arXiv subjects

Fabian Kuhn

Publications and source records attributed to Fabian Kuhn.

At least 37 records · Page 2Linked to original sources

arXiv2023

Toward Online Mobile Facility Location on General Metrics

We introduce an online variant of mobile facility location (MFL) (introduced by Demaine et al. [SODA' 07]). We call this new problem online mobile facility location (OMFL). In the OMFL problem, initially, we are given a set of $k$ mobile facilities with their starting locations. One by one, requests are added. After each request arrives, one can make some changes to the facility locations before the subsequent request arrives. Each request is always assigned to the nearest facility. The cost of this assignment is the distance from the request to the facility. The objective is to minimize the total cost, which consists of the relocation cost of facilities and the distance cost of requests to their nearest facilities. We provide a lower bound for the OMFL problem that even holds on uniform metrics. A natural approach to solve the OMFL problem for general metric spaces is to utilize hierarchically well-separated trees (HSTs) and directly solve the OMFL problem on HSTs. In this paper, we provide the first step in this direction by solving a generalized variant of the OMFL problem on uniform metrics that we call G-OMFL. We devise a simple deterministic online algorithm and provide a tight analysis for the algorithm. The second step remains an open question. Inspired by the $k$-server problem, we introduce a new variant of the OMFL problem that focuses solely on minimizing movement cost. We refer to this variant as M-OMFL. Additionally, we provide a lower bound for M-OMFL that is applicable even on uniform metrics.

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arXiv2023

Time and Space Optimal Massively Parallel Algorithm for the 2-Ruling Set Problem

In this work, we present a constant-round algorithm for the $2$-ruling set problem in the Congested Clique model. As a direct consequence, we obtain a constant round algorithm in the MPC model with linear space-per-machine and optimal total space. Our results improve on the $O(\log \log \log n)$-round algorithm by [HPS, DISC'14] and the $O(\log \log Δ)$-round algorithm by [GGKMR, PODC'18]. Our techniques can also be applied to the semi-streaming model to obtain an $O(1)$-pass algorithm. Our main technical contribution is a novel sampling procedure that returns a small subgraph such that almost all nodes in the input graph are adjacent to the sampled subgraph. An MIS on the sampled subgraph provides a $2$-ruling set for a large fraction of the input graph. As a technical challenge, we must handle the remaining part of the graph, which might still be relatively large. We overcome this challenge by showing useful structural properties of the remaining graph and show that running our process twice yields a $2$-ruling set of the original input graph with high probability.

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arXiv2023

Coloring Fast with Broadcasts

We present an $O(\log^3\log n)$-round distributed algorithm for the $(Δ+1)$-coloring problem, where each node broadcasts only one $O(\log n)$-bit message per round to its neighbors. Previously, the best such broadcast-based algorithm required $O(\log n)$ rounds. If $Δ\in Ω(\log^{3} n)$, our algorithm runs in $O(\log^* n)$ rounds. Our algorithm's round complexity matches state-of-the-art in the much more powerful CONGEST model [Halldórsson et al., STOC'21 & PODC'22], where each node sends one different message to each of its neighbors, thus sending up to $Θ(n\log n)$ bits per round. This is the best complexity known, even if message sizes are unbounded. Our algorithm is simple enough to be implemented in even weaker models: we can achieve the same $O(\log^3\log n)$ round complexity if each node reads its received messages in a streaming fashion, using only $O(\log^3 n)$-bit memory. Therefore, we hope that our algorithm opens the road for adopting the recent exciting progress on sublogarithmic-time distributed $(Δ+1)$-coloring algorithms in a wider range of (theoretical or practical) settings.

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arXiv2023

A Distributed Palette Sparsification Theorem

The celebrated palette sparsification result of [Assadi, Chen, and Khanna SODA'19] shows that to compute a $Δ+1$ coloring of the graph, where $Δ$ denotes the maximum degree, it suffices if each node limits its color choice to $O(\log n)$ independently sampled colors in $\{1, 2, \dots, Δ+1\}$. They showed that it is possible to color the resulting sparsified graph -- the spanning subgraph with edges between neighbors that sampled a common color, which are only $\tilde{O}(n)$ edges -- and obtain a $Δ+1$ coloring for the original graph. However, to compute the actual coloring, that information must be gathered at a single location for centralized processing. We seek instead a local algorithm to compute such a coloring in the sparsified graph. The question is if this can be achieved in $\operatorname{poly}(\log n)$ distributed rounds with small messages. Our main result is an algorithm that computes a $Δ+1$-coloring after palette sparsification with $O(\log^2 n)$ random colors per node and runs in $O(\log^2 Δ+ \log^3 \log n)$ rounds on the sparsified graph, using $O(\log n)$-bit messages. We show that this is close to the best possible: any distributed $Δ+1$-coloring algorithm that runs in the LOCAL model on the sparsified graph, given by palette sparsification, for any $\operatorname{poly}(\log n)$ colors per node, requires $Ω(\log Δ/ \log\log n)$ rounds. This distributed palette sparsification result leads to the first $\operatorname{poly}(\log n)$-round algorithms for $Δ+1$-coloring in two previously studied distributed models: the Node Capacitated Clique, and the cluster graph model.

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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

Local Distributed Rounding: Generalized to MIS, Matching, Set Cover, and Beyond

We develop a general deterministic distributed method for locally rounding fractional solutions of graph problems for which the analysis can be broken down into analyzing pairs of vertices. Roughly speaking, the method can transform fractional/probabilistic label assignments of the vertices into integral/deterministic label assignments for the vertices, while approximately preserving a potential function that is a linear combination of functions, each of which depends on at most two vertices (subject to some conditions usually satisfied in pairwise analyses). The method unifies and significantly generalizes prior work on deterministic local rounding techniques [Ghaffari, Kuhn FOCS'21; Harris FOCS'19; Fischer, Ghaffari, Kuhn FOCS'17; Fischer DISC'17] to obtain polylogarithmic-time deterministic distributed solutions for combinatorial graph problems. Our general rounding result enables us to locally and efficiently derandomize a range of distributed algorithms for local graph problems, including maximal independent set (MIS), maximum-weight independent set approximation, and minimum-cost set cover approximation. As a highlight, we in particular obtain a deterministic $O(\log^2Δ\cdot\log n)$-round algorithm for computing an MIS in the LOCAL model and an almost as efficient $O(\log^2Δ\cdot\log\logΔ\cdot\log n)$-round deterministic MIS algorithm in the CONGEST model. As a result, the best known deterministic distributed time complexity of the four most widely studied distributed symmetry breaking problems (MIS, maximal matching, $(Δ+1)$-vertex coloring, and $(2Δ-1)$-edge coloring) is now $O(\log^2Δ\cdot\log n)$. Our new MIS algorithm is also the first direct polylogarithmic-time deterministic distributed MIS algorithm, which is not based on network decomposition.

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arXiv2022

Node and Edge Averaged Complexities of Local Graph Problems

The node-averaged complexity of a distributed algorithm running on a graph $G=(V,E)$ is the average over the times at which the nodes $V$ of $G$ finish their computation and commit to their outputs. We study the node-averaged complexity for some distributed symmetry breaking problems and provide the following results (among others): - The randomized node-averaged complexity of computing a maximal independent set (MIS) in $n$-node graphs of maximum degree $Δ$ is at least $Ω\big(\min\big\{\frac{\logΔ}{\log\logΔ},\sqrt{\frac{\log n}{\log\log n}}\big\}\big)$. This bound is obtained by a novel adaptation of the well-known KMW lower bound [JACM'16]. As a side result, we obtain the same lower bound for the worst-case randomized round complexity for computing an MIS in trees -- this essentially answers open problem 11.15 in the book of Barenboim and Elkin and resolves the complexity of MIS on trees up to an $O(\sqrt{\log\log n})$ factor. We also show that, $(2,2)$-ruling sets, which are a minimal relaxation of MIS, have $O(1)$ randomized node-averaged complexity. - For maximal matching, we show that while the randomized node-averaged complexity is $Ω\big(\min\big\{\frac{\logΔ}{\log\logΔ},\sqrt{\frac{\log n}{\log\log n}}\big\}\big)$, the randomized edge-averaged complexity is $O(1)$. Further, we show that the deterministic edge-averaged complexity of maximal matching is $O(\log^2Δ+ \log^* n)$ and the deterministic node-averaged complexity of maximal matching is $O(\log^3Δ+ \log^* n)$. - Finally, we consider the problem of computing a sinkless orientation of a graph. The deterministic worst-case complexity of the problem is known to be $Θ(\log n)$, even on bounded-degree graphs. We show that the problem can be solved deterministically with node-averaged complexity $O(\log^* n)$, while keeping the worst-case complexity in $O(\log n)$.

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arXiv2022

Contention Resolution for Coded Radio Networks

Randomized backoff protocols, such as exponential backoff, are a powerful tool for managing access to a shared resource, often a wireless communication channel (e.g., [1]). For a wireless device to transmit successfully, it uses a backoff protocol to ensure exclusive access to the channel. Modern radios, however, do not need exclusive access to the channel to communicate; in particular, they have the ability to receive useful information even when more than one device transmits at the same time. These capabilities have now been exploited for many years by systems that rely on interference cancellation, physical layer network coding and analog network coding to improve efficiency. For example, Zigzag decoding [56] demonstrated how a base station can decode messages sent by multiple devices simultaneously. In this paper, we address the following question: Can we design a backoff protocol that is better than exponential backoff when exclusive channel access is not required. We define the Coded Radio Network Model, which generalizes traditional radio network models (e.g., [30]). We then introduce the Decodable Backoff Algorithm, a randomized backoff protocol that achieves an optimal throughput of $1-o(1)$. (Throughput $1$ is optimal, as simultaneous reception does not increase the channel capacity.) The algorithm breaks the constant throughput lower bound for traditional radio networks [47-49], showing the power of these new hardware capabilities.

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arXiv2022

Sinkless Orientation Made Simple

The sinkless orientation problem plays a key role in understanding the foundations of distributed computing. The problem can be used to separate two fundamental models of distributed graph algorithms, LOCAL and SLOCAL: the locality of sinkless orientation is $Ω(\log n)$ in the deterministic LOCAL model and $O(\log \log n)$ in the deterministic SLOCAL model. Both of these results are known by prior work, but here we give new simple, self-contained proofs for them.

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arXiv2022

Distributed $Δ$-Coloring Plays Hide-and-Seek

We prove several new tight distributed lower bounds for classic symmetry breaking graph problems. As a basic tool, we first provide a new insightful proof that any deterministic distributed algorithm that computes a $Δ$-coloring on $Δ$-regular trees requires $Ω(\log_Δn)$ rounds and any randomized algorithm requires $Ω(\log_Δ\log n)$ rounds. We prove this result by showing that a natural relaxation of the $Δ$-coloring problem is a fixed point in the round elimination framework. As a first application, we show that our $Δ$-coloring lower bound proof directly extends to arbdefective colorings. We exactly characterize which variants of the arbdefective coloring problem are "easy", and which of them instead are "hard". As a second application, which we see as our main contribution, we use the structure of the fixed point as a building block to prove lower bounds as a function of $Δ$ for a large class of distributed symmetry breaking problems. For example, we obtain a tight lower bound for the fundamental problem of computing a $(2,β)$-ruling set. This is an exponential improvement over the best existing lower bound for the problem, which was proven in [FOCS '20]. Our lower bound even applies to a much more general family of problems that allows for almost arbitrary combinations of natural constraints from coloring problems, orientation problems, and independent set problems, and provides a single unified proof for known and new lower bound results for these types of problems. Our lower bounds as a function of $Δ$ also imply lower bounds as a function of $n$. We obtain, for example, that maximal independent set, on trees, requires $Ω(\log n / \log \log n)$ rounds for deterministic algorithms, which is tight.

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arXiv2022

Distributed Edge Coloring in Time Polylogarithmic in $Δ$

We provide new deterministic algorithms for the edge coloring problem, which is one of the classic and highly studied distributed local symmetry breaking problems. As our main result, we show that a $(2Δ-1)$-edge coloring can be computed in time $\mathrm{poly}\logΔ+ O(\log^* n)$ in the LOCAL model. This improves a result of Balliu, Kuhn, and Olivetti [PODC '20], who gave an algorithm with a quasi-polylogarithmic dependency on $Δ$. We further show that in the CONGEST model, an $(8+\varepsilon)Δ$-edge coloring can be computed in $\mathrm{poly}\logΔ+ O(\log^* n)$ rounds. The best previous $O(Δ)$-edge coloring algorithm that can be implemented in the CONGEST model is by Barenboim and Elkin [PODC '11] and it computes a $2^{O(1/\varepsilon)}Δ$-edge coloring in time $O(Δ^\varepsilon + \log^* n)$ for any $\varepsilon\in(0,1]$.

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arXiv2022

Near-Shortest Path Routing in Hybrid Communication Networks

Hybrid networks, i.e., networks that leverage different means of communication, become ever more widespread. To allow theoretical study of such networks, [Augustine et al., SODA'20] introduced the $\mathsf{HYBRID}$ model, which is based on the concept of synchronous message passing and uses two fundamentally different principles of communication: a local mode, which allows every node to exchange one message per round with each neighbor in a local communication graph; and a global mode where any pair of nodes can exchange messages, but only few such exchanges can take place per round. A sizable portion of the previous research for the $\mathsf{HYBRID}$ model revolves around basic communication primitives and computing distances or shortest paths in networks. In this paper, we extend this study to a related fundamental problem of computing compact routing schemes for near-shortest paths in the local communication graph. We demonstrate that, for the case where the local communication graph is a unit-disc graph with $n$ nodes that is realized in the plane and has no radio holes, we can deterministically compute a routing scheme that has constant stretch and uses labels and local routing tables of size $O(\log n)$ bits in only $O(\log n)$ rounds.

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arXiv2022

Routing Schemes and Distance Oracles in the Hybrid Model

The $\mathsf{HYBRID}$ model was introduced as a means for theoretical study of distributed networks that use various communication modes. Conceptually, it is a synchronous message passing model with a local communication mode, where in each round each node can send large messages to all its neighbors in a local network (a graph), and a global communication mode, where each node is allotted limited (polylogarithmic) bandwidth per round which it can use to communicate with any node in the network. Prior work has often focused on shortest paths problems in the local network, as their global nature makes these an interesting case study how combining communication modes in the $\mathsf{HYBRID}$ model can overcome the individual lower bounds of either mode. In this work we consider a similar problem, namely computation of distance oracles and routing schemes. In the former, all nodes have to compute local tables, which allows them to look up the distance (estimates) to any target node in the local network when provided with the label of the target. In the latter, it suffices that nodes give the next node on an (approximately) shortest path to the target. Our goal is to compute these local tables as fast as possible with labels as small as possible. We show that this can be done exactly in $\widetilde O(n^{1/3})$ communication rounds and labels of size $Θ(n^{2/3})$ bits. For constant stretch approximations we achieve labels of size $O(\log n)$ in the same time. Further, as our main technical contribution, we provide computational lower bounds for a variety of problem parameters. For instance, we show that computing solutions with stretch below a certain constant takes $\widetilde Ω(n^{1/3})$ rounds even for labels of size $O(n^{2/3})$.

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arXiv2021

Improved Distributed Fractional Coloring Algorithms

We prove new bounds on the distributed fractional coloring problem in the LOCAL model. Fractional $c$-colorings can be understood as multicolorings as follows. For some natural numbers $p$ and $q$ such that $p/q\leq c$, each node $v$ is assigned a set of at least $q$ colors from $\{1,\dots,p\}$ such that adjacent nodes are assigned disjoint sets of colors. The minimum $c$ for which a fractional $c$-coloring of a graph $G$ exists is called the fractional chromatic number $χ_f(G)$ of $G$. Recently, [Bousquet, Esperet, and Pirot; SIROCCO '21] showed that for any constant $ε>0$, a fractional $(Δ+ε)$-coloring can be computed in $Δ^{O(Δ)} + O(Δ\cdot\log^* n)$ rounds. We show that such a coloring can be computed in only $O(\log^2 Δ)$ rounds, without any dependency on $n$. We further show that in $O\big(\frac{\log n}ε\big)$ rounds, it is possible to compute a fractional $(1+ε)χ_f(G)$-coloring, even if the fractional chromatic number $χ_f(G)$ is not known. That is, this problem can be approximated arbitrarily well by an efficient algorithm in the LOCAL model. For the standard coloring problem, it is only known that an $O\big(\frac{\log n}{\log\log n}\big)$-approximation can be computed in polylogarithmic time in the LOCAL model. We also show that our distributed fractional coloring approximation algorithm is best possible. We show that in trees, which have fractional chromatic number $2$, computing a fractional $(2+ε)$-coloring requires at least $Ω\big(\frac{\log n}ε\big)$ rounds. We finally study fractional colorings of regular grids. In [Bousquet, Esperet, and Pirot; SIROCCO '21], it is shown that in regular grids of bounded dimension, a fractional $(2+ε)$-coloring can be computed in time $O(\log^* n)$. We show that such a coloring can even be computed in $O(1)$ rounds in the LOCAL model.

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arXiv2021

Near-Optimal Distributed Degree+1 Coloring

We present a new approach to randomized distributed graph coloring that is simpler and more efficient than previous ones. In particular, it allows us to tackle the $(\operatorname{deg}+1)$-list-coloring (D1LC) problem, where each node $v$ of degree $d_v$ is assigned a palette of $d_v+1$ colors, and the objective is to find a proper coloring using these palettes. While for $(Δ+1)$-coloring (where $Δ$ is the maximum degree), there is a fast randomized distributed $O(\log^3\log n)$-round algorithm (Chang, Li, and Pettie [SIAM J. Comp. 2020]), no $o(\log n)$-round algorithms are known for the D1LC problem. We give a randomized distributed algorithm for D1LC that is optimal under plausible assumptions about the deterministic complexity of the problem. Using the recent deterministic algorithm of Ghaffari and Kuhn [FOCS2021], our algorithm runs in $O(\log^3 \log n)$ time, matching the best bound known for $(Δ+1)$-coloring. In addition, it colors all nodes of degree $Ω(\log^7 n)$ in $O(\log^* n)$ rounds. A key contribution is a subroutine to generate slack for D1LC. When placed into the framework of Assadi, Chen, and Khanna [SODA2019] and Alon and Assadi [APPROX/RANDOM2020], this almost immediately leads to a palette sparsification theorem for D1LC, generalizing previous results. That gives fast algorithms for D1LC in three different models: an $O(1)$-round algorithm in the MPC model with $\tilde{O}(n)$ memory per machine; a single-pass semi-streaming algorithm in dynamic streams; and an $\tilde{O}(n\sqrt{n})$-time algorithm in the standard query model.

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arXiv2021

Distributed CONGEST Approximation of Weighted Vertex Covers and Matchings

We provide CONGEST model algorithms for approximating minimum weighted vertex cover and the maximum weighted matching. For bipartite graphs, we show that a $(1+\varepsilon)$-approximate weighted vertex cover can be computed deterministically in polylogarithmic time. This generalizes a corresponding result for the unweighted vertex cover problem shown in [Faour, Kuhn; OPODIS '20]. Moreover, we show that in general weighted graph families that are closed under taking subgraphs and in which we can compute an independent set of weight at least a $λ$-fraction of the total weight, one can compute a $(2-2λ+\varepsilon)$-approximate weighted vertex cover in polylogarithmic time in the CONGEST model. Our result in particular implies that in graphs of arboricity $a$, one can compute a $(2-1/a+\varepsilon)$-approximate weighted vertex cover. For maximum weighted matchings, we show that a $(1-\varepsilon)$-approximate solution can be computed deterministically in polylogarithmic CONGEST rounds (for constant $\varepsilon$). We also provide a more efficient randomized algorithm. Our algorithm generalizes results of [Lotker, Patt-Shamir, Pettie; SPAA '08] and [Bar-Yehuda, Hillel, Ghaffari, Schwartzman; PODC '17] for the unweighted case. Finally, we show that even in the LOCAL model and in bipartite graphs of degree $\leq 3$, if $\varepsilon<\varepsilon_0$ for some constant $\varepsilon_0>0$, then computing a $(1+\varepsilon)$-approximation for the unweighted minimum vertex cover problem requires $Ω\big(\frac{\log n}{\varepsilon}\big)$ rounds. This generalizes aresult of [Göös, Suomela; DISC '12], who showed that computing a $(1+\varepsilon_0)$-approximation in such graphs requires $Ω(\log n)$ rounds.

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arXiv2021

Deterministic Distributed Vertex Coloring: Simpler, Faster, and without Network Decomposition

We present a simple deterministic distributed algorithm that computes a $(Δ+1)$-vertex coloring in $O(\log^2 Δ\cdot \log n)$ rounds. The algorithm can be implemented with $O(\log n)$-bit messages. The algorithm can also be extended to the more general $(degree+1)$-list coloring problem. Obtaining a polylogarithmic-time deterministic algorithm for $(Δ+1)$-vertex coloring had remained a central open question in the area of distributed graph algorithms since the 1980s, until a recent network decomposition algorithm of Rozhoň and Ghaffari [STOC'20]. The current state of the art is based on an improved variant of their decomposition, which leads to an $O(\log^5 n)$-round algorithm for $(Δ+1)$-vertex coloring. Our coloring algorithm is completely different and considerably simpler and faster. It solves the coloring problem in a direct way, without using network decomposition, by gradually rounding a certain fractional color assignment until reaching an integral color assignments. Moreover, via the approach of Chang, Li, and Pettie [STOC'18], this improved deterministic algorithm also leads to an improvement in the complexity of randomized algorithms for $(Δ+1)$-coloring, now reaching the bound of $O(\log^3\log n)$ rounds. As a further application, we also provide faster deterministic distributed algorithms for the following variants of the vertex coloring problem. In graphs of arboricity $a$, we show that a $(2+ε)a$-vertex coloring can be computed in $O(\log^3 a\cdot\log n)$ rounds. We also show that for $Δ\geq 3$, a $Δ$-coloring of a $Δ$-colorable graph $G$ can be computed in $O(\log^2 Δ\cdot\log^2 n)$ rounds.

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arXiv2021

Improved Distributed Lower Bounds for MIS and Bounded (Out-)Degree Dominating Sets in Trees

Recently, Balliu, Brandt, and Olivetti [FOCS '20] showed the first $ω(\log^* n)$ lower bound for the maximal independent set (MIS) problem in trees. In this work we prove lower bounds for a much more relaxed family of distributed symmetry breaking problems. As a by-product, we obtain improved lower bounds for the distributed MIS problem in trees. For a parameter $k$ and an orientation of the edges of a graph $G$, we say that a subset $S$ of the nodes of $G$ is a $k$-outdegree dominating set if $S$ is a dominating set of $G$ and if in the induced subgraph $G[S]$, every node in $S$ has outdegree at most $k$. Note that for $k=0$, this definition coincides with the definition of an MIS. For a given $k$, we consider the problem of computing a $k$-outdegree dominating set. We show that, even in regular trees of degree at most $Δ$, in the standard \LOCAL model, there exists a constant $ε>0$ such that for $k\leq Δ^ε$, for the problem of computing a $k$-outdegree dominating set, any randomized algorithm requires at least $Ω(\min\{\logΔ,\sqrt{\log\log n}\})$ rounds and any deterministic algorithm requires at least $Ω(\min\{\logΔ,\sqrt{\log n}\})$ rounds. The proof of our lower bounds is based on the recently highly successful round elimination technique. We provide a novel way to do simplifications for round elimination, which we expect to be of independent interest. Our new proof is considerably simpler than the lower bound proof in [FOCS '20]. In particular, our round elimination proof uses a family of problems that can be described by only a constant number of labels. The existence of such a proof for the MIS problem was believed impossible by the authors of [FOCS '20].

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