arXiv subjects
Publications and source records attributed to Magnús M. Halldórsson.
We provide a $\operatorname{poly}(\log\log n)$-round randomized LOCAL algorithm for the constructive Lovász Local Lemma with polynomial criterion $pd^{c} < 1$, for a sufficiently large constant $c$. The best previous general round complexity was $\widetilde O(\log n)$. The key ingredient is a local reduction that amplifies the LLL probability exponent quadratically, while increasing the dependency radius only by a constant factor.
Graph shattering is a central technique underlying sublogarithmic-time distributed algorithms in the LOCAL model. Its analysis typically relies on bounding the probability that large sets of distant nodes remain unresolved, often via independence assumptions justified by locality. We show that these assumptions fail for pre-shattering procedures that run for super-constant rounds, where dependencies accumulate over time. As a result, several standard shattering arguments in the literature are incomplete, including those for maximal independent set, $(Δ+1)$-coloring, and the distributed Lovász Local Lemma (LLL). We provide a systematic repair of these analyses. Our main contribution is a corrected shattering analysis of the Fischer--Ghaffari LLL algorithm. In addition, we develop general tools that capture common patterns in modern algorithms and yield the required decay bounds without relying on independence. We also present explicit counterexamples to commonly used shattering lemmas. Overall, we establish a robust and reusable foundation for shattering arguments in the presence of long-range dependencies.
Reed [J.~Comb.~Theory B, 1999] showed that graphs of maximum degree $Δ\geq 10^{14}$ without $Δ$-cliques are $(Δ-1)$-colorable. We design a one-pass semi-streaming algorithm for computing such a coloring. Additionally, we prove that any one-pass $(Δ-k)$-coloring algorithm for $0\leq k < (Δ+1)/2$ requires $Ω(n(k+1))$ space.
For any $Δ$, let $k_Δ$ be the maximum integer $k$ such that $(k+1)(k+2)\le Δ$. We give a distributed \LOCAL algorithm that, given an integer $k < k_Δ$, computes a valid $Δ-k$-coloring if one exists. The algorithm runs in $\tilde{O}(\log^4 \log n)$ rounds, which is within a polynomial factor of the $Ω(\log\log n)$ lower bound, which already applies to the case $k=0$. It is also best possible in the sense that if $k \ge k_Δ$, the problem requires $Ω(n/Δ)$ distributed rounds [Molloy, Reed, '14, Bamas, Esperet '19]. For $Δ$ at most polylogarithmic, the algorithm is an exponential improvement over the current state of the art of $O(\log^{49/12} n)$ rounds. When $Δ\ge (\log n)^{50}$, our algorithm achieves an even faster runtime of $O(\log^* n)$ rounds.
We consider the \textsf{Unit Interval Selection} problem in the one-pass random order streaming model. Here, an algorithm is presented a sequence of $n$ unit-length intervals on the line that arrive in uniform random order, and the objective is to output a largest set of disjoint intervals using space linear in the size of an optimal solution. Previous work only considered adversarially ordered streams and established that, in this space constraint, a $(2/3)$-approximation can be achieved, and this is also best possible, i.e. any improvement requires space $Ω(n)$ [Emek et al., TALG'16]. In this work, we show that an improved expected approximation factor can be achieved if the input stream is in uniform random order, with the expectation taken over the stream order. Specifically, we give a one-pass streaming algorithm with expected approximation factor $0.7401$ using space $O(|OPT|)$, where $OPT$ denotes an optimal solution. We also show that algorithms with expected approximation factor above $8/9$ require space $Ω(n)$, and algorithms that compute a better than $2/3$-approximation with probability above $2/3$ also require $Ω(n)$ space. On a technical note, we design an algorithm for the restricted domain $[0,Δ)$, for some constant $Δ$, and use standard techniques to obtain an algorithm for unrestricted domains. For the restricted domain $[0,Δ)$, we run $O(Δ)$ recursive instances of our algorithm, with each instance targeting the situation where a specific interval from $OPT$ arrives first. We establish the interesting property that our algorithm performs worst when the input stream is precisely a set of independent intervals. We then analyse the algorithm on these instances. Our lower bound is proved via communication complexity arguments, similar in spirit to the robust communication lower bounds by [Chakrabarti et al., Theory Comput. 2016].
We consider the problem of maintaining a proper $(Δ+ 1)$-vertex coloring in a graph on $n$-vertices and maximum degree $Δ$ undergoing edge insertions and deletions. We give a randomized algorithm with amortized update time $\widetilde{O}( n^{2/3} )$ against adaptive adversaries, meaning that updates may depend on past decisions by the algorithm. This improves on the very recent $\widetilde{O}( n^{8/9} )$-update-time algorithm by Behnezhad, Rajaraman, and Wasim (SODA 2025) and matches a natural barrier for dynamic $(Δ+1)$-coloring algorithms. The main improvements are in the densest regions of the graph, where we use structural hints from the study of distributed graph algorithms.
We improve the space bound for streaming approximation of Diameter but also of Farthest Neighbor queries, Minimum Enclosing Ball and its Coreset, in high-dimensional Euclidean spaces. In particular, our deterministic streaming algorithms store $\mathcal{O}(\varepsilon^{-2}\log(\frac{1}{\varepsilon}))$ points. This improves by a factor of $\varepsilon^{-1}$ the previous space bound of Agarwal and Sharathkumar (SODA 2010), while offering a simpler and more complete argument. We also show that storing $Ω(\varepsilon^{-1})$ points is necessary for a $(\sqrt{2}+\varepsilon)$-approximation of Farthest Pair or Farthest Neighbor queries.
Two of the most fundamental distributed symmetry-breaking problems are that of finding a maximal independent set (MIS) and a maximal matching (MM) in a graph. It is a major open question whether these problems can be solved in constant rounds of the all-to-all communication model of \textsf{Congested\ Clique}, with $O(\log\log Δ)$ being the best upper bound known (where $Δ$ is the maximum degree). We explore in this paper the boundary of the feasible, asking for \emph{which graphs} we can solve the problems in constant rounds. We find that for several graph parameters, ranging from sparse to highly dense graphs, the problems do have a constant-round solution. In particular, we give algorithms that run in constant rounds when: (1) the average degree is at most $d(G) \le 2^{O(\sqrt{\log n})}$, (2) the neighborhood independence number is at most $β(G) \le 2^{O(\sqrt{\log n})}$, or (3) the independence number is at most $α(G) \le |V(G)|/d(G)^μ$, for any constant $μ> 0$. Further, we establish that these are tight bounds for the known methods, for all three parameters, suggesting that new ideas are needed for further progress.
Graph coloring is fundamental to distributed computing. We give the first general treatment of the coloring of virtual graphs, where the graph $H$ to be colored is locally embedded within the communication graph $G$. Besides generalizing classical distributed graph coloring (where $H=G$), this captures other previously studied settings, including cluster graphs and power graphs. We find that the complexity of coloring a virtual graph depends on the edge congestion of its embedding. The main question of interest is how fast we can color virtual graphs of constant congestion. We find that, surprisingly, these graphs can be colored nearly as fast as ordinary graphs. Namely, we give a $O(\log^4\log n)$-round algorithm for the deg+1-coloring problem, where each node is assigned more colors than its degree. This can be viewed as a case where a distributed graph problem can be solved even when the operation of each node is decentralized.
We consider the problem of coloring graphs of maximum degree $Δ$ with $Δ$ colors in the distributed setting with limited bandwidth. Specifically, we give a $\mathsf{poly}\log\log n$-round randomized algorithm in the CONGEST model. This is close to the lower bound of $Ω(\log \log n)$ rounds from [Brandt et al., STOC '16], which holds also in the more powerful LOCAL model. The core of our algorithm is a reduction to several special instances of the constructive Lovász local lemma (LLL) and the $deg+1$-list coloring problem.
The constructive Lovász Local Lemma has become a central tool for designing efficient distributed algorithms. While it has been extensively studied in the classic LOCAL model that uses unlimited bandwidth, much less is known in the bandwidth-restricted CONGEST model. In this paper, we present bandwidth- and time-efficient algorithms for various subclasses of LLL problems, including a large class of subgraph sampling problems that are naturally formulated as LLLs. Lastly, we use our LLLs to design efficient CONGEST algorithms for coloring sparse and triangle-free graphs with few colors. These coloring algorithms are exponentially faster than previous LOCAL model algorithms.
We provide a $O(\log^6 \log n)$-round randomized algorithm for distance-2 coloring in CONGEST with $Δ^2+1$ colors. For $Δ\gg\operatorname{poly}\log n$, this improves exponentially on the $O(\logΔ+\operatorname{poly}\log\log n)$ algorithm of [Halldórsson, Kuhn, Maus, Nolin, DISC'20]. Our study is motivated by the ubiquity and hardness of local reductions in CONGEST. For instance, algorithms for the Local Lovász Lemma [Moser, Tardos, JACM'10; Fischer, Ghaffari, DISC'17; Davies, SODA'23] usually assume communication on the conflict graph, which can be simulated in LOCAL with only constant overhead, while this may be prohibitively expensive in CONGEST. We hope our techniques help tackle in CONGEST other coloring problems defined by local relations.
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.
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.
We give a randomized $Δ$-coloring algorithm in the LOCAL model that runs in $\text{poly} \log \log n$ rounds, where $n$ is the number of nodes of the input graph and $Δ$ is its maximum degree. This means that randomized $Δ$-coloring is a rare distributed coloring problem with an upper and lower bound in the same ballpark, $\text{poly}\log\log n$, given the known $Ω(\log_Δ\log n)$ lower bound [Brandt et al., STOC '16]. Our main technical contribution is a constant time reduction to a constant number of $(\text{deg}+1)$-list coloring instances, for $Δ= ω(\log^4 n)$, resulting in a $\text{poly} \log\log n$-round CONGEST algorithm for such graphs. This reduction is of independent interest for other settings, including providing a new proof of Brooks' theorem for high degree graphs, and leading to a constant-round Congested Clique algorithm in such graphs. When $Δ=ω(\log^{21} n)$, our algorithm even runs in $O(\log^* n)$ rounds, showing that the base in the $Ω(\log_Δ\log n)$ lower bound is unavoidable. Previously, the best LOCAL algorithm for all considered settings used a logarithmic number of rounds. Our result is the first CONGEST algorithm for $Δ$-coloring non-constant degree graphs.
We present ${\rm poly\log\log n}$-round randomized distributed algorithms to compute vertex splittings, a partition of the vertices of a graph into $k$ parts such that a node of degree $d(u)$ has $\approx d(u)/k$ neighbors in each part. Our techniques can be seen as the first progress towards general ${\rm poly\log\log n}$-round algorithms for the Lovász Local Lemma. As the main application of our result, we obtain a randomized ${\rm poly\log\log n}$-round CONGEST algorithm for $(1+ε)Δ$-edge coloring $n$-node graphs of sufficiently large constant maximum degree $Δ$, for any $ε>0$. Further, our results improve the computation of defective colorings and certain tight list coloring problems. All the results improve the state-of-the-art round complexity exponentially, even in the LOCAL model.
We present a new technique to efficiently sample and communicate a large number of elements from a distributed sampling space. When used in the context of a recent LOCAL algorithm for $(\operatorname{degree}+1)$-list-coloring (D1LC), this allows us to solve D1LC in $O(\log^5 \log n)$ CONGEST rounds, and in only $O(\log^* n)$ rounds when the graph has minimum degree $Ω(\log^7 n)$, w.h.p. The technique also has immediate applications in testing some graph properties locally, and for estimating the sparsity/density of local subgraphs in $O(1)$ CONGEST rounds, w.h.p.
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.