Small maximal matchings of random cubic graphs

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<ul><li><p>Small Maximal Matchingsof Random Cubic Graphs</p><p>H. Assiyatun1 and W. Duckworth2</p><p>1DEPARTMENT OF MATHEMATICSINSTITUT TEKNOLOGI BANDUNG</p><p>BANDUNG 40132, INDONESIAE-mail: hilda@math.itb.ac.id</p><p>2MATHEMATICAL SCIENCES INSTITUTEAUSTRALIAN NATIONAL UNIVERSITY, CANBERRA</p><p>ACT 2000, AUSTRALIAE-mail: Billy.Duckworth@maths.anu.edu.au</p><p>Received August 13, 2002; Revised March 30, 2008</p><p>Published online 14 October 2009 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/jgt.20434</p><p>Abstract: We consider the expected size of a smallest maximal matchingof cubic graphs. Firstly, we present a randomized greedy algorithm forfinding a small maximal matching of cubic graphs. We analyze the average-case performance of this heuristic on random n-vertex cubic graphs usingdifferential equations. In this way, we prove that the expected size of themaximalmatching returned by the algorithm is asymptotically almost surely(a.a.s.) less than 0.34623n. We also give an existence proof which showsthat the size of a smallest maximal matching of a random n-vertex cubicgraph is a.a.s. less than 0.3214n. It is known that the size of a smallestmaximal matching of a random n-vertex cubic graph is a.a.s. larger than0.3158n. 2009 Wiley Periodicals, Inc. J Graph Theory 62: 293323, 2009</p><p>Keywords: matchings; random; cubic; graphs</p><p>Journal of Graph Theory 2009 Wiley Periodicals, Inc.</p><p>293</p></li><li><p>294 JOURNAL OF GRAPH THEORY</p><p>1. INTRODUCTION</p><p>A maximal matching of a graph G is a set of edgesME(G) such that no two edges ofM share a common end-point and every edge in E(G)\M shares at least one end-pointwith an edge of M. The problem of finding a minimum maximal matching of a graphis encompassed by the minimum maximal flow problem which plays an important rolein the investigation of how inefficiently a network may be organized [14].</p><p>Yannakakis and Gavril [21] showed that the problem of finding a minimum maximalmatching of a graph is NP-hard, even when restricted to planar or bipartite graphs ofmaximum degree 3. In the same article they also gave a polynomial time algorithmthat finds a minimum maximal matching of trees. Horton and Kilakos [10] showedthat the problem of finding a minimum maximal matching of a graph remains NP-hard for planar bipartite graphs. They also gave a polynomial time algorithm thatfinds a minimum maximal matching for various classes of chordal graphs. Zito [22]extended these NP-hardness results to include (ks,3s)-graphs for every integer s&gt;0 andfor k{1,2}. (A (,)-graph has minimum degree and maximum degree .)</p><p>A graph G is said to be d-regular if every vertex of G has degree d. We considersimple, connected, cubic (i.e. 3-regular) graphs on n vertices (n must be even). For otherbasic graph theory definitions, not defined here, we refer the reader (as an example) toDiestel [5].</p><p>Duckworth [7] showed that the size of a minimum maximal matching of ann-vertex cubic graph is at most 9n/ 20+O(1). Their proof uses a linear programmingtechnique to analyze the worst-case performance of a simple greedy algorithm. Theuse of this technique also illustrates that there exist infinite families of cubic graphsfor which their algorithm only achieves this bound. In the same article they alsoshowed that there exist infinite families of n-vertex cubic graphs for which the size ofa minimum maximal matching is at least 3n/ 8. Note that it is simple to verify that thesize of a minimum maximal matching of an n-vertex cubic graph is at least 3n/ 10 andat most n/ 2.</p><p>As we consider random cubic graphs that are generated uniformly at random (u.a.r.),we need some notation. Throughout this article, we use the notation P (probability), E(expectation) and say that a property, B=Bn, of a random graph on n vertices holdsasymptotically almost surely (a.a.s.) if limn P(Bn)=1. For other basic random graphtheory definitions not defined here, we refer the reader (as an example) to Jansonet al. [13].</p><p>Zito [23] showed that the size of a smallest maximal matching, M, of a randomn-vertex cubic graph, a.a.s., satisfies 0.3158n</p></li><li><p>SMALL MAXIMAL MATCHINGS OF RANDOM CUBIC GRAPHS 295</p><p>describe a known technique, the differential equation method, for analyzing the perfor-mance of algorithms on random regular graphs. We analyze our algorithm on randomcubic graphs as detailed in Section 4. We prove that the expected size of the maximalmatching returned by the algorithm is a.a.s. less than 0.34623n in Section 5. We useanother known technique, the small subgraph conditioning method, to give an existenceproof that shows that the size of a minimum maximal matching of a random n-vertexcubic graph is a.a.s. less than 0.3214n. The details of this are given in Section 6.Although this upper bound is lower, the use of this method does not indicate an actualalgorithm that is expected to achieve this result. Both results are therefore of interest.</p><p>2. GREEDY ALGORITHMS</p><p>There are many ways in which one may design a greedy algorithm for finding a smallmaximal matching of a cubic graph. These differ in the way each subsequent edge isselected to be part of the matching. In all cases, once an edge has been selected to bepart of the matching, the graph may be updated by removing the selected edge andall edges incident with its end-points. The algorithm we introduce for finding a smallmaximal matching of cubic graphs is a greedy algorithm that is based, at each step,on selecting vertices of minimum degree u.a.r. from an ever-shrinking subgraph of theinput graph.</p><p>Each step of the algorithm involves the process of selecting an edge to add to a set,M, and the deletion of the edges incident with the end-points of the selected edge. Thedeletion of these edges ensures that the matching edges remain vertex disjoint which,in turn, ensures that once no edges remain, the set M is a maximal matching. Therationale behind selecting vertices of current minimum degree is that we do not wantto be in a situation where there are many vertices of small current degree. By selectingsuch a vertex at the start of each step we avoid this build-up of vertices of small degree.If we allowed this to happen, many vertices of small degree would eventually be chosenas an end-point of a matching edge which, in turn, removes a small number of edgesand thus causing the number of steps (and the size of the matching) to increase.</p><p>The algorithm, which we refer to as MMM Greedy, is given in Figure 1. Thealgorithm takes an n-vertex cubic graph G as input and returns a maximal matching,M,of G. We assume the input graph to be connected; otherwise, we apply our algorithmto each connected component. For a vertex v, deg(v) denotes the current degree of vin G and N(v) denotes the current set of neighbors of v. Also, Vi denotes the set ofvertices of current degree i in G and Yi denotes |Vi|.</p><p>The first step of the algorithm involves the process of selecting the first edge of thematching u.a.r. from all the edges of the input graph and deleting the edges incidentwith the end-points of this selected edge. As the input graph is assumed to be connected,we note that after the first step and before the completion of the algorithm, there mustalways exist a vertex of degree 1 or 2. For each subsequent step, if Y1 is non-zero, weselect a vertex, w, u.a.r. from V1. Otherwise, w is selected u.a.r. from V2. A vertex, u,is then selected u.a.r. from the neighbors of w. If u has degree 1, then so does w which</p><p>Journal of Graph Theory DOI 10.1002/jgt</p></li><li><p>296 JOURNAL OF GRAPH THEORY</p><p>FIGURE 1. Algorithm MMM Greedy.</p><p>means that this edge is isolated in G so we simply add the edge uw to M. Otherwise, adifferent edge incident with u is selected to become part of M. We select a neighbor, v,of u u.a.r. from those vertices that have current maximum degree in the set {N(u)\w}.We then add the edge uv to M and delete all edges incident with u and v.</p><p>3. RANDOM GRAPHS AND DIFFERENTIAL EQUATIONS</p><p>In this section we introduce the model we use to generate a cubic graph u.a.r. and givean overview of an established method of analyzing the performance of randomizedalgorithms on regular graphs.</p><p>The standard model for random d-regular graphs is as follows. Take a set of dnpoints in n buckets labeled 1,2, . . . ,n, with d points in each bucket, and choose u.a.r. apairing P=p1, . . . ,pdn/2 of the points such that each pi is an unordered pair of pointsand each point is in precisely one pair pi. The resulting probability space of pairings isdenoted by Pn,d. Form a d-regular pseudograph (i.e. a non-simple graph in which loopsand multiple edges are allowed) on n vertices by placing an edge between vertices iand j for each pair in P having one point in bucket i and one in bucket j. In order toprove that a property is a.a.s. true of a uniformly distributed random d-regular (simple)graph, it is enough to prove that it is a.a.s. true of the pseudograph corresponding to arandom pairing (see, for example, Bollobas [4] and Wormald [19, Section 2]).</p><p>One method of analyzing the performance of a randomized algorithm is to use asystem of differential equations to express the expected changes in variables describingthe state of the algorithm during its execution. Wormald [20] gives an exposition of thismethod and this method has been applied to several other graph-theoretic optimizationproblems including (amongst others) independent dominating set [8], independent set[20] and 2-independent set [6].</p><p>In order to analyze the algorithm given in Figure 1, we incorporate the algorithmas part of a pairing process that generates a random regular graph. In this way, wegenerate the random graph in the order that the edges are examined by the algorithm.</p><p>Journal of Graph Theory DOI 10.1002/jgt</p></li><li><p>SMALL MAXIMAL MATCHINGS OF RANDOM CUBIC GRAPHS 297</p><p>We may consider the generation process as follows. Initially, all vertices have degreezero. Throughout the execution of the generation process vertices will increase in degreeuntil all vertices have degree 3. Once the degree of a vertex reaches 3, the vertex is saidto be saturated and the generation is complete when all vertices are saturated. Duringthe generation process, we refer to the graph being generated as the evolving graph asthe graph evolves from all vertices having degree zero at the start of the process to allvertices having degree 3 at the end of the process.</p><p>During the generation of a random cubic graph we choose the pairs sequentially.The first point, pi, of a pair may be selected by any rule but in order to ensure thatthe cubic graph is generated u.a.r., the second point, pj, of that pair must be selectedu.a.r. from all the remaining free (i.e. unpaired) points. This preserves the uniformdistribution of the final pairing. The freedom of choice of pi enables us to select it u.a.r.from the vertices of a particular degree in the evolving graph and we say that selectingpj u.a.r. from all the remaining free points denotes selecting a mate for pi. Using B(pk)to denote the bucket that the point pk belongs to, we say that the edge from B(pi) toB(pj) is exposed and B(pj) is hit by this edge. We may then determine the degree ofthe vertex represented by the bucket B(pj) without exposing further edges.</p><p>In what follows, we denote the set of vertices of current degree i of the evolvinggraph, at time t, by Vi =Vi(t) and let Yi =Yi(t) denote |Vi|. We may express the state ofthe evolving graph at any point during the execution of the algorithm by considering Y0,Y1 and Y2. In order to analyze our randomized algorithm for finding a small maximalmatching, M, of cubic graphs, we calculate the expected change in this state over apredefined unit of time in relation to the expected change in the size of M.</p><p>Let M=M(t) denote |M| at any stage of the algorithm (time t) and let EX denotethe expected change in a random variable X conditional upon the history of the process.Equations representing EYi and EM are then used to derive a system of differentialequations. The solutions to the differential equations describe functions which representthe behavior of the variables Yi. Wormald [20, Theorem 6.1] describes a generalresult which guarantees that the solutions of the differential equations almost surelyapproximate the variables Yi and M with error o(n). The expected size of M may bededuced from these results.</p><p>4. INCORPORATING THE PAIRING PROCESS</p><p>The incorporated algorithm and pairing process are given in Figure 2. In the algorithm,the function isolate(u,v) exposes all remaining edges incident with the vertices repre-sented by the buckets u and v. This ensures that the set of edges returned is vertexdisjoint.</p><p>The algorithm terminates when there are no vertices of degree 1 or 2 remainingwhich means that a connected component has been completely generated and a maximalmatching has been found in that component. It was shown independently by Bollobas [3]and Wormald [17] that for fixed d3, random d-regular graphs are a.a.s. d-connected,so the result is a.a.s. a maximal matching in the whole graph.</p><p>Journal of Graph Theory DOI 10.1002/jgt</p></li><li><p>298 JOURNAL OF GRAPH THEORY</p><p>FIGURE 2. MMM Greedy incorporated with a pairing process.</p><p>We say that the combined process proceeds in operations where each operationdenotes the selection of an edge to add to the matching along with exposing all edgesincident with its end-points. The first operation of the algorithm involves randomlyselecting the first edge of M and exposing the appropriate edges. After this firstoperation we note that, before the end of the process, there always exists a vertex ofdegree 1 or 2. This follows as random regular graphs are a.a.s. connected.</p><p>There are two basic types of operation performed by the algorithm which we referto as Type 1 and Type 2 operations. A Type 1 operation refers to an operation whereY2 =0 and a vertex is selected u.a.r. from V1. Similarly, a Type 2 operation refers toan operation where Y2&gt;0 and a vertex is selected u.a.r. from V2.</p><p>Once w has been selected, we expose an edge incident with w and let u denote thevertex hit by this exposed edge. If both u and w are saturated after this edge is exposed,we add the edge uw to M. Otherwise, we expose all remaining edges incident with u.Using S to denote the set of vertices hit by these exposed edges, we select a vertex vu.a.r. from the vertices of minimum degree in S. We then add the edge uv to M andexpose all remaining edges incident with the vertices u and v.</p><p>5. ALGORITHM ANALYSIS</p><p>We analyze the average-case performance of MMM Greedy using the differential equa-tion method and in this way prove the following theorem.</p><p>Theorem 1. For a random cubic graph on n vertices, the size of a minimum maximalmatching is a.a.s. less than 0.34623n.</p><p>Proof. After the first operation, we split the remainder of the analysis into twodistinct phases. We informally define Phase 1 as the period of time where any verticesin V2 that are created are used up almost immediately and Y2 remains small. Once therate of generating vertices in V2 becomes larger than the rate that they are used up,</p><p>Journal of Graph Theory DOI 10.1002/j...</p></li></ul>