are the first to explore the class of linear programs with both packing and covering constraints by providing a general-purpose CONGEST solver for such LPs and by studying the sequentially well-studied Santa Claus problem as a central representative. This NP-hard problem can be modeled as a bipartite graph of children and gifts where an edge indicates that a child desires a gift. The goal is to assign the gifts to the children such that the least happy child is as happy as possible. Even though this is a well-studied problem in the sequential setting, we provide the first results in the distributed setting. In particular, we show that the complexity of computing an $\mathcal{O}(\log n/\log \log n)$-approximation is $\widehat Θ(\sqrt n+D)$ rounds, where our $\widetildeΩ(\sqrt n+D)$-round lower bound even holds for any approximation.

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arXiv2026

Tree-Packing Revisited: Faster Fully Dynamic Min-Cut and Arboricity

A tree-packing is a collection of spanning trees of a graph. It has been a useful tool for computing the minimum cut in static, dynamic, and distributed settings. In particular, [Thorup, Comb. 2007] used them to obtain his dynamic min-cut algorithm with $\tilde O(λ^{14.5}\sqrt{n})$ worst-case update time. We reexamine this relationship, showing that we need to maintain fewer spanning trees for such a result; we show that we only need to pack $Θ(λ^3 \log m)$ greedy trees to guarantee a 1-respecting cut or a trivial cut in some contracted graph. Based on this structural result, we then provide a deterministic algorithm for fully dynamic exact min-cut, that has $\tilde O(λ^{5.5}\sqrt{n})$ worst-case update time, for min-cut value bounded by $λ$. In particular, this also leads to an algorithm for general fully dynamic exact min-cut with $\tilde O(m^{1-1/12})$ amortized update time, improving upon $\tilde O(m^{1-1/31})$ [Goranci et al., SODA 2023]. We also give the first fully dynamic algorithm that maintains a $(1+\varepsilon)$-approximation of the fractional arboricity -- which is strictly harder than the integral arboricity. Our algorithm is deterministic and has $O(α\log^6m/\varepsilon^4)$ amortized update time, for arboricity at most $α$. We extend these results to a Monte Carlo algorithm with $O(\text{poly}(\log m,\varepsilon^{-1}))$ amortized update time against an adaptive adversary. Our algorithms work on multi-graphs as well. Both result are obtained by exploring the connection between the min-cut/arboricity and (greedy) tree-packing. We investigate tree-packing in a broader sense; including a lower bound for greedy tree-packing, which - to the best of our knowledge - is the first progress on this topic since [Thorup, Comb. 2007].

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arXiv2026

Parallel Minimum Cost Flow in Near-Linear Work and Square Root Depth for Dense Instances

For $n$-vertex $m$-edge graphs with integer polynomially-bounded costs and capacities, we provide a randomized parallel algorithm for the minimum cost flow problem with $\tilde O(m+n^ {1.5})$ work and $\tilde O(\sqrt{n})$ depth. On moderately dense graphs ($m>n^{1.5}$), our algorithm is the first one to achieve both near-linear work and sub-linear depth. Previous algorithms are either achieving almost optimal work but are highly sequential [Chen, Kyng, Liu, Peng, Gutenberg, Sachdev, FOCS'22], or achieving sub-linear depth but use super-linear work, [Lee, Sidford, FOCS'14], [Orlin, Stein, Oper. Res. Lett.'93]. Our result also leads to improvements for the special cases of max flow, bipartite maximum matching, shortest paths, and reachability. Notably, the previous algorithms achieving near-linear work for shortest paths and reachability all have depth $n^{o(1)}\cdot \sqrt{n}$ [Fischer, Haeupler, Latypov, Roeyskoe, Sulser, SOSA'25], [Liu, Jambulapati, Sidford, FOCS'19]. Our algorithm consists of a parallel implementation of [van den Brand, Lee, Liu, Saranurak, Sidford, Song, Wang, STOC'21]. One important building block is a \emph{dynamic} parallel expander decomposition, which we show how to obtain from the recent parallel expander decomposition of [Chen, Meierhans, Probst Gutenberh, Saranurak, SODA'25].

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arXiv2026

Deterministic Edge Coloring with few Colors in CONGEST

As the main contribution of this work we present deterministic edge coloring algorithms in the CONGEST model. In particular, we present an algorithm that edge colors any $n$-node graph with maximum degree $Δ$ with with $(1+\varepsilon)Δ+O(\sqrt{\log n})$ colors in $\tilde{O}(\log^{2.5} n+\log^2 Δ\log n)$ rounds. This brings the upper bound polynomially close to the lower bound of $Ω(\log n/\log\log n)$ rounds that also holds in the more powerful LOCAL model [Chang, He, Li, Pettie, Uitto; SODA'18]. As long as $Δ\geq c\sqrt{\log n}$ our algorithm uses fewer than $2Δ-1$ colors and to the best of our knowledge is the first polylogarithmic-round CONGEST algorithm achieving this for any range of $Δ$. As a corollary we also improve the complexity of edge coloring with $2Δ-1$ colors for all ranges of $Δ$ to $\tilde{O}(\log^{2.5} n+\log^2 Δ\log n)$. This improves upon the previous $O(\log^8 n)$-round algorithm from [Fischer, Ghaffari, Kuhn; FOCS'17]. Our approach builds on a refined analysis and extension of the online edge-coloring algorithm of Blikstad, Svensson, Vintan, and Wajc [FOCS'25], and more broadly on new connections between online and distributed graph algorithms. We show that their algorithm exhibits very low locality and, if it can additionally have limited local access to future edges (as distributed algorithms can), it can be derandomized for smaller degrees. Under this additional power, we are able to bypass classical online lower bounds and translate the results to efficient distributed algorithms. This leads to our CONGEST algorithm for $(1+\varepsilon)Δ+O(\sqrt{\log n})$-edge coloring. Since the modified online algorithm can be implemented more efficiently in the LOCAL model, we also obtain (marginally) improved complexity bounds in that model.

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arXiv2026

Distributed Sparsest Cut via Eigenvalue Estimation

We give new, improved bounds for approximating the sparsest cut value or in other words the conductance $ϕ$ of a graph in the CONGEST model. As our main result, we present an algorithm running in $O(\log^2 n/ϕ)$ rounds in which every vertex outputs a value $\tilde ϕ$ satisfying $ϕ\le \tilde ϕ\le \sqrt{2.01ϕ}$. In most regimes, our algorithm improves significantly over the previously fastest algorithm for the problem [Chen, Meierhans, Probst Gutenberg, Saranurak; SODA 25]. Additionally, our result generalizes to $k$-way conductance. We obtain these results, by approximating the eigenvalues of the normalized Laplacian matrix $L:=I-{\rm Deg}^{-1/2}A{\rm Deg}^ {-1/2}$, where, $A$ is the adjacency matrix and Deg is the diagonal matrix with the weighted degrees on the diagonal. We show our algorithms are near-optimal by proving a lower bound for computing the smallest non-trivial eigenvalue of $L$, even in the stronger LOCAL model The previous state of the art sparsest cut algorithm is in the technical realm of expander decompositions. Our algorithms, on the other hand, are relatively simple and easy to implement. At the core, they rely on the well-known power method, which comes down to repeatedly multiplying the Laplacian with a vector. This operation can be performed in a single round in the CONGEST model. All our algorithms apply to weighted, undirected graphs. Our lower bounds apply even in unweighted graphs.

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arXiv2026

Deterministic Distance Approximation in MPC via Improved Hitting Sets

In this paper, we provide the first deterministic algorithms with sublogarithmic round complexity for spanners and approximate shortest paths in various MPC models. Moreover, we significantly improve upon the state of the art in the deterministic Congested Clique. In particular, we obtain the following four results on undirected graphs: 1. In both linear MPC and Congested Clique, we obtain an $O(k)$ stretch-spanner of a weighted graph of size $O(n^{1+1/k})$ in $O(1)$ rounds, for some parameter $k\ge 0$. For $k=O(\log{n})$, this leads to an $O(\log n)$ approximation of APSP in constant rounds in both models. 2. In sublinear MPC, we obtain an $O(k^{1+\varepsilon})$-stretch spanner of a weighted graph of size $O(n^{1+1/k})$ in $O(\log k)$ rounds, for any fixed constant $\varepsilon>0$. 3. In Congested Clique, we obtain $O(1)$-approximate APSP for weighted graphs in $O(\log \log \log n)$ rounds. 4. In near-linear MPC, we obtain $(1+\varepsilon)$-approximate single-source shortest paths and $O(1)$-approximate all-pairs shortest paths for unweighted graphs in $\textsf{poly}\log \log n$ rounds. Our algorithm only requires a single near-linear memory machine, where the rest can have sublinear memory. Our deterministic algorithms obtain similar guarantees to the state of the art randomized algorithms without incurring additional factors in the round complexity. To obtain these results, we inspect the randomized algorithms and isolate a randomized sampling routine. Then we derandomize these sampling routines by using a deterministic hitting set. Hereto, we develop a versatile deterministic hitting set algorithm, which we hope will have further derandomization applications.

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arXiv2026

Dynamic Matroids: Base Packing and Covering

In this paper, we consider dynamic matroids, where elements can be inserted to or deleted from the ground set over time. The independent sets change to reflect the current ground set. As matroids are central to the study of many combinatorial optimization problems, it is a natural next step to also consider them in a dynamic setting. The study of dynamic matroids has the potential to generalize several dynamic graph problems, including, but not limited to, arboricity and maximum bipartite matching. We contribute by providing efficient algorithms for some fundamental matroid questions. In particular, we study the most basic question of maintaining a base dynamically, providing an essential building block for future algorithms. We further utilize this result and consider the elementary problems of base packing and base covering. We provide a deterministic algorithm that maintains a $(1\pm \varepsilon)$-approximation of the base packing number $Φ$ in $O(Φ\cdot \text{poly}(\log n, \varepsilon^{-1}))$ queries per update. Similarly, we provide a deterministic algorithm that maintains a $(1\pm \varepsilon)$-approximation of the base covering number $β$ in $O(β\cdot \text{poly}(\log n, \varepsilon^{-1}))$ queries per update. Moreover, we give an algorithm that maintains a $(1\pm \varepsilon)$-approximation of the base covering number $β$ in $O(\text{poly}(\log n, \varepsilon^{-1}))$ queries per update against an oblivious adversary. These results are obtained by exploring the relationship between base collections, a generalization of tree-packings, and base packing and covering respectively. We provide structural theorems to formalize these connections, and show how they lead to simple dynamic algorithms.

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arXiv2025

Towards Constant Time Multi-Call Rumor Spreading on Small-Set Expanders

We study a multi-call variant of the classic PUSH&PULL rumor spreading process where nodes can contact $k$ of their neighbors instead of a single one during both PUSH and PULL operations. We show that rumor spreading can be made faster at the cost of an increased amount of communication between the nodes. As a motivating example, consider the process on a complete graph of $n$ nodes: while the standard PUSH&PULL protocol takes $Θ(\log n)$ rounds, we prove that our $k$-PUSH&PULL variant completes in $Θ(\log_{k} n)$ rounds, with high probability. We generalize this result in an expansion-sensitive way, as has been done for the classic PUSH&PULL protocol for different notions of expansion, e.g., conductance and vertex expansion. We consider small-set vertex expanders, graphs in which every sufficiently small subset of nodes has a large neighborhood, ensuring strong local connectivity. In particular, when the expansion parameter satisfies $ϕ> 1$, these graphs have a diameter of $o(\log n)$, as opposed to other standard notions of expansion. Since the graph's diameter is a lower bound on the number of rounds required for rumor spreading, this makes small-set expanders particularly well-suited for fast information dissemination. We prove that $k$-PUSH&PULL takes $O(\log_ϕ n \cdot \log_{k} n)$ rounds in these expanders, with high probability. We complement this with a simple lower bound of $Ω(\log_ϕ n+ \log_{k} n)$ rounds.

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arXiv2024

New Tradeoffs for Decremental Approximate All-Pairs Shortest Paths

We provide new tradeoffs between approximation and running time for the decremental all-pairs shortest paths (APSP) problem. For undirected graphs with $m$ edges and $n$ nodes undergoing edge deletions, we provide four new approximate decremental APSP algorithms, two for weighted and two for unweighted graphs. Our first result is $(2+ ε)$-APSP with total update time $\tilde{O}(m^{1/2}n^{3/2})$ (when $m= n^{1+c}$ for any constant $0 0$). For comparison, in the special case of $(1+ε, 2)$-approximation, this improves over the state-of-the-art algorithm by [Henzinger, Krinninger, Nanongkai, SICOMP 2016] with total update time of $\tilde{O}(n^{2.5})$. All of our results are randomized, work against an oblivious adversary, and have constant query time.

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arXiv2023

Fast Algorithms for Energy Games in Special Cases

In this paper, we study algorithms for special cases of energy games, a class of turn-based games on graphs that show up in the quantitative analysis of reactive systems. In an energy game, the vertices of a weighted directed graph belong either to Alice or to Bob. A token is moved to a next vertex by the player controlling its current location, and its energy is changed by the weight of the edge. Given a fixed starting vertex and initial energy, Alice wins the game if the energy of the token remains nonnegative at every moment. If the energy goes below zero at some point, then Bob wins. The problem of determining the winner in an energy game lies in $\mathsf{NP} \cap \mathsf{coNP}$. It is a long standing open problem whether a polynomial time algorithm for this problem exists. We devise new algorithms for three special cases of the problem. The first two results focus on the single-player version, where either Alice or Bob controls the whole game graph. We develop an $\tilde{O}(n^ωW^ω)$ time algorithm for a game graph controlled by Alice, by providing a reduction to the All-Pairs Nonnegative Prefix Paths problem (APNP). Thus we study the APNP problem separately, for which we develop an $\tilde{O}(n^ωW^ω)$ time algorithm. For both problems, we improve over the state of the art of $\tilde O(mn)$ for small $W$. For the APNP problem, we also provide a conditional lower bound which states that there is no $O(n^{3-ε})$ time algorithm for any $ε> 0$, unless the APSP Hypothesis fails. For a game graph controlled by Bob, we obtain a near-linear time algorithm. Regarding our third result, we present a variant of the value iteration algorithm, and we prove that it gives an $O(mn)$ time algorithm for game graphs without negative cycles, which improves a previous upper bound.

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arXiv2023

Fast 2-Approximate All-Pairs Shortest Paths

In this paper, we revisit the classic approximate All-Pairs Shortest Paths (APSP) problem in undirected graphs. For unweighted graphs, we provide an algorithm for $2$-approximate APSP in $\tilde O(n^{2.5-r}+n^{ω(r)})$ time, for any $r\in[0,1]$. This is $O(n^{2.032})$ time, using known bounds for rectangular matrix multiplication $n^{ω(r)}$ [Le Gall, Urrutia, SODA 2018]. Our result improves on the $\tilde{O}(n^{2.25})$ bound of [Roditty, STOC 2023], and on the $\tilde{O}(m\sqrt n+n^2)$ bound of [Baswana, Kavitha, SICOMP 2010] for graphs with $m\geq n^{1.532}$ edges. For weighted graphs, we obtain $(2+ε)$-approximate APSP in $\tilde O(n^{3-r}+n^{ω(r)})$ time, for any $r\in [0,1]$. This is $O(n^{2.214})$ time using known bounds for $ω(r)$. It improves on the state of the art bound of $O(n^{2.25})$ by [Kavitha, Algorithmica 2012]. Our techniques further lead to improved bounds in a wide range of density for weighted graphs. In particular, for the sparse regime we construct a distance oracle in $\tilde O(mn^{2/3})$ time that supports $2$-approximate queries in constant time. For sparse graphs, the preprocessing time of the algorithm matches conditional lower bounds [Patrascu, Roditty, Thorup, FOCS 2012; Abboud, Bringmann, Fischer, STOC 2023]. To the best of our knowledge, this is the first 2-approximate distance oracle that has subquadratic preprocessing time in sparse graphs. We also obtain new bounds in the near additive regime for unweighted graphs. We give faster algorithms for $(1+ε,k)$-approximate APSP, for $k=2,4,6,8$. We obtain these results by incorporating fast rectangular matrix multiplications into various combinatorial algorithms that carefully balance out distance computation on layers of sparse graphs preserving certain distance information.

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arXiv2023

The Laplacian Paradigm in Deterministic Congested Clique

In this paper, we bring the techniques of the Laplacian paradigm to the congested clique, while further restricting ourselves to deterministic algorithms. In particular, we show how to solve a Laplacian system up to precision $ε$ in $n^{o(1)}\log(1/ε)$ rounds. We show how to leverage this result within existing interior point methods for solving flow problems. We obtain an $m^{3/7+o(1)}U^{1/7}$ round algorithm for maximum flow on a weighted directed graph with maximum weight $U$, and we obtain an $\tilde{O}(m^{3/7}(n^{0.158}+n^{o(1)}\text{poly}\log W))$ round algorithm for unit capacity minimum cost flow on a directed graph with maximum cost $W$. Hereto, we give a novel routine for computing Eulerian orientations in $O(\log n \log^* n)$ rounds, which we believe may be of separate interest.

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arXiv2023

Minimum Cost Flow in the CONGEST Model

We consider the CONGEST model on a network with $n$ nodes, $m$ edges, diameter $D$, and integer costs and capacities bounded by $\text{poly} n$. In this paper, we show how to find an exact solution to the minimum cost flow problem in $n^{1/2+o(1)}(\sqrt{n}+D)$ rounds, improving the state of the art algorithm with running time $m^{3/7+o(1)}(\sqrt nD^{1/4}+D)$ [Forster et al. FOCS 2021], which only holds for the special case of unit capacity graphs. For certain graphs, we achieve even better results. In particular, for planar graphs, expander graphs, $n^{o(1)}$-genus graphs, $n^{o(1)}$-treewidth graphs, and excluded-minor graphs our algorithm takes $n^{1/2+o(1)}D$ rounds. We obtain this result by combining recent results on Laplacian solvers in the CONGEST model [Forster et al. FOCS 2021, Anagnostides et al. DISC 2022] with a CONGEST implementation of the LP solver of Lee and Sidford [FOCS 2014], and finally show that we can round the approximate solution to an exact solution. Our algorithm solves certain linear programs, that generalize minimum cost flow, up to additive error $ε$ in $n^{1/2+o(1)}(\sqrt{n}+D)\log^3 (1/ε)$ rounds.

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arXiv2022

A Framework for Distributed Quantum Queries in the CONGEST Model

The Quantum CONGEST model is a variant of the CONGEST model, where messages consist of $O(\log(n))$ qubits. We give a general framework for implementing quantum query algorithms in Quantum CONGEST, using the concept of parallel-queries. We apply our framework for distributed quantum queries in two settings: when data is distributed over the network, and graph theoretical problems where the network defines the input. The first is slightly unusual in CONGEST but our results follow almost directly. The second is more traditional for the CONGEST model but here we require some classical CONGEST steps to get our results. In the setting with distributed data, we show how a network can schedule a meeting in one of $k$ dates using $\tilde{O}(\sqrt{kD}+D)$ rounds, with $D$ the network diameter. We also give an algorithm for element distinctness: if all nodes together hold a list of $k$ numbers, they can find a duplicate in $\tilde O(k^{2/3}D^{1/3}+D)$ rounds. We also generalize the protocol for the distributed Deutsch-Jozsa problem from the two-party setting considered in [arXiv:quant-ph/9802040] to general networks, giving a novel separation between exact classical and exact quantum protocols in CONGEST. When the input is the network structure itself, we almost directly recover the $O(\sqrt{nD})$ round diameter computation algorithm of Le Gall and Magniez [arXiv:1804.02917]. We also compute the radius in the same number of rounds, and give an $ε$-additive approximation of the average eccentricity in $\tilde{O}(D+D^{3/2}/ε)$ rounds. Finally, we give quantum speedups for the problems of cycle detection and girth computation. We detect whether a graph has a cycle of length at most $k$ in $O(k+(kn)^{1/2-1/Θ(k)})$ rounds. For girth computation we give an $\tilde{O}(g+(gn)^{1/2-1/Θ(g)})$ round algorithm for graphs with girth $g$, beating the known classical lower bound.

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arXiv2022

The Laplacian Paradigm in the Broadcast Congested Clique

In this paper, we bring the main tools of the Laplacian paradigm to the Broadcast Congested Clique. We introduce an algorithm to compute spectral sparsifiers in a polylogarithmic number of rounds, which directly leads to an efficient Laplacian solver. Based on this primitive, we consider the linear program solver of Lee and Sidford (FOCS 2014). We show how to solve certain linear programs up to additive error $ε$ with $n$ constraints on an $n$-vertex Broadcast Congested Clique network in $\tilde O(\sqrt{n}\log(1/ε))$ rounds. Using this, we show how to find an exact solution to the minimum cost flow problem in $\tilde O(\sqrt{n})$ rounds.

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arXiv2022

Faster Cut Sparsification of Weighted Graphs

A cut sparsifier is a reweighted subgraph that maintains the weights of the cuts of the original graph up to a multiplicative factor of $(1\pmε)$. This paper considers computing cut sparsifiers of weighted graphs of size $O(n\log (n)/ε^2)$. Our algorithm computes such a sparsifier in time $O(m\cdot\min(α(n)\log(m/n),\log (n)))$, both for graphs with polynomially bounded and unbounded integer weights, where $α(\cdot)$ is the functional inverse of Ackermann's function. This improves upon the state of the art by Benczúr and Karger (SICOMP 2015), which takes $O(m\log^2 (n))$ time. For unbounded weights, this directly gives the best known result for cut sparsification. Together with preprocessing by an algorithm of Fung et al. (SICOMP 2019), this also gives the best known result for polynomially-weighted graphs. Consequently, this implies the fastest approximate min-cut algorithm, both for graphs with polynomial and unbounded weights. In particular, we show that it is possible to adapt the state of the art algorithm of Fung et al. for unweighted graphs to weighted graphs, by letting the partial maximum spanning forest (MSF) packing take the place of the Nagamochi-Ibaraki (NI) forest packing. MSF packings have previously been used by Abraham at al. (FOCS 2016) in the dynamic setting, and are defined as follows: an $M$-partial MSF packing of $G$ is a set $\mathcal{F}=\{F_1, \dots, F_M\}$, where $F_i$ is a maximum spanning forest in $G\setminus \bigcup_{j=1}^{i-1}F_j$. Our method for computing (a sufficient estimation of) the MSF packing is the bottleneck in the running time of our sparsification algorithm.

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arXiv2021

An Improved Random Shift Algorithm for Spanners and Low Diameter Decompositions

Spanners have been shown to be a powerful tool in graph algorithms. Many spanner constructions use a certain type of clustering at their core, where each cluster has small diameter and there are relatively few spanner edges between clusters. In this paper, we provide a clustering algorithm that, given $k\geq 2$, can be used to compute a spanner of stretch $2k-1$ and expected size $O(n^{1+1/k})$ in $k$ rounds in the CONGEST model. This improves upon the state of the art (by Elkin, and Neiman [TALG'19]) by making the bounds on both running time and stretch independent of the random choices of the algorithm, whereas they only hold with high probability in previous results. Spanners are used in certain synchronizers, thus our improvement directly carries over to such synchronizers. Furthermore, for keeping the \emph{total} number of inter-cluster edges small in low diameter decompositions, our clustering algorithm provides the following guarantees. Given $β\in (0,1]$, we compute a low diameter decomposition with diameter bound $O\left(\frac{\log n}β\right)$ such that each edge $e\in E$ is an inter-cluster edge with probability at most $β\cdot w(e)$ in $O\left(\frac{\log n}β\right)$ rounds in the CONGEST model. Again, this improves upon the state of the art (by Miller, Peng, and Xu [SPAA'13]) by making the bounds on both running time and diameter independent of the random choices of the algorithm, whereas they only hold with high probability in previous results.

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