arXiv subjects

Alkida Balliu

Publications and source records attributed to Alkida Balliu.

At least 19 recordsLinked to original sources

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

New Hardness Results for the LOCAL Model via a Simple Self-Reduction

Very recently, Khoury and Schild [FOCS 2025] showed that any randomized LOCAL algorithm that solves maximal matching requires $\Omega(\min\{\log \Delta, \log_\Delta n\})$ rounds, where $n$ is the number of nodes in the graph and $\Delta$ is the maximum degree. This result is shown through a new technique, called round elimination via self-reduction. The lower bound proof is beautiful and presents very nice ideas. However, it spans more than 25 pages of technical details, and hence it is hard to digest and generalize to other problems. Historically, the simplification of proofs and techniques has marked an important turning point in our understanding of the complexity of graph problems. Our paper makes a step forward towards this direction, and provides the following contributions. 1. We present a short and simplified version of the round elimination via self-reduction technique. The simplification of this technique enables us to obtain the following two hardness results. 2. We show that any randomized LOCAL algorithm that solves the maximal $b$-matching problem requires $\Omega(\min\{\log_{1+b}\Delta, \log_\Delta n\})$ and $\Omega(\sqrt{\log_{1+b} n})$ rounds. We recall that the $b$-matching problem is a generalization of the matching problem where each vertex can have up to $b$ incident edges in the matching. As a corollary, for $b=1$, we obtain a short proof for the maximal matching lower bound shown by Khoury and Schild. 3. We show that any randomized LOCAL algorithm that properly colors the edges of a graph with $\Delta + k$ colors requires $\Omega(\min\{\log \Delta, \log_\Delta n\})$ and $\Omega(\sqrt{\log n})$ rounds, for any $k\le \Delta^{1-\varepsilon}$ and any constant $\varepsilon > 0$.

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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

Distributed Algorithms for Potential Problems

In this work, we present a fast distributed algorithm for local potential problems: these are graph problems where the task is to find a locally optimal solution where no node can unilaterally improve the utility in its local neighborhood by changing its own label. A simple example of such a problem is the task of finding a locally optimal cut, i.e., a cut where for each node at least half of its incident edges are cut edges. The distributed round complexity of the locally optimal cut problem has been wide open; the problem is known to require $\Omega(\log n)$ rounds in the deterministic LOCAL model and $\Omega(\log \log n)$ rounds in the randomized LOCAL model, but the only known upper bound is the trivial brute-force solution of $O(n)$ rounds. Locally optimal cut in constant-degree graphs is perhaps the simplest example of a locally checkable labeling problem for which there is still such a large gap between current upper and lower bounds. We show that in constant-degree graphs, all local potential problems, including locally optimal cut, can be solved in $\log^{O(1)} n$ rounds, both in the deterministic and randomized LOCAL models. In particular, the deterministic round complexity of the locally optimal cut problem is now settled to $\log^{\Theta(1)} n$. Our algorithms also apply to the general case of graphs of maximum degree $\Delta$. For the special case of locally optimal cut, we obtain a randomized algorithm that runs in $O(\Delta^{2} \log^{6} n)$ rounds, which can be derandomized at polylogarithmic cost with standard techniques. Furthermore, we show that a dependence in $\Delta$ is necessary: we prove a lower bound of $\Omega(\min\{\Delta,\sqrt{n}\})$ rounds, even in the quantum-LOCAL model; in particular, there is no polylogarithmic-round algorithm for the general case.

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arXiv2025

New Limits on Distributed Quantum Advantage: Dequantizing Linear Programs

In this work, we give two results that put new limits on distributed quantum advantage in the context of the LOCAL model of distributed computing. First, we show that there is no distributed quantum advantage for any linear program. Put otherwise, if there is a quantum-LOCAL algorithm $\mathcal{A}$ that finds an $\alpha$-approximation of some linear optimization problem $\Pi$ in $T$ communication rounds, we can construct a classical, deterministic LOCAL algorithm $\mathcal{A}'$ that finds an $\alpha$-approximation of $\Pi$ in $T$ rounds. As a corollary, all classical lower bounds for linear programs, including the KMW bound, hold verbatim in quantum-LOCAL. Second, using the above result, we show that there exists a locally checkable labeling problem (LCL) for which quantum-LOCAL is strictly weaker than the classical deterministic SLOCAL model. Our results extend from quantum-LOCAL also to finitely dependent and non-signaling distributions, and one of the corollaries of our work is that the non-signaling model and the SLOCAL model are incomparable in the context of LCL problems: By prior work, there exists an LCL problem for which SLOCAL is strictly weaker than the non-signaling model, and our work provides a separation in the opposite direction.

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arXiv2025

Distributed Quantum Advantage in Locally Checkable Labeling Problems

In this paper, we present the first known example of a locally checkable labeling problem (LCL) that admits asymptotic distributed quantum advantage in the LOCAL model of distributed computing: our problem can be solved in $O(\log n)$ communication rounds in the quantum-LOCAL model, but it requires $\Omega(\log n \cdot \log^{0.99} \log n)$ communication rounds in the classical randomized-LOCAL model. We also show that distributed quantum advantage cannot be arbitrarily large: if an LCL problem can be solved in $T(n)$ rounds in the quantum-LOCAL model, it can also be solved in $\tilde O(\sqrt{n T(n)})$ rounds in the classical randomized-LOCAL model. In particular, a problem that is strictly global classically is also almost-global in quantum-LOCAL. Our second result also holds for $T(n)$-dependent probability distributions. As a corollary, if there exists a finitely dependent distribution over valid labelings of some LCL problem $\Pi$, then the same problem $\Pi$ can also be solved in $\tilde O(\sqrt{n})$ rounds in the classical randomized-LOCAL and deterministic-LOCAL models. That is, finitely dependent distributions cannot exist for global LCL problems.

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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

Solving Sequential Greedy Problems Distributedly with Sub-Logarithmic Energy Cost

We study the awake complexity of graph problems that belong to the class O-LOCAL, which includes a subset of problems solvable by sequential greedy algorithms, such as $(\Delta+1)$-coloring and maximal independent set. It is known from previous work that, in $n$-node graphs of maximum degree $\Delta$, any problem in the class O-LOCAL can be solved by a deterministic distributed algorithm with awake complexity $O(\log\Delta+\log^\star n)$. In this paper, we show that any problem belonging to the class O-LOCAL can be solved by a deterministic distributed algorithm with awake complexity $O(\sqrt{\log n}\cdot\log^\star n)$. This leads to a polynomial improvement over the state of the art when $\Delta\gg 2^{\sqrt{\log n}}$, e.g., $\Delta=n^\epsilon$ for some arbitrarily small $\epsilon>0$. The key ingredient for achieving our results is the computation of a network decomposition, that uses a small-enough number of colors, in sub-logarithmic time in the Sleeping model, which can be of independent interest.

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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

Asynchronous Fault-Tolerant Distributed Proper Coloring of Graphs

We revisit asynchronous computing in networks of crash-prone processes, under the asynchronous variant of the standard LOCAL model, recently introduced by Fraigniaud et al. [DISC 2022]. We focus on the vertex coloring problem, and our contributions concern both lower and upper bounds for this problem. On the upper bound side, we design an algorithm tolerating an arbitrarily large number of crash failures that computes an $O(\Delta^2)$-coloring of any $n$-node graph of maximum degree $\Delta$, in $O(\log^\star n)$ rounds. This extends Linial's seminal result from the (synchronous failure-free) LOCAL model to its asynchronous crash-prone variant. Then, by allowing a dependency on $\Delta$ on the runtime, we show that we can reduce the colors to $\big(\frac12(\Delta+1)(\Delta+2)-1 \big)$. For cycles (i.e., for $\Delta=2$), our algorithm achieves a 5-coloring of any $n$-node cycle, in $O(\log^\star n)$ rounds. This improves the known 6-coloring algorithm by Fraigniaud et al., and fixes a bug in their algorithm, which was erroneously claimed to produce a 5-coloring. On the lower bound side, we show that, for $k<5$, and for every prime integer~$n$, no algorithm can $k$-color the $n$-node cycle in the asynchronous crash-prone variant of LOCAL, independently from the round-complexities of the algorithms. This lower bound is obtained by reduction from an original extension of the impossibility of solving weak symmetry-breaking in the wait-free shared-memory model. We show that this impossibility still holds even if the processes are provided with inputs susceptible to help breaking symmetry.

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arXiv2024

Shared Randomness Helps with Local Distributed Problems

By prior work, we have many results related to distributed graph algorithms for problems that can be defined with local constraints; the formal framework used in prior work is locally checkable labeling problems (LCLs), introduced by Naor and Stockmeyer in the 1990s. It is known, for example, that if we have a deterministic algorithm that solves an LCL in $o(\log n)$ rounds, we can speed it up to $O(\log^*n)$ rounds, and if we have a randomized $O(\log^*n)$ rounds algorithm, we can derandomize it for free. It is also known that randomness helps with some LCL problems: there are LCL problems with randomized complexity $\Theta(\log\log n)$ and deterministic complexity $\Theta(\log n)$. However, so far there have not been any LCL problems in which the use of shared randomness has been necessary; in all prior algorithms it has been enough that the nodes have access to their own private sources of randomness. Could it be the case that shared randomness never helps with LCLs? Could we have a general technique that takes any distributed graph algorithm for any LCL that uses shared randomness, and turns it into an equally fast algorithm where private randomness is enough? In this work we show that the answer is no. We present an LCL problem $\Pi$ such that the round complexity of $\Pi$ is $\Omega(\sqrt n)$ in the usual randomized \local model with private randomness, but if the nodes have access to a source of shared randomness, then the complexity drops to $O(\log n)$. As corollaries, we also resolve several other open questions related to the landscape of distributed computing in the context of LCL problems. In particular, problem $\Pi$ demonstrates that distributed quantum algorithms for LCL problems strictly benefit from a shared quantum state. Problem $\Pi$ also gives a separation between finitely dependent distributions and non-signaling distributions.

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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

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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arXiv2023

A Note on the Complexity of Maximizing Temporal Reachability via Edge Temporalisation of Directed Graphs

A temporal graph is a graph in which edges are assigned a time label. Two nodes u and v of a temporal graph are connected one to the other if there exists a path from u to v with increasing edge time labels. We consider the problem of assigning time labels to the edges of a digraph in order to maximize the total reachability of the resulting temporal graph (that is, the number of pairs of nodes which are connected one to the other). In particular, we prove that this problem is NP-hard. We then conjecture that the problem is approximable within a constant approximation ratio. This conjecture is a consequence of the following graph theoretic conjecture: any strongly connected directed graph with n nodes admits an out-arborescence and an in-arborescence that are edge-disjoint, have the same root, and each spans $\Omega$(n) nodes.

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arXiv2022

Optimal Deterministic Massively Parallel Connectivity on Forests

We show fast deterministic algorithms for fundamental problems on forests in the challenging low-space regime of the well-known Massive Parallel Computation (MPC) model. A recent breakthrough result by Coy and Czumaj [STOC'22] shows that, in this setting, it is possible to deterministically identify connected components on graphs in $O(\log D + \log\log n)$ rounds, where $D$ is the diameter of the graph and $n$ the number of nodes. The authors left open a major question: is it possible to get rid of the additive $\log\log n$ factor and deterministically identify connected components in a runtime that is completely independent of $n$? We answer the above question in the affirmative in the case of forests. We give an algorithm that identifies connected components in $O(\log D)$ deterministic rounds. The total memory required is $O(n+m)$ words, where $m$ is the number of edges in the input graph, which is optimal as it is only enough to store the input graph. We complement our upper bound results by showing that $Ω(\log D)$ time is necessary even for component-unstable algorithms, conditioned on the widely believed 1 vs. 2 cycles conjecture. Our techniques also yield a deterministic forest-rooting algorithm with the same runtime and memory bounds. Furthermore, we consider Locally Checkable Labeling problems (LCLs), whose solution can be verified by checking the $O(1)$-radius neighborhood of each node. We show that any LCL problem on forests can be solved in $O(\log D)$ rounds with a canonical deterministic algorithm, improving over the $O(\log n)$ runtime of Brandt, Latypov and Uitto [DISC'21]. We also show that there is no algorithm that solves all LCL problems on trees asymptotically faster.

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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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