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Planar Branch Decompositions I: the Ratcatcher INFORMS Journal on Computing informs Vol. 17, No. 4, Fall 2005, pp. 402–412 ® issn 1091-9856 eissn 1526-5528 05 1704 0402 doi 10.1287/ijoc.1040.0075 © 2005 INFORMS Planar Branch Decompositions I: The Ratcatcher Illya V. Hicks Department of Industrial Engineering, Texas A&M University, College Station, Texas 77843-3131, USA, [email protected] he notion of branch decompositions and its related connectivity invariant for graphs, branchwidth, were Tintroduced by Robertson and Seymour in their series of papers that proved Wagner’s conjecture. Branch decompositions can be used to solve NP-hard problems modeled on graphs, but finding optimal branch decom- positions of graphs is also NP-hard. This is the first of two papers dealing with the relationship of branchwidth and planar graphs. A practical implementation of an algorithm of Seymour and Thomas for only computing the branchwidth (not optimal branch decomposition) of any planar hypergraph is proposed. This implementation is used in a practical implementation of an algorithm of Seymour and Thomas for computing the optimal branch decompositions for planar hypergraphs that is presented in the second paper. Since memory requirements can become an issue with this algorithm, two other variations of the algorithm to handle larger hypergraphs are also presented. Key words: planar graph; branchwidth; branch decomposition; carvingwidth History: Accepted by Jan Karel Lenstra, former Area Editor for Design and Analysis of Algorithms; received December 2002; revised August 2003, January 2004; accepted February 2004. 1. Introduction Seymour conceived of two new ways to decompose a A planar graph is a graph that canbe drawnona graph that were beneficial toward the proof. Decom- sphere or plane without having edges that cross. posing a graph by a branch decomposition is one of A subdivisionof a graph G is a graph obtained from these ideas. G by replacing its edges by internally vertex disjoint Branch decompositions open algorithmic possibil- paths. Inthe 1930s, Kuratowski (1930) proved that a ities for intractable problems that can be modeled graph G is planar if and only if G does not contain a ongraphs. Arnborg et al. (1991) showed that sev- eral NP-complete problems canbe solved inpoly- subdivisionof K5 or K3 3. Wagner (1937) later proved that a graph G is planar if and only if G does not con- nomial time using dynamic-programming techniques on input graphs with bounded treewidth, another tain K5 or K3 3 as a minor of G. A graph H is a minor of a graph G if H canbe obtainedfrom a subgraph related connectivity invariant for graphs introduced by RobertsonandSeymour (1983). Actually, the work of G by contracting edges. Let be a class of graphs. of Arnborg et al. was spurred from the work of Cour- is minor closed when all the minors of any member celle (1990). Inaddition,similar results have been of also belong to . Givena minorclosed class of obtained by Borie et al. (1992). The Arnborg et al. graphs , the obstruction set of is the set of minor result is also equivalent to graphs with bounded minimal graphs that are not elements of . Clearly, branchwidth since the branchwidth and treewidth of a anyclass of graphs embeddable ona givensurface graph bound each other by constants (Robertson and is a minor closed class. The question of characteriz- Seymour 1991). Inaddition,there is a plethora of the- ing the obstruction set for other surfaces remained ory about branchwidth and branch decompositions. openuntil1979–1980 whenArchdeacon(1980) and For example, branchwidth and branch decomposi- Glover et al. (1979) solved the case for the projec- tionshave beenshownuseful inproducingmatroid tive plane where they proved that there are 35 minor analogs of the graph minors theorem and shorter minimal “nonprojective-planar” graphs. In the 1930s, proofs of the graph minors theorem related to graphs Erdös posed the questionof whether the obstruc- with bounded branchwidth/treewidth (Geelen et al. tionset for a givensurface is finite.Archdeaconand 2002). Also, Seymour and Thomas (1994) proposed Huneke (1989) proved that the obstruction set for a polynomial-time algorithm to compute the branch- any nonorientable surface is finite, and Robertson and width for planar graphs, which will be referred to as Seymour (1985) proved the case for any surface as a the ratcatcher method. corollary of the graph minors theorem, which proved Despite the number of theoretical results about Wagner’s conjecture, every minor closed class has branch decompositions and branch decomposition- a finiteobstructionset. Inaddition,Robertsonand based algorithms for solving NP-hard problems, there 402 Hicks: Planar Branch Decompositions I: The Ratcatcher INFORMS Journal on Computing 17(4), pp. 402–412, © 2005 INFORMS 403 has beenlittle effort to develop practical algorithms. for the details of the algorithms with a proof of cor- Examples of practical branch decomposition-based rectness for the additional algorithms. Finally, §§5 algorithms are the work of Cook and Seymour (2003) and 6 contain computational results and conclusions, onthe travelingsalesmanproblem (TSP) andthe work respectively. The reader is referred to West (2001) for of Hicks (2004a) on the minor containment problem. background material on graph theory. One is also referred to the work of Christian (2003) for other examples of branch decomposition-based algo- 2. Preliminaries rithms. Similar results for tree decomposition-based algorithms have beengivenby Koster et al. (2002) 2.1. Branch Decompositions for their work onpartial constrainsatisfactionprob- Throughout this paper, a graph is only considered to lems, and Alber and Neidermeier (2002) for their be undirected and may have loops and multiple edges work ondominationproblems. These examples exem- unless stated otherwise. Also, we may assume that all plify the possible usefulness of branch decomposition graphs are biconnected. For instance, for a graph G, and tree decomposition-based algorithms for other onecanderive anoptimal branchdecompositionfor NP-hard problems. G from the optimal branch decompositions of G’s Seymour and Thomas (1994) proved that testing connected components. In addition, for a connected graph G , onecanderive anoptimal branchdecompo- whether a general graph has branchwidth at most k, o sitionfor G from the optimal branch decompositions some input integer, is NP-complete. However, the o of the biconnected components of G . aforementioned ratcatcher method is a polynomial- o A hypergraph H consists of a finite set VH of time algorithm used to compute the branchwidth nodes, a finite set EH of edges, and an incidence of planargraphs. Givenaninteger k, the ratcatcher relationbetweenthem that is notrestricted to two method tests whether the branchwidth of the graph is ends for each edge. Thus, hypergraphs are general- at least k by finding a structure in the graph that pro- izations of graphs where edges can have any num- hibits finding a branch decomposition of the graph ber of ends. Since this paper is about planar graphs with width at most k−1. Seymour and Thomas (1994) and planar hypergraphs, let us define planarity for also proposed a polynomial-time algorithm using the hypergraphs. Let H be a hypergraph; then IH , the ratcatcher method to compute optimal branch decom- incidence graph of H, is the simple bipartite graph positions for planar graphs, which will be referred with vertex set VH ∪ EH such that v ∈ VH is to as the edge-contraction method. Thus, the current adjacent to e ∈ EH if and only if v is anendof e knowledge base to compute optimal branch decom- in H. Seymour and Thomas (1994) define a hyper- positions for planar graphs is a two-stage process. The graph H as being planar if and only if IH is planar. two-stage process involves first finding the branch- In addition, we will denote G∗ as a planar dual of an width of the input planar graph using the ratcatcher embedding of the graph G inthe sphere or plane.Fur- method, then using the edge-contraction method in thermore, for a planar graph G drawninthe sphere, conjunction with the ratcatcher method to examine RG will denote the set of regions of G. planar hypergraphs needed to construct the optimal Let us now define branch decompositions and branch decomposition for the input planar graph. branchwidth of a graph. Let G be a graph with node A more detailed discussionof the implementationof set VG and edge set EG . Let T be a tree having the edge-contractionmethodwith the additionof a EG leaves in which every nonleaf node has degree heuristic, the cycle method, for runtime speedup is three. Let be a bijectionbetweenthe edges of G presented in the related paper (Hicks 2005). and the leaves of T . The pair T is called a branch This paper is spurred from the Seymour and decomposition of G. Notice that removing an edge e Thomas (1994) paper; a practical implementation of of T partitions the edges of G into two subsets Ae the ratcatcher method is presented. Unfortunately, the and Be. The middle set of e, denoted by mide , is the set algorithm requires a significant amount of memory to VAe ∩ VBe . The width of a branch decomposition achieve the proposed complexity of Oe2 where e is T is the maximum cardinality of the middle sets the number of edges in the input graph. This mem- over all edges in T . The branchwidth of G, denoted by ory requirement limits the size of the input graphs for G , is the minimum width over all branch decom- this algorithm (up to 5,000 edges). Thus, two addi- positions of G. A branch decomposition of G is opti- tional methods, the comprat method and the memrat mal if its width is equal to the branchwidth of G.If method, with complexity Oe3 , are presented to han- there exists a nonleaf node in T with degree greater dle medium (up to 20,000 edges) and large (up to thanthree, thenthe pair T is called a partial branch 70,000 edges) graphs respectively.
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