arXiv subjects

Yannic Maus

Publications and source records attributed to Yannic Maus.

At least 55 records · Page 3Linked to original sources

arXiv2020

Local Conflict Coloring Revisited: Linial for Lists

Linial's famous color reduction algorithm reduces a given $m$-coloring of a graph with maximum degree $Δ$ to a $O(Δ^2\log m)$-coloring, in a single round in the LOCAL model. We show a similar result when nodes are restricted to choose their color from a list of allowed colors: given an $m$-coloring in a directed graph of maximum outdegree $β$, if every node has a list of size $Ω(β^2 (\log β+\log\log m + \log \log |\mathcal{C}|))$ from a color space $\mathcal{C}$ then they can select a color in two rounds in the LOCAL model. Moreover, the communication of a node essentially consists of sending its list to the neighbors. This is obtained as part of a framework that also contains Linial's color reduction (with an alternative proof) as a special case. Our result also leads to a defective list coloring algorithm. As a corollary, we improve the state-of-the-art truly local $(deg+1)$-list coloring algorithm from Barenboim et al. [PODC'18] by slightly reducing the runtime to $O(\sqrt{Δ\logΔ})+\log^* n$ and significantly reducing the message size (from huge to roughly $Δ$). Our techniques are inspired by the local conflict coloring framework of Fraigniaud et al. [FOCS'16].

cs.DS↗

arXiv2020

Distributed Approximation on Power Graphs

We investigate graph problems in the following setting: we are given a graph $G$ and we are required to solve a problem on $G^2$. While we focus mostly on exploring this theme in the distributed CONGEST model, we show new results and surprising connections to the centralized model of computation. In the CONGEST model, it is natural to expect that problems on $G^2$ would be quite difficult to solve efficiently on $G$, due to congestion. However, we show that the picture is both more complicated and more interesting. Specifically, we encounter two phenomena acting in opposing directions: (i) slowdown due to congestion and (ii) speedup due to structural properties of $G^2$. We demonstrate these two phenomena via two fundamental graph problems, namely, Minimum Vertex Cover (MVC) and Minimum Dominating Set (MDS). Among our many contributions, the highlights are the following. - In the CONGEST model, we show an $O(n/ε)$-round $(1+ε)$-approximation algorithm for MVC on $G^2$, while no $o(n^2)$-round algorithm is known for any better-than-2 approximation for MVC on $G$. - We show a centralized polynomial time $5/3$-approximation algorithm for MVC on $G^2$, whereas a better-than-2 approximation is UGC-hard for $G$. - In contrast, for MDS, in the CONGEST model, we show an $\tildeΩ(n^2)$ lower bound for a constant approximation factor for MDS on $G^2$, whereas an $Ω(n^2)$ lower bound for MDS on $G$ is known only for exact computation. In addition to these highlighted results, we prove a number of other results in the distributed CONGEST model including an $\tildeΩ(n^2)$ lower bound for computing an exact solution to MVC on $G^2$, a conditional hardness result for obtaining a $(1+ε)$-approximation to MVC on $G^2$, and an $O(\log Δ)$-approximation to the MDS problem on $G^2$ in $\mbox{poly}\log n$ rounds.

cs.DS↗

arXiv2020

Distance-2 Coloring in the CONGEST Model

We give efficient randomized and deterministic distributed algorithms for computing a distance-$2$ vertex coloring of a graph $G$ in the CONGEST model. In particular, if $Δ$ is the maximum degree of $G$, we show that there is a randomized CONGEST model algorithm to compute a distance-$2$ coloring of $G$ with $Δ^2+1$ colors in $O(\logΔ\cdot\log n)$ rounds. Further if the number of colors is slightly increased to $(1+ε)Δ^2$ for some $ε>1/{\rm polylog}(n)$, we show that it is even possible to compute a distance-$2$ coloring deterministically in polylog$(n)$ time in the CONGEST model. Finally, we give a $O(Δ^2 + \log^* n)$-round deterministic CONGEST algorithm to compute distance-$2$ coloring with $Δ^2+1$ colors.

cs.DC↗

arXiv2020

Efficient Deterministic Distributed Coloring with Small Bandwidth

We show that the $(degree+1)$-list coloring problem can be solved deterministically in $O(D \cdot \log n \cdot\log^2Δ)$ rounds in the \CONGEST model, where $D$ is the diameter of the graph, $n$ the number of nodes, and $Δ$ the maximum degree. Using the recent polylogarithmic-time deterministic network decomposition algorithm by Rozhoň and Ghaffari [STOC 2020], this implies the first efficient (i.e., $\poly\log n$-time) deterministic \CONGEST algorithm for the $(Δ+1)$-coloring and the $(\mathit{degree}+1)$-list coloring problem. Previously the best known algorithm required $2^{O(\sqrt{\log n})}$ rounds and was not based on network decompositions. Our techniques also lead to deterministic $(\mathit{degree}+1)$-list coloring algorithms for the congested clique and the massively parallel computation (MPC) model. For the congested clique, we obtain an algorithm with time complexity $O(\logΔ\cdot\log\logΔ)$, for the MPC model, we obtain algorithms with round complexity $O(\log^2Δ)$ for the linear-memory regime and $O(\log^2Δ+ \log n)$ for the sublinear memory regime.

cs.DC↗

arXiv2020

Classification of distributed binary labeling problems

We present a complete classification of the deterministic distributed time complexity for a family of graph problems: binary labeling problems in trees. These are locally checkable problems that can be encoded with an alphabet of size two in the edge labeling formalism. Examples of binary labeling problems include sinkless orientation, sinkless and sourceless orientation, 2-vertex coloring, perfect matching, and the task of coloring edges red and blue such that all nodes are incident to at least one red and at least one blue edge. More generally, we can encode e.g. any cardinality constraints on indegrees and outdegrees. We study the deterministic time complexity of solving a given binary labeling problem in trees, in the usual LOCAL model of distributed computing. We show that the complexity of any such problem is in one of the following classes: $O(1)$, $Θ(\log n)$, $Θ(n)$, or unsolvable. In particular, a problem that can be represented in the binary labeling formalism cannot have time complexity $Θ(\log^* n)$, and hence we know that e.g. any encoding of maximal matchings has to use at least three labels (which is tight). Furthermore, given the description of any binary labeling problem, we can easily determine in which of the four classes it is and what is an asymptotically optimal algorithm for solving it. Hence the distributed time complexity of binary labeling problems is decidable, not only in principle, but also in practice: there is a simple and efficient algorithm that takes the description of a binary labeling problem and outputs its distributed time complexity.

cs.DC↗

arXiv2019

Deterministic Distributed Dominating Set Approximation in the CONGEST Model

We develop deterministic approximation algorithms for the minimum dominating set problem in the CONGEST model with an almost optimal approximation guarantee. For $ε>1/{\text{poly}}\log Δ$ we obtain two algorithms with approximation factor $(1+ε)(1+\ln (Δ+1))$ and with runtimes $2^{O(\sqrt{\log n \log\log n})}$ and $O(Δ\cdot\text{poly}\log Δ+\text{poly}\log Δ\log^{*} n)$, respectively. Further we show how dominating set approximations can be deterministically transformed into a connected dominating set in the \CONGEST model while only increasing the approximation guarantee by a constant factor. This results in a deterministic $O(\log Δ)$-approximation algorithm for the minimum connected dominating set with time complexity $2^{O(\sqrt{\log n \log\log n})}$.

cs.DS↗

arXiv2019

P-SLOCAL-Completeness of Maximum Independent Set Approximation

We prove that the maximum independent set approximation problem with polylogarithmic approximation factor is P-SLOCAL-complete. Thus an efficient algorithm for the maximum independent set approximation in the LOCAL model implies efficient algorithms for many problems in the LOCAL model including the computation of (polylog n, polylog n) network decompositions.

cs.DS↗

arXiv2019

A Sharp Threshold Phenomenon for the Distributed Complexity of the Lovász Local Lemma

The Lovász Local Lemma (LLL) says that, given a set of bad events that depend on the values of some random variables and where each event happens with probability at most $p$ and depends on at most $d$ other events, there is an assignment of the variables that avoids all bad events if the LLL criterion $ep(d+1)<1$ is satisfied. In this paper, we study the dependency of the distributed complexity of the LLL problem on the chosen LLL criterion. We show that for the fundamental case of each random variable of the considered LLL instance being associated with an edge of the input graph, that is, each random variable influences at most two events, a sharp threshold phenomenon occurs at $p = 2^{-d}$: we provide a simple deterministic (!) algorithm that matches a known $Ω(\log^* n)$ lower bound in bounded degree graphs, if $p < 2^{-d}$, whereas for $p \geq 2^{-d}$, a known $Ω(\log \log n)$ randomized and a known $Ω(\log n)$ deterministic lower bounds hold. In many applications variables affect more than two events; our main contribution is to extend our algorithm to the case where random variables influence at most three different bad events. We show that, surprisingly, the sharp threshold occurs at the exact same spot, providing evidence for our conjecture that this phenomenon always occurs at $p = 2^{-d}$, independent of the number $r$ of events that are affected by a variable. Almost all steps of the proof framework we provide for the case $r=3$ extend directly to the case of arbitrary $r$; consequently, our approach serves as a step towards characterizing the complexity of the LLL under different exponential criteria.

cs.DS↗

arXiv2019

On the Complexity of Distributed Splitting Problems

One of the fundamental open problems in the area of distributed graph algorithms is the question of whether randomization is needed for efficient symmetry breaking. While there are fast, $\text{poly}\log n$-time randomized distributed algorithms for all of the classic symmetry breaking problems, for many of them, the best deterministic algorithms are almost exponentially slower. The following basic local splitting problem, which is known as the \emph{weak splitting} problem takes a central role in this context: Each node of a graph $G=(V,E)$ has to be colored red or blue such that each node of sufficiently large degree has at least one node of each color among its neighbors. Ghaffari, Kuhn, and Maus [STOC '17] showed that this seemingly simple problem is complete w.r.t. the above fundamental open question in the following sense: If there is an efficient $\text{poly}\log n$-time determinstic distributed algorithm for weak splitting, then there is such an algorithm for all locally checkable graph problems for which an efficient randomized algorithm exists. In this paper, we investigate the distributed complexity of weak splitting and some closely related problems. E.g., we obtain efficient algorithms for special cases of weak splitting, where the graph is nearly regular. In particular, we show that if $δ$ and $Δ$ are the minimum and maximum degrees of $G$ and if $δ=Ω(\log n)$, weak splitting can be solved deterministically in time $O\big(\fracΔδ\cdot\text{poly}(\log n)\big)$. Further, if $δ= Ω(\log\log n)$ and $Δ\leq 2^{\varepsilonδ}$, there is a randomized algorithm with time complexity $O\big(\fracΔδ\cdot\text{poly}(\log\log n)\big)$.

cs.DC↗

arXiv2019

Noidy Conmunixatipn: On the Convergence of the Averaging Population Protocol

We study a process of \emph{averaging} in a distributed system with \emph{noisy communication}. Each of the agents in the system starts with some value and the goal of each agent is to compute the average of all the initial values. In each round, one pair of agents is drawn uniformly at random from the whole population, communicates with each other and each of these two agents updates their local value based on their own value and the received message. The communication is noisy and whenever an agent sends any value $v$, the receiving agent receives $v+N$, where $N$ is a zero-mean Gaussian random variable. The two quality measures of interest are (i) the total sum of squares $TSS(t)$, which measures the sum of square distances from the average load to the \emph{initial average} and (ii) $\barϕ(t)$, measures the sum of square distances from the average load to the \emph{running average} (average at time $t$). It is known that the simple averaging protocol---in which an agent sends its current value and sets its new value to the average of the received value and its current value---converges eventually to a state where $\barϕ(t)$ is small. It has been observed that $TSS(t)$, due to the noise, eventually diverges and previous research---mostly in control theory---has focused on showing eventual convergence w.r.t. the running average. We obtain the first probabilistic bounds on the convergence time of $\barϕ(t)$ and precise bounds on the drift of $TSS(t)$ that show that albeit $TSS(t)$ eventually diverges, for a wide and interesting range of parameters, $TSS(t)$ stays small for a number of rounds that is polynomial in the number of agents. Our results extend to the synchronous setting and settings where the agents are restricted to discrete values and perform rounding.

cs.DC↗

arXiv2019

Improved Distributed Degree Splitting and Edge Coloring

The degree splitting problem requires coloring the edges of a graph red or blue such that each node has almost the same number of edges in each color, up to a small additive discrepancy. The directed variant of the problem requires orienting the edges such that each node has almost the same number of incoming and outgoing edges, again up to a small additive discrepancy. We present deterministic distributed algorithms for both variants, which improve on their counterparts presented by Ghaffari and Su [SODA'17]: our algorithms are significantly simpler and faster, and have a much smaller discrepancy. This also leads to a faster and simpler deterministic algorithm for $(2+o(1))Δ$-edge-coloring, improving on that of Ghaffari and Su.

cs.DC↗

arXiv2018

Local Distributed Algorithms in Highly Dynamic Networks

The present paper studies local distributed graph problems in highly dynamic networks. Communication and changes of the graph happen in synchronous rounds and our algorithms always, i.e., in every round, satisfy non-trivial guarantees, no matter how dynamic the network is. We define a (in our view) natural generalization of static graph problems to the dynamic graph setting. Throughout the execution of an algorithm we consider a sliding window over the last $T$, e.g., polylogarithmic, rounds. Then, in some round, the feasibility of an output only depends on the topology of the graphs in the current sliding window and we call a feasible output a $T$-dynamic solution. The guarantees of a $T$-dynamic solution become stronger the more stable the graph is during this sliding window and, in particular, they coincide with the definition of the static graph problem if the graph is static throughout the window. We further present an abstract framework that allows to develop algorithms that output $T$-dynamic solutions in all rounds. The resulting algorithms have another desirable property: If a constant neighborhood around some part of the graph is stable during an interval $[t_1,t_2]$, the algorithms compute a static solution for this part of the graph throughout the interval $[t_1+T',t_2]$ for some (small) $T'>0$. We demonstrate our generic framework with two sample problems that abstract basic operations in dynamic networks, namely $\textit{(degree+1)-vertex coloring}$ and $\textit{maximal independent set (MIS)}$. To illustrate the given guarantees consider the vertex coloring problem: The sliding window of our (randomized) algorithm is of length $T=O(\log n)$ and any conflict between two nodes caused by a newly inserted edge is resolved within that time. During this conflict resolving both nodes always output colors that are not in conflict with their respective 'old' neighbors.

cs.DS↗

arXiv2018

Deterministic Distributed Ruling Sets of Line Graphs

An $(α,β)$-ruling set of a graph $G=(V,E)$ is a set $R\subseteq V$ such that for any node $v\in V$ there is a node $u\in R$ in distance at most $β$ from $v$ and such that any two nodes in $R$ are at distance at least $α$ from each other. The concept of ruling sets can naturally be extended to edges, i.e., a subset $F\subseteq E$ is an $(α,β)$-ruling edge set of a graph $G=(V,E)$ if the corresponding nodes form an $(α,β)$-ruling set in the line graph of $G$. This paper presents a simple deterministic, distributed algorithm, in the $\mathsf{CONGEST}$ model, for computing $(2,2)$-ruling edge sets in $O(\log^* n)$ rounds. Furthermore, we extend the algorithm to compute ruling sets of graphs with bounded diversity. Roughly speaking, the diversity of a graph is the maximum number of maximal cliques a vertex belongs to. We devise $(2,O(\mathcal{D}))$-ruling sets on graphs with diversity $\mathcal{D}$ in $O(\mathcal{D}+\log^* n)$ rounds. This also implies a fast, deterministic $(2,O(\ell))$-ruling edge set algorithm for hypergraphs with rank at most $\ell$. Furthermore, we provide a ruling set algorithm for general graphs that for any $B\geq 2$ computes an $\big(α, α\lceil \log_B n \rceil \big)$-ruling set in $O(α\cdot B \cdot \log_B n)$ rounds in the $\mathsf{CONGEST}$ model. The algorithm can be modified to compute a $\big(2, β\big)$-ruling set in $O(βΔ^{2/β} + \log^* n)$ rounds in the $\mathsf{CONGEST}$~ model, which matches the currently best known such algorithm in the more general $\mathsf{LOCAL}$ model.

cs.DS↗

arXiv2017

Deterministic Distributed Edge-Coloring with Fewer Colors

We present a deterministic distributed algorithm, in the LOCAL model, that computes a $(1+o(1))Δ$-edge-coloring in polylogarithmic-time, so long as the maximum degree $Δ=\tildeΩ(\log n)$. For smaller $Δ$, we give a polylogarithmic-time $3Δ/2$-edge-coloring. These are the first deterministic algorithms to go below the natural barrier of $2Δ-1$ colors, and they improve significantly on the recent polylogarithmic-time $(2Δ-1)(1+o(1))$-edge-coloring of Ghaffari and Su [SODA'17] and the $(2Δ-1)$-edge-coloring of Fischer, Ghaffari, and Kuhn [FOCS'17], positively answering the main open question of the latter. The key technical ingredient of our algorithm is a simple and novel gradual packing of judiciously chosen near-maximum matchings, each of which becomes one of the color classes.

cs.DS↗

arXiv2017

On the Complexity of Local Distributed Graph Problems

This paper is centered on the complexity of graph problems in the well-studied LOCAL model of distributed computing, introduced by Linial [FOCS '87]. It is widely known that for many of the classic distributed graph problems (including maximal independent set (MIS) and $(Δ+1)$-vertex coloring), the randomized complexity is at most polylogarithmic in the size $n$ of the network, while the best deterministic complexity is typically $2^{O(\sqrt{\log n})}$. Understanding and narrowing down this exponential gap is considered to be one of the central long-standing open questions in the area of distributed graph algorithms. We investigate the problem by introducing a complexity-theoretic framework that allows us to shed some light on the role of randomness in the LOCAL model. We define the SLOCAL model as a sequential version of the LOCAL model. Our framework allows us to prove completeness results with respect to the class of problems which can be solved efficiently in the SLOCAL model, implying that if any of the complete problems can be solved deterministically in $\log^{O(1)} n$ rounds in the LOCAL model, we can deterministically solve all efficient SLOCAL-problems (including MIS and $(Δ+1)$-coloring) in $\log^{O(1)} n$ rounds in the LOCAL model. We show that a rather rudimentary looking graph coloring problem is complete in the above sense: Color the nodes of a graph with colors red and blue such that each node of sufficiently large polylogarithmic degree has at least one neighbor of each color. The problem admits a trivial zero-round randomized solution. The result can be viewed as showing that the only obstacle to getting efficient determinstic algorithms in the LOCAL model is an efficient algorithm to approximately round fractional values into integer values.

cs.DS↗

arXiv2017

Greedy Routing and the Algorithmic Small-World Phenomenom

The algorithmic small-world phenomenon, empirically established by Milgram's letter forwarding experiments from the 60s, was theoretically explained by Kleinberg in 2000. However, from today's perspective his model has several severe shortcomings that limit the applicability to real-world networks. In order to give a more convincing explanation of the algorithmic small-world phenomenon, we study decentralized greedy routing in a more flexible random graph model (geometric inhomogeneous random graphs) which overcomes all previous shortcomings. Apart from exhibiting good properties in theory, it has also been extensively experimentally validated that this model reasonably captures real-world networks. In this model, the greedy routing protocol is purely distributed as each vertex only needs to know information about its direct neighbors. We prove that it succeeds with constant probability, and in case of success almost surely finds an almost shortest path of length θ(loglog n), where our bound is tight including the leading constant. Moreover, we study natural local patching methods which augment greedy routing by backtracking and which do not require any global knowledge. We show that such methods can ensure success probability 1 in an asymptotically tight number of steps. These results also address the question of Krioukov et al. whether there are efficient local routing protocols for the internet graph. There were promising experimental studies, but the question remained unsolved theoretically. Our results give for the first time a rigorous and analytical affirmative answer.

cs.SI↗

arXiv2016

Polynomial Lower Bound for Distributed Graph Coloring in a Weak LOCAL Model

We show an $Ω\big(Δ^{\frac{1}{3}-\fracη{3}}\big)$ lower bound on the runtime of any deterministic distributed $\mathcal{O}\big(Δ^{1+η}\big)$-graph coloring algorithm in a weak variant of the \LOCAL\ model. In particular, given a network graph \mbox{$G=(V,E)$}, in the weak \LOCAL\ model nodes communicate in synchronous rounds and they can use unbounded local computation. We assume that the nodes have no identifiers, but that instead, the computation starts with an initial valid vertex coloring. A node can \textbf{broadcast} a \textbf{single} message of \textbf{unbounded} size to its neighbors and receives the \textbf{set of messages} sent to it by its neighbors. That is, if two neighbors of a node $v\in V$ send the same message to $v$, $v$ will receive this message only a single time; without any further knowledge, $v$ cannot know whether a received message was sent by only one or more than one neighbor. Neighborhood graphs have been essential in the proof of lower bounds for distributed coloring algorithms, e.g., \cite{linial92,Kuhn2006On}. Our proof analyzes the recursive structure of the neighborhood graph of the respective model to devise an $Ω\big(Δ^{\frac{1}{3}-\fracη{3}}\big)$ lower bound on the runtime for any deterministic distributed $\mathcal{O}\big(Δ^{1+η}\big)$-graph coloring algorithm. Furthermore, we hope that the proof technique improves the understanding of neighborhood graphs in general and that it will help towards finding a lower (runtime) bound for distributed graph coloring in the standard \LOCAL\ model. Our proof technique works for one-round algorithms in the standard \LOCAL\ model and provides a simpler and more intuitive proof for an existing $Ω(Δ^2)$ lower bound.

cs.DC↗

arXiv2015

Wavelet Riesz bases associated to nonisotropic dilations

A bounded, Riemann integrable and measurable set $K\subset \mathbb{R}^d$, which fulfills \[\sum\limits_{γ\inΓ}\mathbb{1}_K(x-γ)=k\text{ almost everywhere, $x\in\mathbb{R}^d$}\] for a lattice $Γ\subset\mathbb{R}^d$ is called $k$-tiling. If $K\subset\mathbb{R}^d$ is $k$-tiling $L^2(K)$ will admit a Riesz basis of exponentials. We use this result to construct generalized Riesz wavelet bases of $L^2(\mathbb{R}^2)$, arising from the action of suitable subsets of the affine group. One example of our construction is the first known shearlet Riesz basis.

math.FA↗