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Branch Decompositions and Their Applications RICE UNIVERSITY Branch Decomp ositions and their Applications by Illya V Hicks A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree Do ctor of Philosophy Approved Thesis Committee William J Co ok Chairman Noah G Harding Professor of Computational and Applied Mathematics Nathaniel Dean Asso ciate Professor of Computational and Applied Mathematics Alan Cox Asso ciate Professor of Computer Science Da vid Applegate Asso ciate Professor of Computational and Applied Mathematics Houston Texas April Abstract Branch Decomp ositions and their Applications by Illya V Hicks Many reallife problems can b e mo deled as optimization or decision problems on graphs Also many of those reallife problems are NPhard One traditional metho d to solve these problems is by branch and b ound while another metho d is by graph decomp ositions In the s Rob ertson and Seymour conceived of two new ways to decomp ose the graph in order to solve these problems These ingenious ideas were only byproducts of their work proving Wagners Conjecture A branch decomp osition is one of these ideas A pap er by Arnborg Lagergren and Seese showed that many NPcomplete problems can b e solved in p olynomial time using divide and conquer techniques on input graphs with b ounded branchwidth but a pap er by Seymour and Thomas proved that computing an optimal branch decomp osition is also NPcomplete Although computing optimal branch decomp ositions is NPcomplete there is a plethora of theory ab out branchwidth and branch decomp ositions For example a pap er by Seymour and Thomas oered a p olynomial time algorithm to compute the branchwidth and optimal branch decomp osition for planar graphs This do ctoral research is concentrated on constructing branch decomp ositions for graphs and using branch decomp ositions to solve NPcomplete problems mo deled on graphs In particular a heuristic to compute nearoptimal branch decomp ositions is presented iii and the heuristic is compared to previous heuristics in the sub ject Furthermore a practical implementation of an algorithm given in a pap er by Seymour and Thomas for computing optimal branch decomp ositions of planar graphs is implemented with the addition of heuristics to give the algorithm a divide and conquer design In addition this work includes a theoretical result relating the branchwidth of planar graphs to their duals characterizations of branchwidth for Halin and chordal graphs Also this work presents an algorithm for minor containment using a branch decomp osition and a parallel implementation of the heuristic for general graphs using pthreads Acknowledgments First and foremost I would like to thank Go d for blessing me to b e in this situation He has b een there with me from Waco through SWT and to Rice and I pray that he will continue to bless and strengthen me SecondlyIwould liketo thank mywife Casmin for her patience and endurance I would also like to thank my parents Mr and Mrs Lewis and Lorine Hicks my brother Sedrick Hicks and his family and the rest of my extended family and friends for their continued supp ort Next I would like to thank Dr Co ok and Dr Tapia for mentoring me through these graduate years Your advice and consultation have b een valuable FinallyI would like to thank my committee members and the rest of the departmentthat gave me a family atmosphere for these past years Iwould also like to take the time to acknowledge my grandparents who have passed away b efore they could see me earn a do ctorate degree I would like to dedicate this thesis to them b ecause I stand on their shoulders and hop efully my descendants will stand on mine So I dedicate this thesis to Mr and Mrs Willie and Isab ella Hicks and Rev and Mrs Andrew and Gilb erta Go o den Other p eople that have passed awaythatIwould also liketo acknowledge are Mrs Dorothy Collins Mrs Johnetta Willie Mrs Ruth Evans Mrs Shirley Eldridge and Mrs Lonnie B Ho dges Contents Abstract ii Acknowledgments iv List of Illustrations vii List of Tables ix Introduction Branchwidth Treewidth and other Denitions Branchwidth Branchwidth and The Petersen Graph Treewidth Finding Tree Decomp ositions The Graph Minors Pro ject Motivation Finding Branch Decomp ositions Tree Building Eigenvector Metho d Diameter Metho d Test Instances Computational Results vi Planar Branch Decomp ositions Ratcatcher Metho d Carvingwidth Antipo dality The Algorithm CycleMethod Edge Contraction Metho d Computational Results Planar Graphs and Their Duals Characterizing Branchwidth Halin Branchwidth Chordal Branchwidth Branch Decomp osition Applications Minimum Fillin Minor Containment Edge Discrimination Maximum Clique Computational Results Parallel Branch Decomp ositions Conclusions and Future Work Bibliography Illustrations Example graph Optimal branch decomp osition of width for Figure Obstruction set for graphs with branchwidth at most Q Tangle of order for Figure The Petersen graph and an optimal branch decomp osition Tangle of order for the Petersen graph Maximal prop er minors of the Petersen graph Optimal branch decomp osition for pm Optimal tree decomp osition for Figure Obstruction set for graphs with treewidth at most Optimal branch decomp osition for Figure The initial split Subsequent splits if jX j Subsequent splits if jX j Example G a Example G and G x y Planar Test Instances with their branchwidth Compiler Test Instances with their branchwidth Fillin Test Instances with their branchwidth viii Q from Chapter Medial graph of Figure Example hypergraph H v Medial graph of Figure Corresp onding dual graph of Figure Resulting hypergraphs from using path P Some Test Instances with their corresp onding branchwidth Some Test Instances with their corresp onding branchwidth Some Test Instances with their corresp onding branchwidth M has branchwidth The branchwidthofHalingraphsis H H The minor set for a leaf Joining minor set graphs for a nonleaf no de p Edge discrimination The maximum clique in the conict graph of this minor set graph is Do es tele contain the Petersen graph P as a minor The toroidal grid C x C rkfs has branchwidth Tables Planar graphs Some more planar graphs Compiler graphs More compiler graphs More compiler graphs More compiler graphs Minimum Fillin graphs Cycle Metho d versus Edge Contraction Metho d Cycle Metho d Results Classes of Graphs and their branchwidth Minimum Fillin Results Minor Containment Results Minor Containment Results Parallel implementation for planar graphs Parallel implementation for more planar graphs Chapter Intro duction Many reallife problems can b e mo deled as optimization or decision problems on graphs Consider for instance the problem in whicha spy has to visit a number of cities and return to home base The spy can not revisit a city in fear of capture and b ecause of budget cuts the spy has to follow the shortest route visiting all of the sp ecied cities and returning to home base The problem is formally known as the traveling salesman problem The input can b e mo deled as a weighted graph in which the no des represent the dierent cities and home base and an edge b etween twonodes represents the path b etween the corresp onding cities and has weight that corresp onds to the cost of the spytraveling on that particular route The problem is to nd a circuit in the graph whichcontains all no des and has minimum total weight Unfortunately many graph problems like the one mentioned ab ove are hard in the sense that there are probably no ecient algorithms to solve them One wayto overcome this disadvantage is by decomp osing the graph into subgraphs such that the structure of the input graph would assure that the problem on the subgraphs are easy to solve Such a decomp osition structure might help in nding a more ecient algorithm that computes an optimal solution or a go o d approximate solution for the complete problem One suitable structure is the tree structure b ecause many graph problems that are hard in general are eciently solvable on trees One type of decomp osition that has a tree structure is the branch decomp osition Branch decomp ositions were introduced in Rob ertson and Seymour as part of their Graph Minors Pro ject a series of pap ers containing the pro of of Wagners conjec ture Branch decomp ositions op en algorithmic p ossibilities for intractable
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