Deterministic CONGEST Algorithm for MDS on Bounded Arboricity Graphs
aa r X i v : . [ c s . D S ] F e b Deterministic CONGEST Algorithm for MDSon Bounded Arboricity Graphs
Saeed Akhoondian Amiri ∗ February 23, 2021
Abstract
We provide a deterministic CONGEST algorithm to constant factor approximatethe minimum dominating set on graphs of bounded arboricity in O (log n ) rounds. Thisimproves over the well known randomized algorithm of Lenzen and Wattenhofer [19]by making it a deterministic algorithm. The dominating set problem is to find a set of vertices whose closed neighborhood is theentire graph. Naturally, we are interested in a set of minimum size, or a so called minimumdominating set (MDS). The problem has been deeply studied in the sequential model aswell as distributed models.In this work, we are interested in algorithm design for LOCAL and CONGEST models.In these models every node of the network can be seen as a processor that communicates withits neighbors in synchronous communication rounds via communication links (the edges ofthe graph). In the LOCAL model there is no bandwidth limitation for communication links,but in the CONGEST model every link can carry a message of size O (log n ) per round, where n is the number of processors in the network. The aim of a distributed algorithm in suchmodels is to provide an approximate solution for the given problem while minimizing thenumber of communication rounds. At the end every node outputs its share of the solution.For instance for the dominating set problem, every node outputs whether it belongs to thecomputed dominating set.From the distributed perspective, Kuhn et al. [18] showed that in general graphs it isimpossible to find a logarithmic approximate of MDS in roughly speaking sublogarithmicrounds. From the positive side, Deurer et al. [12] provided a deterministic CONGESTalgorithm with O (log ∆)-approximation guarantee in polylogarithmic number of rounds (byemploying the recent breakthrough result for network decompositions [22]).The status of approximation algorithms for the MDS problem in sparse graphs is muchbrighter. In particular, Lenzen et al. [19], provided a deterministic constant factor approx-imation LOCAL algorithm for planar graphs in a constant number of rounds. Wawrzy-niak [23] extended that result to the CONGEST model and improved the approximationguarantee. The earlier work on planar graphs have been extended to bounded genus graphsby Amiri et al [3, 4, 5]. Then Czygrinow et al., first improved that algorithm to the CON-GEST model, then they extended it to excluded minor graphs [10, 11]. Amiri et al. [2],additionally provided a constant factor approximation on bounded expansion graphs in log-arithmic rounds for a generalized version of the problem: the distance- r MDS. The latterrecently has been improved in two directions: Kublenz et al. [17], reduced the number ofrounds to a constant but only for the standard MDS-problem, Amiri and Wiederhake [6]showed that for high girth graphs the approximation algorithm for distance r -MDS can beobtained by O ( r ) rounds. The minimum girth requirement showed to be useful in a recentwork of Alipour and Jafari [1], where they showed that in C -free planar graphs there is abetter approximation guarantee for the MDS problem.In another work, Lenzen and Wattenhofer provided a constant factor approximationrandomized algorithm with a logarithmic number of rounds on graphs of bounded arboric-ity [19]. This is the most generic result on sparse graphs, in a sense that it is a superclass ofall aforementioned sparse graphs. Although the number of rounds is not constant and thealgorithm is not deterministic, it performs much better than the existing work for generalgraphs. They also provided a deterministic O ( α log ∆)-approximation algorithm in O (log ∆)rounds, where α is the arboricity of the input graph and ∆ is the maximum degree, theyalso show that their algorithm can fall in a worst case with approximation factor O (log ∆).In this work we will extend their first result and show that it actually can be turned into adeterministic algorithm. ∗ University of Cologne, Germany, [email protected]. ottlenecks of Designing Algorithms for Bounded Arboricity Graphs The edge set of a graph of bounded arboricity can be decomposed into a bounded numberof forests. Such a decomposition is called
Forest Decomposition and it can be computed inlogarithmic number of rounds. This is where the logarithmic barrier appears in designingalgorithms on such graphs.In the following we briefly explain why we need such a, relatively slow to compute,decomposition and why we cannot simply employ the known techniques for other sparsegraph classes, at least for the case of MDS problem.All of the existing analysis methods for approximate MDS on planar, bounded genus,excluded minor, and bounded expansion graphs, are basically contraction-based: e.g. aftercontraction of subgraphs of a planar graph the resulting graph is still planar. In other words,all of the above-mentioned classes are closed under contraction. This facilitates analyzingdomination problems: to our best of understanding in all of such existing analysis, a crucialpoint of the analysis is to contract certain subgraphs to a single vertex and since the newgraph has the same structural attributes as the original one (e.g. it is still planar), we canemploy the structural properties of them in the rest of analysis, for instance we can arguethe resulting graph has a linear number of edges w.r.t. remaining vertices.Thus, contraction based arguments enabled researchers to employ the locality at its best.Such a delightful behavior does not appear in graphs of bounded arboricity. One famousexample for this is the subdivided clique. Subdivided clique is a graph obtained by firsttaking a complete graph, then placing one vertex in the middle of each edge, or in otherwords, subdivide every edge once. Subdivided clique has arboricity 2 but contracting halfof the edges, makes the remaining a complete graph.Hence, the logarithmic barrier for graphs of bounded arboricity seems to be inevitable byusing existing techniques. There is a progress in providing a decomposition with as fewestforests as possible [14, 15]. Breaking the logarithmic round complexity barrier for a constantnumber of forests contradicts the known lower bound of Linial [20] for coloring unorientedtrees. Working on a non-constant number of forests, happens to be useful in designingdistributed coloring algorithms, however for approximation algorithms, in particular thecase of the dominating set problem, it blows up the approximation guarantee.
Our Contribution and the Algorithm of Lenzen and Wattenhofer
Our algorithms is the continuation of the idea that appeared first in [19]. Lenzen andWattenhofer did the following: first, they have computed the forest decomposition of theinput graph. Based on this decomposition they create a specific auxiliary graph and concludethat the problem boils down to computing a maximal independent set (MIS) on that graph.Then by employing the randomized algorithm of Luby [21] for MIS, they solved the problemin O (log n ) rounds. Interestingly in their work, they started to use arguments based on setcover, however, due to the lack of existing distributed algorithms for set cover at the timethey employed MIS .In this work, we employ the distributed set cover directly and consequently provide aconstant factor approximation for MDS in O (log n ) deterministic rounds of the CONGESTmodel. We also provide a simpler analysis than the previous work. Their analysis was basedon separate counting arguments for child and parent nodes of MDS. We carry them out alltogether, which makes the proof of our following main theorem simpler. At the end we breakthe logarithmic barrier of the number of rounds by sacrificing the approximation guarantee. Theorem 1.
There is a CONGEST algorithm that runs in O (log n ) rounds and computesan O ( α ) -approximation of MDS in graphs of arboricity at most α . We assume familiarity of reader with basic graph notations. First, we briefly explain theconcept of forest decompositions.A graph G has arboricity at most α if there are at most α spanning forests (or similarlyspanning trees) such that their union spans all the edges of G . The set of such forests F is called a forest decomposition of G . Barenboim and Elkin [8] provided a CONGESTalgorithm that in O (log n/ǫ ) rounds computes a forest decomposition with at most (2 + ǫ ) α -forests in graphs of arboricity α (Algorithm 2 in the mentioned paper). The algorithmnot only computes the decomposition but also calculates the directed rooted trees of eachforest, hence by the end of the algorithm, every node knows to which forests it belongs.Additionally, every node knows its children and parents in the forests. However, in [13] authors mentioned that computing an approximate set cover in an instance withbounded frequency in logarithmic rounds is simple and they believe that it was known by the communityprior to their paper, thus their focus was on breaking the logarithmic barrier. U and a set S of subsets of U . The question is to find a subset of S such that the union of its elementscovers U . Clearly, we would like to minimize the size of such a subset. In the boundedfrequency variant of the problem, every element of U belongs to at most f elements of S ,for some constant f .In the distributed variant we model the problem with a bipartite graph H = ( A, B, E ):elements of S form the partition A , elements of U are the partition B and there is an edgebetween a vertex v in partition A and a vertex u in partition B if v appears in the set u . The network is the graph H and the problem is to find a dominating set/set cover, ofa minimum size from partition A . This problem has a deterministic O ( f )-approximationCONGEST algorithm in O ( f log ∆) rounds, when the frequency of each element is boundedby f . We will employ the following known result. Theorem 2 (Theorem 1 of Even et al. [13]) . There is a deterministic distributed algorithmin the CONGEST model that computes a f (1 + ǫ ) approximation of minimum set-cover, in O ( log( f ∆) ǫ log log( f ∆) ) rounds, in any set-system of frequency f and maximum set size ∆ , and forany < ǫ < . As already explained in the introduction a similar idea to what we will explain in thefollowing was described in the work of Lenzen and Wattenhofer, but they used Luby’salgorithm as a subroutine to obtain the desired approximation guarantee, here we will usethe set cover problem to provide a deterministic CONGEST algorithm.To approximate MDS in rooted trees we can follow the following simple greedy choice:every parent chooses itself as a dominator. This gives a constant factor approximation forMDS on trees. However, for graphs of bounded arboricity, when there are multiple forests,this can cause an issue: too many parents are chosen for a specific set of children.To resolve this issue we convert the problem to the bounded frequency set cover problem.In the following, we explain how to perform this transformation in the CONGEST model.
The universe U is the set of all nodes. We construct a bipartite graph H = ( A, B, E ) (asan instance of distributed set cover) as follows.Let F be a forest decomposition with at most f forests. For a node u we construct a set S u = { u } S C ( u ), here C ( u ) is the set of children of u in the forests of F . We say u is the representative of set S u . Then, the partition A consists of all the possible sets of form S v for v ∈ V ( G ). The part B is just V ( G ). The edge set E is defined as the natural way: if anelement (in part B ) is in a specific set (of part A ), then there is an edge between them.To simulate the graph H in the CONGEST model, every node v simulates S v , its imageas an element in part B , and all of the edges from the set S v to the corresponding elements(children) in part B as well as all edges from its image in part B to all its container sets(parents) in part A . Since the forest decomposition is given in advance, every node canconstruct the corresponding subgraph of H without an extra communication. Later whenwe invoke the set cover algorithm, whenever we are dealing with the set S v or an element v in part B the node v of G will perform the computational tasks.Observe that the degree of nodes in part B is at most f + 1, thus the instance has boundedfrequency for a constant f . Lemma 3.
Every element of B appears in at most f + 1 elements of A .Proof. Since there are f forests, a vertex u ∈ B is connected only to its own set S u or thecorresponding sets of its parents in part A , so it has at most f + 1 edges. Consequently itappears in at most f + 1 sets. We may assume that the input graph G has arboricity at most α . Then the algorithm is asfollows. 3 lgorithm 1: CONGEST approximation of MDS on bounded arboricity graphs.Input is a graph G with arboricity at most α . Construct a (2 + 1) α -forest decomposition F by algorithm in [8]. Construct H from F as explained in Section 3.1. Compute S , an O ( α )-approximate set cover of H by Theorem 2. Output the representative nodes of S .For the number of rounds we have the following lemma. Lemma 4.
The Algorithm 1 runs in O (log n ) rounds of the CONGEST model.Proof. Since every step of the algorithm can be performed in the CONGEST model bymessages of size O (log n ), the algorithm as a whole is a CONGEST algorithm. For therunning time, the first line requires O (log n ) rounds to compute the forest decomposition.Once the forest decomposition is calculated, we have the second line for free, i.e. it can becomputed without any new communication. For the third line, the running time depends onthe choice of ǫ – preciseness of the approximation for set cover. Hence, for constant valuesof ǫ , it takes O ( log( α ∆)log log( α ∆) ) rounds to perform the third line. Since ∆ ≤ n , the dominatingfactor in the round complexity is the first line, hence the algorithm performs in O (log n )rounds as claimed.Next we analyze the correctness and the approximation guarantee of algorithm. Lemma 5.
The Algorithm 1 outputs a dominating set of G , which is an O ( α ) -approximationfor the MDS problem.Proof. Since every set cover of H corresponds to a dominating set in G , the algorithm returnsa valid dominating set. It remains to prove the correctness of approximation guarantee.Let M be an MDS of G ; we construct a dominating set D that represents a set cover of H . By keeping the size of D within O ( α | M | ) we ensure that every O ( α )-approximate setcover in H is an O ( α )-approximation for MDS in G .Let u ∈ M then u dominates subset S ′ u ∈ V ( G ). Since it dominates all nodes in S ′ u ,it has edges to all of them. W.r.t. F these edges are either parent to child edges ( u is theparent) or child to parent edges ( u is a child). For the first case, u has all its children in itscorresponding bag S u in the set cover instance, it remains to add the latter nodes to ourdominating set D so that the union of their corresponding sets is the entire graph. Define D := M S u ∈ M P ( u ), where P ( u ) is the set of parents of u w.r.t. forests of F . Hence, wehave S v ∈ D S v = V ( G ) = U . Thus D is representing a set cover of H .Since by construction every node has at most 3 α parents, the size of D is at most(3 α + 1) | M | . Therefore, as D is representing a set cover of H , the optimal set cover hasat most | D | ≤ (3 α + 1) | M | sets. Consequently, the O ( α )-approximation of set cover inline 3 of Algorithm 1 has size at most O ( α (3 α + 1) | M | ), thus it corresponds to an O ( α )-approximation of MDS for G .Our main theorem is a follow up of the previous lemmas. Theorem 1
There is a CONGEST algorithm that runs in O (log n ) rounds and computesan O ( α )-approximation of MDS in graphs of arboricity at most α . Proof.
Algorithm 1 as proved in Lemma 5 provides an O ( α )-approximation of MDS in adeterministic fashion. On the other hand by Lemma 4, it terminates in O (log n ) rounds.To speed up the algorithm, we can use more forests to stop forest decomposition com-putation earlier, but then the approximation guarantee increases. For instance we can finda forest decomposition with √ log n forests in O (cid:0) log n log log n (cid:1) rounds to obtain the following.Here we only have to replace α with √ log n in the Algorithm 1. The following breaks thelogarithmic round complexity barrier by sacrificing the approximation guarantee. Corollary 6.
There is a CONGEST algorithm that runs in O (cid:0) log n log log n (cid:1) rounds and computesa O (log n ) -approximation of MDS in graphs of bounded arboricity. Can the approximation factor be improved to α + O ( α ) or better? We obtainedalmost 3 α + O ( α )-approximation guarantee and our result was dependent on the existingalgorithms for set cover and forest decomposition. We set ǫ = 1 in the algorithm forcomputation of the forest decomposition thus we got this bound; clearly one can use a smallervalue for epsilon to achieve a better approximation guarantee (in the cost of increasingnumber of rounds proportional to 1 /ǫ ).However, by this method, i.e. transferring the problem to set cover instance, it seemsthat it is impossible to provide a significantly better approximation guarantee. By this4pproach, unless there are logarithmic round algorithms to compute a forest decompositionwith α + O (1) forests and an ( α + O (1))-approximation for the set cover problem, we cannotget a α + O ( α )-approximation; we are not aware of existence of such algorithms. It is worthto mention that although the statement of the Theorem 1 in [19] might be misinterpretedas a ( α + O ( α ))-approximation guarantee; their distributed algorithm with existing toolshas essentially a same approximation guarantee as ours. We provided a deterministic CONGEST algorithm to constant factor approximate MDSon graphs of bounded arboricity and closed the gap between randomized and deterministicalgorithms on these graphs. However, it is not clear what is the gap between possibilitiesand impossibilities for the MDS problem in graphs of bounded arboricity. Bad news is thatthere is no explicit lowerbound for these graphs. All we know about lowerbounds is eitherrestricted to planar graphs [9, 16] or is for general graphs [18], nothing in-between is known.Due to the algorithmic barrier involved with construction of forest decomposition, itmight be the case that any radical improvement in the number of rounds for MDS couldcause a fundamental improvement on the existing distributed tools for bounded arboricitygraphs. However, there seems to be more accessible research directions. Maybe the firststep is to improve the approximation factor: the best known sequential algorithm has O ( α )-approximation guarantee [7] . Any o ( α )-approximation distributed algorithm is makingthe gap between sequential and distributed algorithms smaller. Acknowledgment:
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