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Complexity and Approximation of Some Graph Modi Cation Problems Tel Aviv University The Raymond and Beverly Sackler Faculty of Exact Sciences Scho ol of Mathematical Sciences Complexity and Approximation of Some Graph Mo dication Problems Thesis submitted in partial fulllment of graduate requirements for the degree MSc in Tel Aviv University Department of Computer Science By Assaf Natanzon Prepared under the sup ervision of Prof Ron Shamir Acknowledgments Iwould like to thank my advisor Prof Ron Shamir for directing me and giving me a helping hand and for his clarifying comments and reformulations Iwould liketothankmy b eloved Mom and Dad for their supp ort Abstract In an edge mo dication problem one has to change the edge set of a given graph as little as p ossible so as to satisfy a certain prop ertyWe prove in this pap er the NP hardness of a variety of edge mo dication problems with resp ect to some wellstudied classes of graphs These include p erfect chordal comparability split threshold and cluster graphs Weshow that some of these problems b ecome p olynomial when the input graph has b ounded degree We give a general constant factor approximation algorithm for deletion and editing problems on b ounded degree graphs with resp ect to prop erties that can b e characterized by a nite set of forbidden induced subgraphs We giveanapproximation algorithm for Cluster Deletion and Editing We also prove some combinatorial b ounds on the sizes of the minimum completion and deletion sets of a graph Contents Intro duction and Summary Intro duction Denitions Previous results Summary of thesis results Outline of the thesis NP Hardness results Chain and Chordal Deletion Perfect Mo dications Perfect Completion Perfect Deletion Perfect Editing Comparability Editing Cluster Deletion Cograph Deletion and Completion Split Graphs Deletion and Completion Approximation Algorithms Mo dication problems on b ounded degree graphs for families charac terized by a nite set of forbidden induced subgraphs Edge Deletion approximation algorithm Edge Editing approximation algorithm Cluster Deletion and Editing Cluster Editing Cluster Deletion Polynomial Results On Bounded Degree Graphs Combinatorial b ounds on the size of minimum completion sets A concrete example for Chordal Completion Random graphs The minimum chordal completion of a random graph The minimum p erfect graph containing a random graph The minimum p erfect deletion in a random graph Chapter Intro duction and Summary Intro duction In this thesis we deal with graph mo dication problems Mo dication problems on graphs play an imp ortant role in computer science and have applications in several elds including computer algebra and molecular biology They include edge com pletion problems edge deletion problems edge editing problems and vertex deletion problems Wenow dene the generic problems we shall study in this thesis All problems are dened b elow with resp ect to a family of graphs F that is not a part of the input For example F can b e the set of chordal graphs or the p erfect graphs etc The Minimum Completion problem is dened as follows given an arbitrary graph G V E nd a set of nonedges F such that jF j is minimum and G V E F F ie nd a minimum set of edges whose addition to G will form a graph that b elongs to F For a graph G V E a kcompletion set for a graph family F or edges so that V E F F an F kcompletionisasetF of at most k The Minimum Deletion problem is dened as follows given an arbitrary graph G V E nd a set of edges H such that jH j is minimumand G V E n H F ie nd a minimum set of edges F whose removal from G will form a graph that b elongs to F For a graph G V E a kdeletion set for a graph family F or an F kdeletionisasetF of at most k edges so that V E n F F The Minimum Editing problem is dened as follows given an arbitrary graph G V E nd a set of nonedges F and a set of edges H such that jF H j is minimum and G V E F n H F ie nd a minimum set of edges you need to edit add or delete such that editing of these edges in the graph G will form a graph that b elongs to F For a graph G V E a kediting set for a graph family F or an F keditingisasetK F H of at most k edges so that V E F n H F The Minimum Vertex Deletion problem is dened as follows given an arbitrary graph G V E nd a set of vertices F such that jF j is minimum and the induced subgraph G V n F E V n F V n F b elongs to F V nF In the Maximum Induced Subgraph Problem one wishes to maximize jV n F j rather then to minimize jF j The optim um solutions for b oth problems give the same graph but they dier when approximation is sought In the sequel we shall abbreviate minimum completion problem for prop erty F to F Completion Names of other problems will b e abbreviated in the same manner We shall study for example Chordal Completion Perfect Editing and Chain Deletion The imp ortance of edge mo dication problems is b oth theoretical and practical From the theoretical p oint of view edge mo dication problems are natural problems in graph theory Moreover several fundamental problems in graph theory suchas maximum matching and maxcut can b e formulated as edge mo dication problems From the practical p oint of view edge mo dication problems have imp ortant applications Chordal Completion which is also called the minimum l lin problem arises in numerically p erforming a Gaussian elimination on a sparse symmetric matrix such that a minimum numb er of new nonzero elements is intro duced see Mo dication problems with resp ect to interval and prop er interval graphs have application in physical mapping of DNA Cluster graph mo dication problems arise in numerous application areas where clustering is called for see eg In general mo dication problems arise when the input data that is supp osed to b e represented byanF graph contains errors False negative errors corresp ond to missing edges that should b e added and false p ositive errors corresp ond to edges that must b e removed One seeks a solution minimizing the numb er of errors allowing one or b oth kinds of errors Denitions All graphs in this study are nite and without parallel edges and self lo ops Let G V E denote a graph For w V dene Adj w fxjw x E g Given a subset A V of the vertices we dene the subgraph induced byA tobe G A E A A where E fx y E jx y Ag A graph of G is any graph A E obtained by taking A V and E E If A sub A G is a subgraph of H then we shall also say that H is a sup ergraph of G The complement of G is the graph G V E E fx y V V jx y and x y E g A clique in a graph G is a complete induced subgraph K is a clique graph of size l l A clique is maximum if there is no clique of G of larger cardinality G is the number of vertices in a maximum clique of G wo of which A stable set or an independent set in a graph is a subset of vertices no t are adjacent A prop er ccoloring of a graph is a partition of its vertices V X X X c such that each X is an indep endent set The chromatic number of G denoted G i is the smallest p ossible c for which there exists a prop er ccoloring of G A graph G is bipartite if G has a coloring ie its vertices can b e partitioned into two indep endent sets Asequenceofvertices v v is called a path of length k ifv v E for k i i i k A path v v is called simple if v v for i j A simple path k i j v v iscalledchord less if v v E ji j j P denotes a chordless path k i j k on k vertices Asequenceofvertices v v is called a cycle of length l ifv v E l i i for i l and v v E A cycle v v is called simple if v v for i j l l i j A simple cycle v v is chord less if v v E ji j j or ji j j l C l i j k denotes a k vertex chordless cycle For a graph G and an integer l l G or lG is the graph formed by l indep endent copies of the graph G Let H b e a graph A graph G is called H free if G do es not contain any induced copyof H For example a graph is a clique i it is K freeFor a family F of graphs wesay that a graph G is Ffree if it is F free for every F F A family of graphs F is hereditary if for any graph G F every induced subgraph of G b elongs to F Agraph G V Eis perfect if G G for every A V A A A graph G V EisBerge if it contains no induced chordless cycle of o dd size and no induced complementofa chordless cycle of o dd size e will b e mainly interested in some graph sub classes of p erfect graphs classes or W in closely related classes For a survey on p erfect graphs and for more information ab out these prop erties see A bipartite graph G P Q E is called a chain graph if the vertices of one of the parts say P can b e ordered v v v so that Adj v Adj v Adj v k k recall that Adj v fxjv x E g This prop erty is called the chain property This prop erty is hereditaryasifweremovea vertex from P or Q the chain of containments do es not change For an example of a chain graph see Figure Figure a chain graph An undirected graph G is called chordal or triangulated if it contains no induced chordless cycle of length greater than three An orientation of an undirected graph G V E is an assignment of a single directed arc uv or vu to each undirected edge u v E The resulting directed graph V F is also called an orientation of G An undirected graph G V Eisa comparability graphortransitively
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