You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
The @math -coloring problem is a fundamental symmetry breaking problem in distributed computing. We give a new randomized coloring algorithm for @math -coloring running in @math rounds with probability @math in a graph with @math nodes and maximum degree @math . This implies that the @math -coloring problem is easier than the maximal independent set problem and the maximal matching problem, due to their lower bounds by Kuhn, Moscibroda, and Wattenhofer [PODC'04]. Our algorithm also extends to the list-coloring problem where the palette of each node contains @math colors.

Cited Abstract(s):
[1]: Adecomposition of a graphG=(V,E) is a partition of the vertex set into subsets (calledblocks). Thediameter of a decomposition is the leastd such that any two vertices belonging to the same connected component of a block are at distance ≤d. In this paper we prove (nearly best possible) statements, of the form: Anyn-vertex graph has a decomposition into a small number of blocks each having small diameter. Such decompositions provide a tool for efficiently decentralizing distributed computations. In [4] it was shown that every graph has a decomposition into at mosts(n) blocks of diameter at mosts(n) for (s(n) = n^ O( n n) ). Using a technique of Awerbuch [3] and Awerbuch and Peleg [5], we improve this result by showing that every graph has a decomposition of diameterO (logn) intoO(logn) blocks. In addition, we give a randomized distributed algorithm that produces such a decomposition and runs in timeO(log2n). The construction can be parameterized to provide decompositions that trade-off between the number of blocks and the diameter. We show that this trade-off is nearly best possible, for two families of graphs: the first consists of skeletons of certain triangulations of a simplex and the second consists of grid graphs with added diagonals. The proofs in both cases rely on basic results in combinatorial topology, Sperner's lemma for the first class and Tucker's lemma for the second.

[2]: The authors introduce a concept of network decomposition, a partitioning of an arbitrary graph into small-diameter connected components, such that the graph created by contracting each component into a single node has low chromatic number. They present an efficient distributed algorithm for constructing such a decomposition and demonstrate its use for design of efficient distributed algorithms. The method yields new deterministic distributed algorithms for finding a maximal independent set in an arbitrary graph and for ( Delta +1)-coloring of graphs with maximum degree Delta . These algorithms run in O(n sup epsilon ) time for epsilon =O((log log n log n) sup 1 2 ), whereas the best previously known deterministic algorithms required Omega (n) time. The techniques can also be used to remove randomness from the previously known most distributed breadth-first search algorithm. >

[3]: Abstract A partition ( C 1 , C 2 , … , C q ) of G = ( V , E ) into clusters of strong (respectively, weak) diameter d , such that the supergraph obtained by contracting each C i is l -colorable is called a strong (resp., weak) ( d , l ) -network-decomposition. Network-decompositions were introduced in a seminal paper by Awerbuch, Goldberg, Luby and Plotkin in 1989. showed that strong ( d , l ) -network-decompositions with d = l = exp ⁡ O ( log ⁡ n log ⁡ log ⁡ n ) can be computed in distributed deterministic time O ( d ) . Even more importantly, they demonstrated that network-decompositions can be used for a great variety of applications in the message-passing model of distributed computing. The result of was improved by Panconesi and Srinivasan in 1992: in the latter result d = l = exp ⁡ O ( log ⁡ n ) , and the running time is O ( d ) as well. In another remarkable breakthrough Linial and Saks (in 1992) showed that weak ( O ( log ⁡ n ) , O ( log ⁡ n ) ) -network-decompositions can be computed in distributed randomized time O ( log 2 ⁡ n ) . Much more recently Barenboim (2012) devised a distributed randomized constant-time algorithm for computing strong network decompositions with d = O ( 1 ) . However, the parameter l in his result is O ( n 1 2 + ϵ ) . In this paper we drastically improve the result of Barenboim and devise a distributed randomized constant-time algorithm for computing strong ( O ( 1 ) , O ( n ϵ ) ) -network-decompositions. As a corollary we derive a constant-time randomized O ( n ϵ ) -approximation algorithm for the distributed minimum coloring problem, improving the previously best-known O ( n 1 2 + ϵ ) approximation guarantee. We also derive other improved distributed algorithms for a variety of problems. Most notably, for the extremely well-studied distributed minimum dominating set problem currently there is no known deterministic polylogarithmic-time algorithm. We devise a deterministic polylogarithmic-time approximation algorithm for this problem, addressing an open problem of Lenzen and Wattenhofer (2010).

[4]:
A variety of network decompositions have been developed to solve distributed computing problems. [2] introduced the notion of @math -decompositions where each component has diameter @math and the contracted graph is @math -colorable. They give a deterministic procedure to obtain a @math -decomposition, which can be used to deterministically compute a @math -coloring and MIS in @math rounds. Panconesi and Srinivasan [4] showed how to obtain a @math -decomposition, yielding @math -time algorithms for @math -coloring and MIS. Linial and Saks [1] gave a randomized algorithm for obtaining a @math -decomposition in @math rounds. [3] gave a randomized algorithm for obtaining @math -decompositions in constant rounds.