58th Annual IEEE Symposium on Foundations of Computer Science Deterministic Distributed Edge-Coloring via Hypergraph Maximal Matching Manuela Fischer and Mohsen Ghaffari Fabian Kuhn Department of Computer Science Department of Computer Science ETH Zurich University of Freiburg 8092 Zurich, Switzerland 79110 Freiburg, Germany manuela.fi[email protected] and [email protected] [email protected] Abstract—We present a deterministic distributed algorithm maximum degree Δ, and where each node has a Θ(log n)- that computes a (2Δ − 1)-edge-coloring, or even list-edge- bit unique identifier. Initially, nodes have knowledge of coloring, in any n-node graph with maximum degree Δ,in G n O 8 · n their neighbors in as well as log and log Δ up to (log Δ log ) rounds. This answers one of the long-standing 1 open questions of distributed graph algorithms from the late constants. At the end, each node should know its own 1980s, which asked for a polylogarithmic-time algorithm. See, part of the solution, e.g., the colors of its edges in edge- e.g., Open Problem 4 in the Distributed Graph Coloring book coloring. Communication happens in synchronous rounds, of Barenboim√ and Elkin. The previous best round complexities where in each round each node sends a message to each of O( log n) were√ 2 by Panconesi and Srinivasan [STOC’92] and its neighbors.2 The main complexity measure is the number O˜ O ∗ n ( Δ) + (log ) by Fraigniaud, Heinrich, and Kosowski of rounds needed for solving a given graph problem. The [FOCS’16]. A corollary of our deterministic list-edge-coloring also improves the randomized complexity of (2Δ − 1)-edge- four classic local distributed graph problems are Maximal coloring to poly(log log n) rounds. Independent Set (MIS), (Δ+1)-vertex-coloring, (2Δ − 1)- The key technical ingredient is a deterministic distributed edge-coloring, and maximal matching [PR01], [BE13]. All algorithm for hypergraph maximal matching, which we believe these problems have trivial greedy sequential algorithms, will be of interest beyond this result. In any hypergraph of as well as simple O(log n)-round randomized distributed rank r — where each hyperedge has at most r vertices — with algorithms [Lub86], [ABI86], and even some faster ones n nodes and maximum degree Δ, this algorithm computes a [BEPS12], [EPS15], [Gha16], [HSS16]. But the determin- maximal matching in O r5 6+log r · n rounds. ( log Δ log ) istic distributed complexity remains widely open, despite This hypergraph matching algorithm and its extensions extensive interest (see, e.g., the first five open problems also lead to a number of other results. In particular, we ob- of [BE13]). Particularly, with regards to MIS — which is tain a polylogarithmic-time deterministic distributed maximal independent set (MIS) algorithm for graphs with bounded the hardest of the four problems, as the other three can be neighborhood independence, hence answering Open Problem reduced to MIS locally [Lin87] — Linial [Lin92] asked O /ε 5 of Barenboim and Elkin’s book, a (log Δ/ε) (log 1 ) - “can it [MIS] always be found [deterministically] in ε round deterministic algorithm for (1 + )-approximation of polylogarithmic time?”. maximum matching, and a quasi-polylogarithmic-time deter- λ This remains the most well-known open question√ of the ministic distributed algorithm for orienting -arboricity graphs O n with out-degree at most (1 + ε)λ, for any constant ε>0, area. The best known round complexity is 2 ( log ), due hence partially answering Open Problem 10 of Barenboim and to Panconesi and Srinivasan [PS92]. Panconesi and Rizzi Elkin’s book. pointed out in the year 2001 [PR01] that “while maximal Keywords-distributed graph algorithms; local algorithms; de- matchings can be computed in polylogarithmic time, in n, terministic distributed algorithms; edge-coloring; hypergraph; in the distributed model [HKP98], it is a decade old open maximal matching; rounding linear programs problem whether the same running time is achievable for the remaining 3 structures.” The status remains the same I. INTRODUCTION as of today, after almost two more decades. In particular, Distributed graph algorithms have been studied exten- for edge-coloring which is our main target, Barenboim and sively over the past 30 years, since the seminal work of Elkin stated the following problem in their recent Distributed Linial [Lin87]. Despite this, determining whether there are efficient deterministic distributed algorithms for the most 1If the latter is not the case, it is enough to try exponentially increasing classic problems of the area remains a long-standing open estimates. 2In the LOCAL model, messages might be of arbitrary size. A variant question. of the model where the messages have to be of bounded size is known Distributed graph algorithms are typically studied in a as the CONGEST model [Pel00]. Our edge-coloring algorithms and our MIS and vertex-coloring algorithms for graphs of bounded neighborhood standard synchronous message passing model known as the O n LOCAL independence in fact work with small (log )-bit-size messages. Though, model [Lin87], [Pel00]: the network is abstracted for the sake of the readability, we avoid explicitly discussing the details of as an undirected graph G =(V,E), n = |V |, with this aspect. 0272-5428/17 $31.00 © 2017 IEEE 180 DOI 10.1109/FOCS.2017.25 √ Graph Coloring book [BE13]: (2Δ − 1)-edge-coloring was 2O( log log n) rounds3, due to OPEN PROBLEM 11.4 [BE13]: Devise or rule out a Elkin, Pettie, and Su [EPS15]. By improving this, Corol- deterministic (2Δ − 1)-edge-coloring algorithm that lary I.2 widens the provable gap between the complexity runs in polylogarithmic time. of (2Δ − 1)-edge-coloring, which is now in poly(log log n) rounds, and the complexity of maximal matching, which A. Our Contributions needs Ω( log n/ log log n) rounds [KMW16]. Deterministic Edge-Coloring Algorithm: One of our main end results is a positive resolution of the above question. Unified Formulation as Hypergraph Maximal Matching: Our first step towards proving Theorem I.1 is a simple Theorem I.1. There is a deterministic distributed algorithm 8 unification of all the aforementioned four classic problems: that computes a (2Δ−1)-edge-coloring in O(log Δ·log n) − n MIS, (Δ+1)-vertex-coloring, (2Δ 1)-edge-coloring, and rounds, in any -node graph with maximum degree Δ. More- maximal matching. We can cast each of these problems as over, the same algorithm solves list-edge-coloring, where 4 e ∈ E a maximal matching problem on hypergraphs of some rank each edge must get a color from an arbitrary given r, which depends on the problem, and increases as we move list Le of colors with |Le| = de +1, where de denotes the e from maximal matching to maximal independent set. In number of edges incident to . other words, the hypergraph maximal matching problem can For list-edge-coloring,√ the previously best known round be used to obtain a smooth interpolation between maximal complexity was 2O( log n), by a classic network decom- matching in graphs and maximal independent set in graphs. position of Panconesi and√ Srinivasan [PS92], which it- Recall that the rank of a hypergraph is the maximum number self improved on an 2O( log n·log log n)-round algorithm of of vertices in any of its hyperedges. Moreover, a matching Awerbuch et al. [ALGP89].√ For low-degree graphs, the in a hypergraph is a set of hyperedges, no two of which ∗ best known is an (O˜( Δ) + O(log n))-round algorithm share an endpoint. of Fraigniaud, Heinrich, and Kosowski [FHK16], which is We present a reduction from (2Δ − 1)-edge-coloring to more general and applies also to (Δ+1)-list-vertex-coloring. maximal matching in rank-3 hypergraphs, as we sketch next There are some known poly log n-round deterministic edge- in Lemma I.3. A similar reduction can be used for list-edge- coloring algorithms, but all have two major shortcomings: coloring, as formalized in Lemma III.1. We note that these (A) they require more colors, and (B) they are quite re- reductions are inspired by the well-known reduction of Luby stricted and do not work for list-coloring. These algorithms from (Δ + 1)-vertex-coloring to MIS [Lub86], [Lin87]. are as follows: (I) a ((2+o(1))Δ)-edge-coloring by Ghaffari O log Δ and Su [GS17]; (II) a Δ · 2 ( log log Δ )-edge-coloring by Lemma I.3. Given a deterministic distributed algo- Barenboim and Elkin [BE11]—see also [GS17, Appendix rithm A that computes a maximal matching in N- B] for a short proof; and (III) an O(Δ log n)-edge-coloring vertex hypergraphs of rank 3 and maximum degree d by Czygrinow et al. [CHK01]. See also [BE11, Chapter 8]. in T (N,d) rounds, there is a deterministic distributed B − We also note that a recent work of Barenboim, Elkin, algorithm that computes a (2Δ 1)-edge-coloring of n G V,E and Maimon [BEM17] presents an efficient deterministic any -node graph =( ) with maximum degree T n , − algorithm for computing a (Δ + o(Δ))-edge-coloring in Δ in at most (3 Δ 2Δ 1) rounds. −δ G V,E graphs with arboricity a ≤ Δ1 , for some constant δ>0. Proof Sketch: To edge-color =( ),we H − G In Corollary IV.1, we sketch how a simple combination of generate a hypergraph : Take 2Δ 1 copies of .For e ∈ E e e the list-edge-coloring algorithm of Theorem I.1 with H- each edge , let 1 to 2Δ−1 be its copies. For each e ∈ E w H partitionings [BE13, Chapter 5.1] significantly extends their , add one extra vertex e to and then, change e e result.