On finding spanning eulerian subgraphs

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<ul><li><p>On Finding Spanning Eulerian Subgraphs </p><p>M. B. Richey and R. Gary Parker School of Industrial and Systems Engineering, Georgia Institute of Technology, </p><p>Atlanta, Georgia 30332 </p><p>R. L. Rardin School of Industrial Engineering, Purdue University, W. Lafayette, Indiana 47907 </p><p>In this article, we examine the problem of producing a spanning Eulerian subgraph in an undirected graph. After the bX?-completeness of the general problem is established, we present polynomial-time algorithms for both the maximization and minimization versions where instances are defined on a restricted class of graphs referred to as series-parallel. Some novelties in the minimization case are discussed, as are heuristic ideas. </p><p>1. INTRODUCTION </p><p>Combinatorial optimization problems defined on graphs and networks have long been a source of great interest to practitioners and researchers alike. This is due, in large measure, to the obvious value associated with possession of effective algorithms for treating a host of practical problems arising in areas of distribution, routing, sched- uling, and the like. It is also the case, however, that interest is generated from the general intransigence of such problems regarding the development of efficient schemes for their resolution. </p><p>One of the classic models in this area is the so-called Chinese postmans problem. Required is a minimum length cycle in a graph which includes each edge at least once. Of course, if the input graph is Eulerian (connected with even degree at every vertex), the problem is trivially resolved-we simply trace the existing Eulerian cycle. Alter- natively, if the graph is not Eulerian, a simple but elegant scheme [4] can be used which, in essence, yields a minimum cost edge duplication. The resulting graph is Eulerian and the desired cycle can then be traced. </p><p>From a more abstract perspective, the Chinese postmans problem can be viewed as one of seeking, given some graph GO?,&amp;), a minimum cardinality superset 1 6, such that G@,&amp; is Eulerian. With this in mind, suppose we consider the opposite notion. Given G@,&amp;), find a subset 6 C_ 6, such that G(tl,6) is Eulerian. That is, we now seek a spanning Eulerian subgraph (SES) of Gel,&amp;). In fact, it is with this problem that this article is concerned. </p><p>Clearly, in dealing with the SES problem, it is meaningful to consider both a maximum and minimum cardinality version. Obviously, a maximum version of the Chinese postman problem is ill-defined (assuming nonnegative edge costs). Regardless, in what follows, we shall examine the complexity of the basic problem of simply finding an arbitrary SES, which, as we shall see, is WP-complete. This makes especially legitimate the investigation of special cases. Accordingly, we provide polynomial-time algorithms for the problem defined on the class of graphs known as series-parallel. We conclude with a brief discussion regarding heuristic notions. </p><p>Naval Research Logistics Quarterly, Vol. 32, pp. 443-455 (1985) Copyright 0 1985 by John Wiley &amp; Sons, Inc. CCC O28- 1441 /85/03O443- 13$O4.OO </p></li><li><p>444 Naval Research Logistics Quarterly, Vol. 32 (1985) </p><p>2. PROBLEM COMPLEXITY </p><p>Establishing the difficulty of the minimum (cardinality) SES is a straightforward exercise. Clearly, a suitable algorithm could then be used to decide which graphs are Hamiltonian; that is, a Hamiltonian cycle is an SES and is minimal in this regard. </p><p>The analogous complexity result for the maximum SES case is only slightly less obvious once we have the following lemma: </p><p>LEMMA 2.1: Deciding hamiltonicity in cubic, three-connected planar graphs is %?-complete. </p><p>PROOF: See [ 5 ] . Q.E.D. </p><p>Observe that a graph is called cubic if it is regular with every vertex having degree 3. Thus, minimum or maximum edge cardinality issues notwithstanding, the following theorem can be stated, which establishes that simply finding an SES is difficult. </p><p>THEOREM 2.2: Given an arbitrary undirected graph, G@,&amp;), deciding whether or not G@,G) possesses an SES is 39-complete. </p><p>PROOF: Clearly the problem is in %!? since for any subgraph of G , checking eulericity (including connectivity) is easy. To show the necessary reduction, we employ Lemma 2.1. Let GOY,&amp;) be a graph satisfying the hypothesis of the lemma. Then, obviously, the only SES in G@,&amp;) is also a Hamiltonian cycle for any more dense subgraph cannot have even degree everywhere and, alternatively, the only other even- degree subgraphs are not connected. Hence, an algorithm for finding an SES in arbitrary graphs could be used to decide hamiltonicity in the restricted class specified in Lemma 2.1. The latter problem is c.nt9-complete and the result follows. Q.E.D. </p><p>Obvious in the proof of Theorem 2.2 is the equivalence of the minimum and max- imum SES problems defined on cubic graphs. Simply stated, on such graphs, minimum and maximum SESs are one and the same. This aside, however, it may be instructive to briefly point out the failure of the naive Chinese postman-based algorithm for the maximum SES version. </p><p>Recall in the classic case, some non-Eulerian G@,&amp;) is appended by a least cardinality edge-set, G, such that the supergraph GOY,&amp; U g) is Eulerian. In the maximum SES problem, the aim is the opposite-we seek the least cardinality edge removal that leaves an SES. Important in this context is that the former (Chinese postman) problem is easily solved while the latter, as Theorem 2 .2 shows, is not. </p><p>If GO?,&amp;) is not Eulerian, it can be made so, in an optimal way, by duplicating edges implied by a least-weight perfect matching of odd-degree vertices. This matching is defined on a complete graph the edges of which have weights determined by minimum edge cardinality paths between the stated odd vertex-pairs in G. Unfortunately, the same scheme cannot be employed in order to determine an optimal edge removal. The culprit in this case is the requirement that the resulting (spanning) subgraph be con- nected. The key phenomenon can be demonstrated by considering the graph in Figure 1 a. Suitable matchings of odd-degree vertices and the corresponding edge removals are depicted by the subgraphs in parts b and c. Neither graph is an SES; however, in part d, a matching is found that in fact produces a feasible subgraph. </p></li><li><p>445 Richey, Parker, and Rardin: Spanning Eulerian Subgraphs </p><p>2 </p><p>Matching: 1-2.6-7,8-9,3-4,5-10. </p><p>(a) (b) </p><p>Matching: 1-6,2-7,3-8,4-9,5-10 Matching: 1-2,6-7,3-8.4-9,5-10 </p><p>(d 1 ( C ) </p><p>Figure 1. Even-degree subgraph construction by odd vertex matching. </p><p>We note, of course, that if the input graph has sufficiently high connectivity, this naive edge removal tactic will work. Unfortunately, we do not know a minimum value of connectivity in this regard. </p><p>3. SOLVABLE CASES </p><p>The complexity result of Theorem 2.2 makes unlikely the development of efficient algorithms for producing maximum or minimum, indeed even feasible, SESs. Ac- cordingly, we now turn to the treatment of special cases. </p><p>Series-Parallel Graphs </p><p>The class of graphs referred to as series-parallel form a proper subset of planar graphs. The next two theorems provide characterizations. </p><p>THEOREM 3.1: An undirected graph, GO?,&amp;), is series-parallel if and only if it can be reduced to an edge or family of edges by the repeated application of the following operations: </p><p>A (i) Series reduction: Replace any degree-2 vertex j and its incident edges e = A </p><p>( i , j ) , f A ( j , k ) such that i # k , by a new edge g = ( i , k ) . </p></li><li><p>446 Naval Research Logistics Quarterly, Vol. 32 (1985) </p><p>A A (ii) Parallel reduction: Replace any two edges e = (i,j) and f = (i,j) by a new </p><p>(iii) Jackknife reduction: Replace any degree-1 vertex i , its incident edge e = </p><p>A edge g = (i,j). </p><p>( i , j ) , and any other edge f = ( j , k ) by a new edge g = ( j , k ) . </p><p>A </p><p>A A </p><p>PROOF: See [7]. Q.E.D. </p><p>The elementary operations of the theorem are demonstrated in Figure 2. Perhaps the more familiar specification of series-parallel graphs is the following, </p><p>which is implicit in the work of Duffin. </p><p>THEOREM 3.2: An undirected graph, G@y,6), is series-parallel if and only if it possesses no subgraph homeomorphic from K4, the complete graph on four vertices. </p><p>PROOF: See [3]. Q.E.D. </p><p>The observant reader will note that for biconnected graphs, the jackknife operation of Theorem 3.1 can be eliminated and the equivalence of the two characterizations remains intact. </p><p>Also useful is the following result of Dirac, which we state as a corollary in view of the previous two theorems. </p><p>COROLLARY 3.3: A biconnected, simple graph, G(V,&amp;), in which every vertex possesses the degree of at least 3 has a subgraph homeomorphic from K4. </p><p>PROOF: See [2]. Q.E.D. </p><p>A key observation here is that the graphs obtained by repeated application of the reduction operations of Theorem 3.1 are well defined. This is easy to see in that premature stopping prior to the formation of a single edge yields a (reduced) graph satisfying the condition of the corollary. But this means that the original graph is not free of a K4 homeomorph and thus, from Theorem 3.2, could not have been series- parallel. </p><p>operation (i) (ii) </p><p>wL-a--wvirb- </p><p>(iii) (i) (iii) (ii) </p><p>Figure 2. Reduction operations of Theorem 3.1. </p></li><li><p>Richey , Parker, and Rardin: Spanning Eulerian Subgraphs 447 </p><p>The importance of series-parallel graphs from our perspective is that the SES problem is solvable on instances defined accordingly. It is worth noting that such graphs also provide resolvable cases for a host of other problems as well. The important reference in this regard is Takamizawa et al. [ 101 where it is shown that members of fairly broad, generic problem classes are solvable on series-parallel graphs. Other treatises can be found from works given in the reference list (e.g., [8], [9], and [ll]). </p><p>Following, we provide details of polynomial-time algorithms for each SES case (maximum and minimum). These have been separated in order to expose a particular refinement when the minimum version is related to hamiltonicity testing. </p><p>The Maximum SES Case </p><p>Throughout the reduction process defined by operations (i)-(iii) of Theorem 3.1, pseudoedges are formed. Moreover, each such edge represents a subgraph of the original graph. Let us denote by G,(D,,G,) the current reduced graph, and let Q[e] and &amp;[el be the vertex and edge sets, respectively, which correspond to e E 6,. Setsd[e] and &amp;[el define a specific subgraph, denoted simply as G[e]. Also, we call i a renninal of e if i E V[e] fl 13,. </p><p>The basic notion is straightforward. We simply keep track of SESs or potential SESs relative to subgraphs G[e]. This, in turn, requires that we correctly ascertain the interrelationships between these subgraphs vis-his their series, parallel, and jackknife combinations. The following lemma captures the idea. </p><p>LEMMA 3.4: If maximum edge cardinality subgraphs of G[e] are known relative to the following properties for every e E G,, then the maximum SES of G@,@ can be determined. </p><p>(i) A spanning Eulerian subgraph of G[e] denoted by SES. (ii) The vertex disjoint union of two closed walks, each beginning and ending </p><p>at only one terminal of e (neither passing through both terminals) and which together span 3 [ e ] , denoted by BI. </p><p>(iii) A (possibly nonsimple) path beginning at one terminal of e , ending at the other, and which spans D[e] denoted by PATH. </p><p>(iv),(v) A closed walk beginning and ending at the left (right) terminal of e, say iL( iR) , where iL(iR) is a cut-vertex in G,, denoted by W(RJ). </p><p>PROOF: Since, by definition, a complete solution must be spanning, so too must every corresponding subgraph of G[e] relative to V[e] (which includes the terminals of e). Now, if every vertex in such a (spanning) subgraph has even degree, then the subgraph can necessarily be expressed as the edge-disjoint union of cycles. Cases (i) and (ii) of the lemma capture the only possibilities in this regard. In (i), the subgraph is connected and is thus Eulerian, while in case (ii), it is not. If the subgraph is connected, spanning, and possesses two odd-degree vertices (the terminals of e), then it can be expressed as a (perhaps nonsimple) path beginning at one terminal of e and ending at the other. This semi-Edenan case is exhausted in (iii). Note that, here, we allow the possibility that eulericity results ultimately by augmentation with any non- G[e] path between the terminals of e. Finally, cases (iv) and (v) conform to the jackknife operation, where in general, any SES which includes a cut-vertex of G, must be </p></li><li><p>448 Naval Research Logistics Quarterly, Vol. 32 (1985) </p><p>expressable as a closed walk beginning and ending at the cut-vertex which is the terminal (left or right) of the respective e . </p><p>These are the only ways spanning Eulerian or semi-Edenan subgraphs can arise and the proof is complete. Q.E.D. </p><p>Actual computation regarding maximum values for each of the subgraphs detailed in Lemma 3.4 can be performed through (pseudo)-edge labels given by the 5-tuple, L[e] = (SES,BI,PATH,LJ,RJ) where l(e,*) denotes the value of * E {SES,BI,PATH,LJ,RJ}. Throughout these label updates, one may need to replace LJ by RJ, or vice versa, depending on the conventions adopted relative to leftness and rightness. Thus, for series reduction we have the following lemma. </p><p>LEMMA 3.5: If (pseudo)-edges e and f are series reduced, then for the resulting edge g, L[g] is determined as follows: </p><p>l(g,SES) = Z(e,SES) + l(f,SES) + l(e,RJ), l(g,BI) = max[l(e,SES) + l(f,BI),l(e,BI) + I(f,SES)] + Z(e,RJ), </p><p>l(g,LJ) = KeLJ), &amp;g,RJ) = U,RJ) . </p><p>I(g,PATH) = l(e,PATH) + Z(f,PATH) + I(e,RJ), </p><p>PROOF: First, observe that in the labels for SES, BI, and PATH the I(e,RJ) term can be replaced by Z(f,LJ). Unless stated otherwise, we will adopt the convention where RJ is employed when such a symmetry is apparent. To complete the proof, we can simply consider the following table: </p><p>f SES BI PATH </p><p>SES </p><p>e BI </p><p>PATH </p><p>Interpreting, an SES series reduced with an SES produces an SES. Alternately, a subgraph with property BI for edge e series reduced with an SES for f yields a BI, whereas an SES and a PATH are not even compatible due to parity differences in the vertex degrees. Other cells have similar interpretations which are left to the reader. Observe also that since the jackknife reduction describes interaction entirely through a single (cut) vertex, its update or contribution to one of the other updates is obvious and thus is not included in the table. </p><p>Now, from the table, we see that there is but one way to obtain an SES by series reduction. Adding the edge cardinality implied by RJ preserves eulericity , yielding I(g,SES). The two ways of obtaining BI subgraphs follow by symmetry. Thus, max- </p></li><li><p>Richey, Parker, and Rardin: Spanning Eulerian Subgraphs 449 </p><p>imizing over the two SES-BI cases and appending by RJ gives l(g,BI). Finally, g is a PATH only if e and f are. Hence, adding path values and RJ clearly produces I(g , PATH). Q . E . D . </p><p>A similar result regarding parallel reduction can also be given: </p><p>LEMMA 3.6: If (ps...</p></li></ul>