arXiv subjects

Sebastian Brandt

Publications and source records attributed to Sebastian Brandt.

At least 19 recordsLinked to original sources

arXiv2026

A Fast Deterministic Algorithm for $(\Delta+1)$-edge coloring in CONGEST

Vizing's theorem states that any graph of maximum degree $\Delta$ can be properly edge-colored with $\Delta + 1$ colors (which is optimal in general). A recent breakthrough result by Bernshteyn showed that such a $(\Delta + 1)$-edge coloring can be found deterministically in $poly(\Delta,\log n)$ rounds in the LOCAL model of distributed computing, where $n$ denotes the number of vertices of the input graph [J. Comb. Theory 2022]. Since then, the exponent in the $poly(\log n)$-part of the runtime has been improved by Christiansen [STOC 2023] and Bernshteyn and Dhawan [J. Comb. Theory, Series B, 2025]. However, the algorithms used in all of these works use large messages, leaving open the question for efficient algorithms in the more restrictive CONGEST model. We answer this question by presenting the first $poly(\Delta,\log n)$-round algorithm for $(\Delta + 1)$-edge coloring in the CONGEST model. Our algorithm is deterministic and the $n$-dependency of its runtime, $\tilde{O}(\log^5 n)$, matches the best published dependency in the LOCAL model.

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arXiv2026

The Distributed Complexity Landscape on Trees Depends on the Knowledge About the Network Size

One of the central models in distributed computing is Linial's LOCAL model [SIAM J. Comp. 1992]. Over time, researchers have studied distributed graph problems in the LOCAL model under slightly different assumptions, such as whether nodes know the exact network size $n$, only a polynomial upper bound on $n$, or nothing at all. We ask whether these differences are merely technical or fundamentally affect the theory of Locally Checkable Labelings (LCLs), one of the most studied problem classes. LCLs are graph problems whose valid solutions can be characterized by a finite set of allowed constant-radius neighborhoods. Since their introduction by Naor and Stockmeyer [FOCS 1995], they have become central in distributed computing, and the last decade has seen major progress in understanding their complexity. For example, Chang, Kopelowitz, and Pettie [FOCS 2016] showed that the randomized complexity of any LCL on $n$-node graphs is at least its deterministic complexity on $\sqrt{\log n}$-node graphs. Later, Chang and Pettie [FOCS 2017] showed that any randomized $n^{o(1)}$-round algorithm for LCLs on bounded-degree trees can be turned into a deterministic $O(\log n)$-round algorithm. Then, Balliu et al. [STOC 2018] showed that such automatic speedups are impossible for general bounded-degree graphs. However, these results fundamentally rely on nodes knowing $n$. How much does this assumption affect the theory of LCLs? Our work shows that if nodes are oblivious to $n$, or know only a polynomial upper bound on it, then even on trees, the theory of LCLs changes significantly. While the fundamental classification of problems remains the same, we show the landscape becomes much more complex: for example, for LCLs, randomness helps in more cases; some problems have very unnatural complexities; and some have a lower bound that depends on which definition of $\Omega$ we use!

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arXiv2025

On the Complexity of Distributed Edge Coloring and Orientation Problems

Understanding the role of randomness when solving locally checkable labeling (LCL) problems in the LOCAL model has been one of the top priorities in the research on distributed graph algorithms in recent years. For LCL problems in bounded-degree graphs, it is known that randomness cannot help more than polynomially, except in one case: if the deterministic complexity of an LCL problem is in $\Omega(\log n)$ and its randomized complexity is in $o(\log n)$, then the randomized complexity is guaranteed to be $poly(\log \log n)$. However, the fundamental question of \emph{which} problems with a deterministic complexity of $\Omega(\log n)$ can be solved exponentially faster using randomization still remains wide open. We make a step towards answering this question by studying a simple, but natural class of LCL problems: so-called degree splitting problems. These problems come in two varieties: coloring problems where the edges of a graph have to be colored with $2$ colors and orientation problems where each edge needs to be oriented. For $\Delta$-regular graphs (where $\Delta=O(1)$), we obtain the following results. - We gave an exact characterization of the randomized complexity of all problems where the edges need to be colored with two colors, say red and blue, and which have a deterministic complexity of $O(\log n)$. - For edge orientation problems, we give a partial characterization of the problems that have a randomized complexity of $\Omega(\log n)$ and the problems that have a randomized complexity of $poly\log\log n$. While our results are cleanest to state for the $\Delta$-regular case, all our algorithms naturally generalize to nodes of any degree $d<\Delta$ in general graphs of maximum degree $\Delta$.

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arXiv2025

On the Universality of Round Elimination Fixed Points

Recent work on distributed graph algorithms [e.g. STOC 2022, ITCS 2022, PODC 2020] has drawn attention to the following open question: are round elimination fixed points a universal technique for proving lower bounds? That is, given a locally checkable problem $\Pi$ that requires at least $\Omega(\log n)$ rounds in the deterministic LOCAL model, can we always find a relaxation $\Pi'$ of $\Pi$ that is a nontrivial fixed point for the round elimination technique [see STOC 2016, PODC 2019]? If yes, then a key part of distributed computational complexity would be also decidable. The key obstacle so far has been a certain family of homomorphism problems [ITCS 2022], which require $\Omega(\log n)$ rounds, but the only known proof is based on Marks' technique [J. AMS 2016]. We develop a new technique for constructing round elimination lower bounds systematically. Using so-called tripotent inputs we show that the aforementioned homomorphism problems indeed admit a lower bound proof that is based on round elimination fixed points. Hence we eliminate the only known obstacle for the universality of round elimination. Yet we also present a new obstacle: we show that there are some problems with inputs that require $\Omega(\log n)$ rounds, yet there is no proof that is based on relaxations to nontrivial round elimination fixed points. Hence round elimination cannot be a universal technique for problems with inputs (but it might be universal for problems without inputs). We also prove the first fully general lower bound theorem that is applicable to any problem, with or without inputs, that is a fixed point in round elimination. Prior results of this form were only able to handle certain very restricted inputs.

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arXiv2025

A Post-Quantum Lower Bound for the Distributed Lov\'asz Local Lemma

In this work, we study the Lov\'asz local lemma (LLL) problem in the area of distributed quantum computing, which has been the focus of attention of recent advances in quantum computing [STOC'24, STOC'25, STOC'25]. We prove a lower bound of $2^{\Omega(\log^* n)}$ for the complexity of the distributed LLL in the quantum-LOCAL model. More specifically, we obtain our lower bound already for a very well-studied special case of the LLL, called sinkless orientation, in a stronger model than quantum-LOCAL, called the randomized online-LOCAL model. As a consequence, we obtain the same lower bounds for sinkless orientation and the distributed LLL also in a variety of other models studied across different research communities. Our work provides the first superconstant lower bound for sinkless orientation and the distributed LLL in all of these models, addressing recently stated open questions. Moreover, to obtain our results, we develop an entirely new lower bound technique that we believe has the potential to become the first generic technique for proving post-quantum lower bounds for many of the most important problems studied in the context of locality.

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arXiv2025

Faster Distributed $\Delta$-Coloring via a Reduction to MIS

Recent improvements on the deterministic complexities of fundamental graph problems in the LOCAL model of distributed computing have yielded state-of-the-art upper bounds of $\tilde{O}(\log^{5/3} n)$ rounds for maximal independent set (MIS) and $(\Delta + 1)$-coloring [Ghaffari, Grunau, FOCS'24] and $\tilde{O}(\log^{19/9} n)$ rounds for the more restrictive $\Delta$-coloring problem [Ghaffari, Kuhn, FOCS'21; Ghaffari, Grunau, FOCS'24; Bourreau, Brandt, Nolin, STOC'25]. In our work, we show that $\Delta$-coloring can be solved deterministically in $\tilde{O}(\log^{5/3} n)$ rounds as well, matching the currently best bound for $(\Delta + 1)$-coloring. We achieve our result by developing a reduction from $\Delta$-coloring to MIS that guarantees that the (asymptotic) complexity of $\Delta$-coloring is at most the complexity of MIS, unless MIS can be solved in sublogarithmic time, in which case, due to the $\Omega(\log n)$-round $\Delta$-coloring lower bound from [BFHKLRSU, STOC'16], our reduction implies a tight complexity of $\Theta(\log n)$ for $\Delta$-coloring. In particular, any improvement on the complexity of the MIS problem will yield the same improvement for the complexity of $\Delta$-coloring (up to the true complexity of $\Delta$-coloring). Our reduction yields improvements for $\Delta$-coloring in the randomized LOCAL model and when complexities are parameterized by both $n$ and $\Delta$. We obtain a randomized complexity bound of $\tilde{O}(\log^{5/3} \log n)$ rounds (improving over the state of the art of $\tilde{O}(\log^{8/3} \log n)$ rounds) on general graphs and tight complexities of $\Theta(\log n)$ and $\Theta(\log \log n)$ for the deterministic, resp.\ randomized, complexity on bounded-degree graphs. In the special case of graphs of constant clique number (which for instance include bipartite graphs), we also give a reduction to the $(\Delta+1)$-coloring problem.

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arXiv2025

Towards Optimal Deterministic LOCAL Algorithms on Trees

While obtaining optimal algorithms for the most important problems in the LOCAL model has been one of the central goals in the area of distributed algorithms since its infancy, tight complexity bounds are elusive for many problems even when considering \emph{deterministic} complexities on \emph{trees}. We take a step towards remedying this issue by providing a way to relate the complexity of a problem $\Pi$ on trees to its truly local complexity, which is the (asymptotically) smallest function $f$ such that $\Pi$ can be solved in $O(f(\Delta)+\log^*n)$ rounds. More specifically, we develop a transformation that takes an algorithm $\mathcal A$ for $\Pi$ with a runtime of $O(f(\Delta)+\log^*n)$ rounds as input and transforms it into an $O(f(g(n))+\log^* n)$-round algorithm $\mathcal{A}'$ on trees, where $g$ is the function that satisfies $g(n)^{f(g(n))}=n$. If $f$ is the truly local complexity of $\Pi$ (i.e., if $\mathcal{A}$ is asymptotically optimal), then $\mathcal{A}'$ is an asymptotically optimal algorithm on trees, conditioned on a natural assumption on the nature of the worst-case instances of $\Pi$. Our transformation works for any member of a wide class of problems, including the most important symmetry-breaking problems. As an example of our transformation we obtain the first strongly sublogarithmic algorithm for $(\text{edge-degree+1})$-edge coloring (and therefore also $(2\Delta-1)$-edge coloring) on trees, exhibiting a runtime of $O(\log^{12/13} n)$ rounds. This breaks through the $\Omega(\log n/\log\log n)$-barrier that is a fundamental lower bound for other symmetry-breaking problems such as maximal independent set or maximal matching (that already holds on trees), and proves a separation between these problems and the aforementioned edge coloring problems on trees. We extend a subset of our results to graphs of bounded arboricity.

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arXiv2025

Faster Distributed $\Delta$-Coloring via Ruling Subgraphs

Brooks' theorem states that all connected graphs but odd cycles and cliques can be colored with $\Delta$ colors, where $\Delta$ is the maximum degree of the graph. Such colorings have been shown to admit non-trivial distributed algorithms [Panconesi and Srinivasan, Combinatorica 1995] and have been studied intensively in the distributed literature. In particular, it is known that any deterministic algorithm computing a $\Delta$-coloring requires $\Omega(\log n)$ rounds in the LOCAL model [Chang, Kopelowitz, and Pettie, FOCS 2016], and that this lower bound holds already on constant-degree graphs. In contrast, the best upper bound in this setting is given by an $O(\log^2 n)$-round deterministic algorithm that can be inferred already from the works of [Awerbuch, Goldberg, Luby, and Plotkin, FOCS 1989] and [Panconesi and Srinivasan, Combinatorica 1995] roughly three decades ago, raising the fundamental question about the true complexity of $\Delta$-coloring in the constant-degree setting. We answer this long-standing question almost completely by providing an almost-optimal deterministic $O(\log n \log^* n)$-round algorithm for $\Delta$-coloring, matching the lower bound up to a $\log^* n$-factor. Similarly, in the randomized LOCAL model, we provide an $O(\log \log n \log^* n)$-round algorithm, improving over the state-of-the-art upper bound of $O(\log^2 \log n)$ [Ghaffari, Hirvonen, Kuhn, and Maus, Distributed Computing 2021] and almost matching the $\Omega(\log \log n)$-round lower bound by [BFHKLRSU, STOC 2016]. Our results make progress on several important open problems and conjectures. One key ingredient for obtaining our results is the introduction of ruling subgraph families as a novel tool for breaking symmetry between substructures of a graph, which we expect to be of independent interest.

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arXiv2025

On the Locality of Hall's Theorem

The last five years of research on distributed graph algorithms have seen huge leaps of progress, both regarding algorithmic improvements and impossibility results: new strong lower bounds have emerged for many central problems and exponential improvements over the state of the art have been achieved for the runtimes of many algorithms. Nevertheless, there are still large gaps between the best known upper and lower bounds for many important problems. The current lower bound techniques for deterministic algorithms are often tailored to obtaining a logarithmic bound and essentially cannot be used to prove lower bounds beyond $\Omega(\log n)$. In contrast, the best deterministic upper bounds are often polylogarithmic, raising the fundamental question of how to resolve the gap between logarithmic lower and polylogarithmic upper bounds and finally obtain tight bounds. We develop a novel algorithm design technique aimed at closing this gap. In essence, each node finds a carefully chosen local solution in $O(\log n)$ rounds and we guarantee that this solution is consistent with the other nodes' solutions without coordination. The local solutions are based on a distributed version of Hall's theorem that may be of independent interest and motivates the title of this work. We showcase our framework by improving on the state of the art for the following fundamental problems: edge coloring, bipartite saturating matchings and hypergraph sinkless orientation. In particular, we obtain an asymptotically optimal $O(\log n)$-round algorithm for $3\Delta/2$-edge coloring in bounded degree graphs. The previously best bound for the problem was $O(\log^4 n)$ rounds, obtained by plugging in the state-of-the-art maximal independent set algorithm from arXiv:2303.16043 into the $3\Delta/2$-edge coloring algorithm from arXiv:1711.05469 .

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arXiv2024

Distributed Quantum Advantage for Local Problems

We present the first local problem that shows a super-constant separation between the classical randomized LOCAL model of distributed computing and its quantum counterpart. By prior work, such a separation was known only for an artificial graph problem with an inherently global definition [Le Gall et al. 2019]. We present a problem that we call iterated GHZ, which is defined using only local constraints. Formally, it is a family of locally checkable labeling problems [Naor and Stockmeyer 1995]; in particular, solutions can be verified with a constant-round distributed algorithm. We show that in graphs of maximum degree $\Delta$, any classical (deterministic or randomized) LOCAL model algorithm will require $\Omega(\Delta)$ rounds to solve the iterated GHZ problem, while the problem can be solved in $1$ round in quantum-LOCAL. We use the round elimination technique to prove that the iterated GHZ problem requires $\Omega(\Delta)$ rounds for classical algorithms. This is the first work that shows that round elimination is indeed able to separate the two models, and this also demonstrates that round elimination cannot be used to prove lower bounds for quantum-LOCAL. To apply round elimination, we introduce a new technique that allows us to discover appropriate problem relaxations in a mechanical way; it turns out that this new technique extends beyond the scope of the iterated GHZ problem and can be used to e.g. reproduce prior results on maximal matchings [FOCS 2019, PODC 2020] in a systematic manner.

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arXiv2024

Towards Fully Automatic Distributed Lower Bounds

In the past few years, a successful line of research has lead to lower bounds for several fundamental local graph problems in the distributed setting. These results were obtained via a technique called round elimination. On a high level, the round elimination technique can be seen as a recursive application of a function that takes as input a problem $\Pi$ and outputs a problem $\Pi'$ that is one round easier than $\Pi$. Applying this function recursively to concrete problems of interest can be highly nontrivial, which is one of the reasons that has made the technique difficult to approach. The contribution of our paper is threefold. Firstly, we develop a new and fully automatic method for finding lower bounds of $\Omega(\log_\Delta n)$ and $\Omega(\log_\Delta \log n)$ rounds for deterministic and randomized algorithms, respectively, via round elimination. Secondly, we show that this automatic method is indeed useful, by obtaining lower bounds for defective coloring problems. We show that the problem of coloring the nodes of a graph with $3$ colors and defect at most $(\Delta - 3)/2$ requires $\Omega(\log_\Delta n)$ rounds for deterministic algorithms and $\Omega(\log_\Delta \log n)$ rounds for randomized ones. We note that lower bounds for coloring problems are notoriously challenging to obtain, both in general, and via the round elimination technique. Both the first and (indirectly) the second contribution build on our third contribution -- a new and conceptually simple way to compute the one-round easier problem $\Pi'$ in the round elimination framework. This new procedure provides a clear and easy recipe for applying round elimination, thereby making a substantial step towards the greater goal of having a fully automatic procedure for obtaining lower bounds in the distributed setting.

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arXiv2024

Distributed Computation with Local Advice

In this work we study local computation with advice: the goal is to solve a graph problem $\Pi$ with a distributed algorithm in $T(\Delta)$ communication rounds, for some function $T$ that only depends on the maximum degree $\Delta$ of the graph, and the key question is how many bits of advice per node are needed. Some of our results regard Locally Checkable Labeling problems (LCLs), which are constraint-satisfaction graph problems that can be defined with a finite set of valid input/output-labeled neighborhoods. Our main results are: - Any LCL can be solved with only $1$ bit of advice per node in graphs with sub-exponential growth. Moreover, we can make the set of nodes that carry advice bits arbitrarily sparse. As a corollary, any LCL admits a locally checkable proof with $1$ bit per node in graphs with sub-exponential growth. - The assumption of sub-exponential growth is complemented by a conditional lower bound: assuming the Exponential-Time Hypothesis, there are locally checkable labeling problems that cannot be solved in general with any constant number of bits per node. - In any graph we can find an almost-balanced orientation with $1$ bit of advice per node, and again we can make the advice arbitrarily sparse. As a corollary, we can also compress an arbitrary subset of edges so that a node of degree $d$ stores only $d/2 + 2$ bits, and we can decompress it locally, in $T(\Delta)$ rounds. - In any graph of maximum degree $\Delta$, we can find a $\Delta$-coloring (if it exists) with $1$ bit of advice per node, and again, we can make the advice arbitrarily sparse. - In any $3$-colorable graph, we can find a $3$-coloring with $1$ bit of advice per node. As a corollary, in bounded-degree graphs there is a locally checkable proof that certifies $3$-colorability with $1$ bit of advice per node.

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arXiv2024

Completing the Node-Averaged Complexity Landscape of LCLs on Trees

The node-averaged complexity of a problem captures the number of rounds nodes of a graph have to spend on average to solve the problem in the LOCAL model. A challenging line of research with regards to this new complexity measure is to understand the complexity landscape of locally checkable labelings (LCLs) on families of bounded-degree graphs. Particularly interesting in this context is the family of bounded-degree trees as there, for the worst-case complexity, we know a complete characterization of the possible complexities and structures of LCL problems. A first step for the node-averaged complexity case has been achieved recently [DISC '23], where the authors in particular showed that in bounded-degree trees, there is a large complexity gap: There are no LCL problems with a deterministic node-averaged complexity between $\omega(\log^* n)$ and $n^{o(1)}$. For randomized algorithms, they even showed that the node-averaged complexity is either $O(1)$ or $n^{\Omega(1)}$. In this work we fill in the remaining gaps and give a complete description of the node-averaged complexity landscape of LCLs on bounded-degree trees. Our contributions are threefold. - On bounded-degree trees, there is no LCL with a node-averaged complexity between $\omega(1)$ and $(\log^*n)^{o(1)}$. - For any constants $0 0$, there exists a constant $c$ and an LCL problem with node-averaged complexity between $\Omega((\log^* n)^c)$ and $O((\log^* n)^{c+\varepsilon})$. - For any constants $0<\alpha\leq 1/2$ and $\varepsilon>0$, there exists an LCL problem with node-averaged complexity $\Theta(n^x)$ for some $x\in [\alpha, \alpha+\varepsilon]$.

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arXiv2024

Tight Lower Bounds in the Supported LOCAL Model

We study the complexity of fundamental distributed graph problems in the recently popular setting where information about the input graph is available to the nodes before the start of the computation. We focus on the most common such setting, known as the Supported LOCAL model, where the input graph (on which the studied graph problem has to be solved) is guaranteed to be a subgraph of the underlying communication network. Building on a successful lower bound technique for the LOCAL model called round elimination, we develop a framework for proving complexity lower bounds in the stronger Supported LOCAL model. Our framework reduces the task of proving a (deterministic or randomized) lower bound for a given problem $\Pi$ to the graph-theoretic task of proving non-existence of a solution to another problem $\Pi'$ (on a suitable graph) that can be derived from $\Pi$ in a mechanical manner. We use the developed framework to obtain substantial (and, in the majority of cases, asymptotically tight) Supported LOCAL lower bounds for a variety of fundamental graph problems, including maximal matching, maximal independent set, ruling sets, arbdefective colorings, and generalizations thereof. In a nutshell, for essentially any major lower bound proved in the LOCAL model in recent years, we prove a similar lower bound in the Supported LOCAL model. Our framework also gives rise to a new deterministic version of round elimination in the LOCAL model: while, previous to our work, the general round elimination technique required the use of randomness (even for obtaining deterministic lower bounds), our framework allows to obtain deterministic (and therefore via known lifting techniques also randomized) lower bounds in a purely deterministic manner. Previously, such a purely deterministic application of round elimination was only known for the specific problem of sinkless orientation [SOSA'23].

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arXiv2023

Location Estimation and Recovery using 5G Positioning: Thwarting GNSS Spoofing Attacks

The availability of cheap GNSS spoofers can prevent safe navigation and tracking of road users. It can lead to loss of assets, inaccurate fare estimation, enforcing the wrong speed limit, miscalculated toll tax, passengers reaching an incorrect location, etc. The techniques designed to prevent and detect spoofing by using cryptographic solutions or receivers capable of differentiating legitimate and attack signals are insufficient in detecting GNSS spoofing of road users. Recent studies, testbeds, and 3GPP standards are exploring the possibility of hybrid positioning, where GNSS data will be combined with the 5G-NR positioning to increase the security and accuracy of positioning. We design the Location Estimation and Recovery(LER) systems to estimate the correct absolute position using the combination of GNSS and 5G positioning with other road users, where a subset of road users can be malicious and collude to prevent spoofing detection. Our Location Verification Protocol extends the understanding of Message Time of Arrival Codes (MTAC) to prevent attacks against malicious provers. The novel Recovery and Meta Protocol uses road users' dynamic and unpredictable nature to detect GNSS spoofing. This protocol provides fast detection of GNSS spoofing with a very low rate of false positives and can be customized to a large family of settings. Even in a (highly unrealistic) worst-case scenario where each user is malicious with a probability of as large as 0.3, our protocol detects GNSS spoofing with high probability after communication and ranging with at most 20 road users, with a false positive rate close to 0. SUMO simulations for road traffic show that we can detect GNSS spoofing in 2.6 minutes since its start under moderate traffic conditions.

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arXiv2023

On the Node-Averaged Complexity of Locally Checkable Problems on Trees

Over the past decade, a long line of research has investigated the distributed complexity landscape of locally checkable labeling (LCL) problems on bounded-degree graphs, culminating in an almost-complete classification on general graphs and a complete classification on trees. The latter states that, on bounded-degree trees, any LCL problem has deterministic worst-case time complexity $O(1)$, $\Theta(\log^* n)$, $\Theta(\log n)$, or $\Theta(n^{1/k})$ for some positive integer $k$, and all of those complexity classes are nonempty. Moreover, randomness helps only for (some) problems with deterministic worst-case complexity $\Theta(\log n)$, and if randomness helps (asymptotically), then it helps exponentially. In this work, we study how many distributed rounds are needed on average per node in order to solve an LCL problem on trees. We obtain a partial classification of the deterministic node-averaged complexity landscape for LCL problems. As our main result, we show that every problem with worst-case round complexity $O(\log n)$ has deterministic node-averaged complexity $O(\log^* n)$. Then we show how using randomization we can speed this up and show that every problem with worst case round complexity $O(\log n)$ has randomized node-averaged complexity $O(1)$. We further establish bounds on the node-averaged complexity of problems with worst-case complexity $\Theta(n^{1/k})$: we show that all these problems have node-averaged complexity $\widetilde{\Omega}(n^{1 / (2^k - 1)})$, and that this lower bound is tight for some problems. The lower bound holds even for the randomized case and the upper bound is a deterministic algorithm.

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arXiv2022

Distributed Maximal Matching and Maximal Independent Set on Hypergraphs

We investigate the distributed complexity of maximal matching and maximal independent set (MIS) in hypergraphs in the LOCAL model. A maximal matching of a hypergraph $H=(V_H,E_H)$ is a maximal disjoint set $M\subseteq E_H$ of hyperedges and an MIS $S\subseteq V_H$ is a maximal set of nodes such that no hyperedge is fully contained in $S$. Both problems can be solved by a simple sequential greedy algorithm, which can be implemented naively in $O(Δr + \log^* n)$ rounds, where $Δ$ is the maximum degree, $r$ is the rank, and $n$ is the number of nodes. We show that for maximal matching, this naive algorithm is optimal in the following sense. Any deterministic algorithm for solving the problem requires $Ω(\min\{Δr, \log_{Δr} n\})$ rounds, and any randomized one requires $Ω(\min\{Δr, \log_{Δr} \log n\})$ rounds. Hence, for any algorithm with a complexity of the form $O(f(Δ, r) + g(n))$, we have $f(Δ, r) \in Ω(Δr)$ if $g(n)$ is not too large, and in particular if $g(n) = \log^* n$ (which is the optimal asymptotic dependency on $n$ due to Linial's lower bound [FOCS'87]). Our lower bound proof is based on the round elimination framework, and its structure is inspired by a new round elimination fixed point that we give for the $Δ$-vertex coloring problem in hypergraphs. For the MIS problem on hypergraphs, we show that for $Δ\ll r$, there are significant improvements over the naive $O(Δr + \log^* n)$-round algorithm. We give two deterministic algorithms for the problem. We show that a hypergraph MIS can be computed in $O(Δ^2\cdot\log r + Δ\cdot\log r\cdot \log^* r + \log^* n)$ rounds. We further show that at the cost of a worse dependency on $Δ$, the dependency on $r$ can be removed almost entirely, by giving an algorithm with complexity $Δ^{O(Δ)}\cdot\log^* r + O(\log^* n)$.

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arXiv2022

The Landscape of Distributed Complexities on Trees and Beyond

We study the local complexity landscape of locally checkable labeling (LCL) problems on constant-degree graphs with a focus on complexities below $\log^* n$. Our contribution is threefold: Our main contribution is that we complete the classification of the complexity landscape of LCL problems on trees in the LOCAL model, by proving that every LCL problem with local complexity $o(\log^* n)$ has actually complexity $O(1)$. This result improves upon the previous speedup result from $o(\log \log^* n)$ to $O(1)$ by [Chang, Pettie, FOCS 2017]. In the related LCA and Volume models [Alon, Rubinfeld, Vardi, Xie, SODA 2012, Rubinfeld, Tamir, Vardi, Xie, 2011, Rosenbaum, Suomela, PODC 2020], we prove the same speedup from $o(\log^* n)$ to $O(1)$ for all bounded degree graphs. Similarly, we complete the classification of the LOCAL complexity landscape of oriented $d$-dimensional grids by proving that any LCL problem with local complexity $o(\log^* n)$ has actually complexity $O(1)$. This improves upon the previous speed-up from $o(\sqrt[d]{\log^* n})$ by Suomela in [Chang, Pettie, FOCS 2017].

cs.DS↗