Spanheight, A Natural Extension of Bandwidth and Treedepth

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1 Master s Thesis Spanheight, A Natural Extension of Banwith an Treeepth Author: N. van Roen Supervisor: Prof. r. Hans. L. Bolaener A thesis sumitte in fulfilment of the requirements for the egree of Master of Science in the Faculty of Science Department of Information an Computing Sciences ICA Octoer, 2015

2 Astract In graph theory, the anwith prolem has a long history, an a numer of practical applications. Fomin, Heggernes, an Telle [20] introuce treespan, an extension from anwith to epth first search spanning trees in the context of the occupancy measure in search games. It is the equivalent of a tree-ecomposition, where ajacent ags ifferentiate with at most 1 vertex, an each vertex is only allowe to appear in at most k ags. Dregi [16] explore parameterize algorithms for treespan uner the name ajacencyspan. In this thesis, we efine spanheight as a ifferent extension from anwith to epth first search spanning trees. A spanheight-ecomposition is a DFS spanning tree, in which every pair of ajacent vertices are connecte with a path of length at most k + 2. It is the equivalent of a tree-ecomposition, where ajacent ags ifferentiate with at most 1 vertex, an the su-tree inuce y ags containing a vertex v has height at most k + 2. We proof spanheight to e NP-Complete. We introuce a single exponential algorithm for k-spanheight using O(n 2 9 n ) time an O(n 2 2 n ) space. This algorithm is ase on the ucketassignment technique from Feige [17]. Attempting to fin an FPT algorithm, we argue that the graph minor theorem an Courcelle s theorem cannot e use to proof the FPT memership of spanheight. Instea we proof it to e FPT when restricte to graphs of oune treeepth. Finally we present an O(nt 4 2 7t3 2 t log t ) time an O(n 2 3t3 2 t log t ) space FPT algorithm y parameterizing on the treeepth t of a graph. A secon prolem we stuy is restricte-spanheight, which is a special case of spanheight where the DFS spanning tree is restricte to eges from the input graph. For restrictespanheight we provie similar results to spanheight. As a sie result we look at reconfiguration of DFS spanning trees. The results in this thesis are mainly of theoretical significance, ecause the presente algorithms are very slow. i

3 Acknowlegements I woul like to thank Hans Bolaener for introucing me to the fiel of exact an parameterize algorithms. For suggesting spanheight as a topic for this thesis, an helping me fin great literature. For the countless iscussions an comments on early versions of algorithms. I want to thank all memers of the treewith stuy group for the TACO ay, an the weekly talks on complexity proofs, FPT algorithms, an other topics like measure an conquer. Thank you for listening to my presentation on an early version of this thesis, an giving valuale comments. Finally, I woul like to thank my parents Eugène an Mirana, for everything. ii

4 Contents Acknowlegements ii 1 Introuction Organization of this thesis Preliminaries Definitions Depth first search spanning tree Treewith Treeepth Banwith DFS Spanning Tree Reconfiguration 9 4 Properties of Spanheight Relation to anwith Relation to treewith Relation to treeepth NP-Completeness 14 6 Exact Algorithms Spanheight algorithm Restricte-spanheight algorithm The Graph Minor Theorem 24 8 Courcelle s Theorem Restricte-treeepth Restricte-spanheight Treeepth Spanheight Restricte-spanheight FPT Algorithm Main algorithm Operations Correctness of Algorithm Running time Spanheight FPT algorithm Main algorithm Algorithm operations Correctness of Algorithm Running time Conclusion 51 iii

5 Chapter 1 Introuction Fomin, Heggernes, an Telle [20] introuce the graph property treespan. Their motivation comes from graph ase search games, which strongly resemle well known graph prolems. A search game consist of a single fugitive, an multiple cops. The fugitive is invisile, an is caught when a cop stans on the same vertex of the graph, an the fugitive has no vertex to escape to. In a stanar search the cops move much slower than the fugitive. The prolem of fining the minimum numer of cops for which there exist a guarantee strategy to catch the fugitive is the equivalent of fining a minimal path-ecomposition of the graph. This is trivially true when the cops start searching at a vertex in the root ag, an walk to a vertex in the chil ag, such that every vertex in the chil ag is occupie y a cop. They continue this process until all ags are searche. The fugitive cannot reach the upper sie of the path-ecomposition without walking on the same vertex as a cop, y the efinition of a path-ecomposition. In a special search version calle Inert or Lazy search, the fugitive is only allowe to move just efore the cops are going to walk to his vertex. This version enales the graph to e ecompose into a tree structure, ecause ranches can e searche inepenently. The minimum numer of cops for this type of prolem is equal to the treewith of a graph. A ifferent optimization ojective, minimizes the numer of turns a cop spens on each iniviual vertex. This is calle the occupation time of a vertex, an the prolem is equivalent to the anwith prolem. The anwith prolem is equivalent to fining an optimal pathecomposition with the following properties: The numer of ags that contain a vertex v, is oune y the anwith For each pair of ajacent ags, exactly one vertex is introuce an/or forgotten, such that X 1 \ X 2 = 1. The main interest of Fomin et al was to fin the graph property that is equivalent to a Lazy Search version of the anwith prolem, or equivalently, a Lazy Search minimizing the Occupation time. This prolem is an extension of anwith to trees, an equivalent to fining an optimal tree-ecomposition with the following properties: The numer of ags that contain a vertex v, is oune y the treespan k For each pair of ajacent ags, exactly one vertex is introuce an/or forgotten, such that X 1 \ X 2 = 1. They call this search prolem treespan. Numer of Searchers Occupation Time Stanar Search pathwith anwith Lazy Search treewith treespan Fomin et al. also proof certain properties for treespan of specific graph classes, an show that treespan is NP-Complete on coipartite graphs. Rautenach [25] presents lower ouns for treespan ase on the chromatic numer, connectivity numer an the ratio etween eges an vertices of a graph. In [16], Dregi efines ajacencyspan, as the structural graph prolem equivalent to the graph search prolem treespan. The equivalence follows from the fact that oth prolems are equivalent to the same restricte type of tree-ecomposition of a graph. 1

6 Chapter 1. Introuction 2 a c a c Figure 1.1: A mapping of a graph G to the interval [1..4] with anwith 3. The grey ege enotes that {a, } are ajacent in G. Figure 1.2: A DFS spanning tree of Figure 1.1, where the vertex is place in its own ranch, reucing the istances etween ajacent vertices. The grey eges are ack-eges. In this thesis, we have a ifferent viewpoint on anwith, an a ifferent extension to trees. A solution to the anwith prolem is a mapping of the vertices v V to the interval [1..n]. Such that the istance etween any pair of ajacent vertices, in this interval, is at most (see Figure 1.1). This interval can e seen as a DFS spanning tree consisting of a path, an we formulate the following question. Can we create ranches on this path, to ecrease the anwith? We formally efine this as spanheight: Spanheight Instance: An unirecte connecte graph G = (V, E), an a positive integer k. Question: Can we a eges to E an get the graph H = (V, E ), where H is calle a supergraph of G. Does there exist a DFS spanning tree T = (V, F, r) of H in which every ack-ege in E \ F spans at most k vertices? For the anwith solution in Figure 1.1, it is shown in Figure 1.2, that allowing ranches will ecrease the maximum istance etween the ajacent vertices. Therefore, we elieve that spanheight is a natural extension of anwith to trees. The spanheight prolem has a relation to tree-ecompositions. In fact the spanheight prolem is equivalent to fining an optimal treeecomposition with the following properties: The su-tree, inuce y the ags containing v, has height oune y the spanheight k. For each pair of ajacent ags, exactly one vertex is introuce an/or forgotten, such that X 1 \ X 2 = 1. These efinitions allows us to compare treespan an spanheight. It is trivial that a solution for treespan is also a solution for spanheight, an therefore treespan upper ouns spanheight. Spanheight strongly resemles the treeepth of a graph, which also searches DFS spanning trees. The ojective function of treeepth is to minimize the istance etween the root an every leaf vertex. Therefore, oth prolems minimize the istance etween pairs of vertices on epth first search spanning trees. However, spanheight is much harer ecause its ojective function is local to every vertex. Our motivation to research spanheight comes from the oservation that spanheight seems to e an extension from treeepth to something closer to treewith. An since oth treeepth an treewith have interesting practical applications, maye spanheight will as well. The research ojective of this thesis is to fin fast algorithms for spanheight. In this search we will e using the anwith, treewith an treeepth relations to spanheight. A secon prolem we stuy is restricte-spanheight, which is a special case of spanheight where the epth first search spanning tree is restricte to eges from the input graph. For restricte-spanheight we provie similar results to spanheight. A goo reason for stuying oth prolems is to stuy the effect of this restriction on the complexity. Restricte-spanheight Instance: An unirecte connecte graph G = (V, E), an a positive integer k. Question: Does there exist a DFS spanning tree T = (V, F ) of G in which every ack-ege in E \ F spans at most k vertices?

7 Chapter 1. Introuction 3 The complexity class Fixe Parameter Tractaility (areviate FPT) is concerne with fining faster algorithms for NP-Complete prolems y solving the prolem for a restricte set of instances. This restricte set of instances must amit a certain property, which is calle the parameter. The concept is that we esign an algorithm, that is exponential in the size of the parameter, an polynomial in the length of the input. This gives a running time of O(poly(n) f(k)), where f(k) is allowe to e an exponential function in the parameter k. The function f(k) is often in the form of 2 k, ut larger exponential functions are allowe. Transferring the exponential epenency from 2 n to 2 k is eneficial when k is much smaller. A requirement of FPT algorithms is that k is a constant an inepenent of n. Because k is a small constant, the running time of an FPT algorithm is polynomial with a hien constant (exponential) factor. Kernelization is a metho to reuce the size of the input from n to O(poly(k)) for some parameter k. This effectively creates an FPT algorithm if this reuction takes polynomial time (See [15], chapter 4-5). Assuming the AND-istillation conjecture hols, there oes not exist a polynomial kernel for the graph property treewith. The similarity etween treewith an spanheight makes us assume that same type of argument is applicale to spanheight, an we will not consier Kernelization in this thesis. However, in [6] treewith is shown to have a kernel on graphs of oune vertex cover numer, an this approach may e use to create a kernel for spanheight as well. Dynamic programming on a tree-ecomposition of the input graph is another well researche metho to create FPT algorithms (See [15], chapter 10-12,15). The enefit of this approach is that it works for a lot of prolems. For example, the prolems Inepenent set an Hamiltonian cycle are uner strong assumptions not in FPT, ut the versions restricte to graphs of oune treewith are in FPT. For most local graph prolems there oes not seem to exist a trivial single exponential time 2 k FPT algorithm on treeecompositions. Cygan et al. [14] show that a small group of locality prolems o amit single exponential proailistic algorithms. In [9, 7] Boleaner et al. show that these prolems also amit single exponential eterministic algorithms using representative sets. Uner the Strong Exponential Time Hypothesis assumption, we can fin lower ouns on the minimum complexity of FPT algorithms parameterize y treewith (see [23]). These lower ouns are useful to etermine of known algorithms are optimal. Dynamic programming on a tree-ecompositions requires computing a tree-ecomposition eforehan. Computing a tree-ecomposition of oune treewith can e one in linear time Bolaener [3]. The tutorial [4] y the same author summarizes upper ouns an lower ouns etween treewith an relate prolems, an lists results on special graph classes. There are a numer of FPT algorithms on tree-ecompositions using epth first search. In [5] the epth first search algorithm is use to etect circus graph minors, creating an FPT algorithm for a special case of graph minor tests. [15, chapter 15] expans on this, an shows how to use epth first search as a parameterize algorithm esign tool using the Plehn-Voigt Theorem. The DFS spanning tree prolem treeepth amits an O (2 t2 ) time algorithm on a tree-ecomposition [26]. There are many other graph properties that can e use as a FPT algorithm esign tools. For the anwith prolem, the only known FPT algorithm parameterizes y the vertex cover numer s of the graph (Fellows et al. [18]). When s is small, than all other O(n) vertices of the graph can e categorize into at 2 s categories. This is than use to create an algorithm that focuses on the numer of vertices per category, instea of each vertex iniviually. In [16, chapter 7.2], Dregi introuces an FPT algorithm for treespan with a running time of O(s 2+s 4 s n+(s2 s ) 2.5(s+1)2s +o(s2 s) ), where s enotes the size of a vertex cover of the graph. This algorithm can e trivially change to output a solution for the spanheight prolem, an therefore we informally claim that spanheight is FPT for graphs of oune vertex cover numer. Their algorithm is ase on the similar algorithm for anwith from [18]. It enumerates all structures on the vertex cover, an inserts the other categorize vertices using an FPT version of the Integer Linear Programming prolem, oune y the numer of categories 2 s. The completeness an correctness of their algorithm is questionale, ecause they only give a rough sketch of the ILP moel. However, eing familiar with the original FPT algorithm for anwith, we are ale to recognize that a more refine ILP moel oes exist. The complexity analyses for this algorithm contains small mistakes. Most notaly, the algorithm efines O(s 2 s+1 ) zones, an every zone is assigne one of the O(2 s ) vertex categories. The algorithm must exhaustively enumerate

8 Chapter 1. Introuction 4 all possile cominations, giving a total of O((2 s ) s 2s+1 ) iterations. However, their analyses incorrectly states that they enumerate at most O(s4 s ) ifferent cominations, possily ecause they thought the numer of cominations was multiplicative. In chapter 5 of the same paper, Dregi present an XP algorithm, y oserving that graphs of oune treespan, have oune egree of 2k. Note that this egree oun oes not translate to spanheight. In chapter 7.1, a correct O (s n ) algorithm is given for treespan. Then incorrectly, they ofuscate this into a FPT algorithm for treespan restricte to graphs of oune treespan k an oune vertex cover numer s. Their metho consist of using the egree oun to rewrite s n to the equivalent running time s s2k ecause s2k n. 1.1 Organization of this thesis In Chapter 3 we look at the reconfiguration on DFS spanning trees, which seems to e useful for heuristic purposes an not for exact algorithms for spanheight. In Chapter 4 the relationship etween spanheight an treeepth, treewith an anwith is explore. In Chapter 5 we proof NP-Completeness for oth versions of spanheight with a reuction from treeepth. Then we focus on fining fast exact algorithms. First a single exponential algorithm for graphs of oune spanheight is given in Chapter 6. This algorithm is strongly ase on the ucke-assignment algorithm for anwith. The same algorithm is then aapte to solve restricte-spanheight. In orer to get an exact su-exponential algorithm we proof that our prolems are Fixe Parameter Tractale. Chapter 7 an 8 look at meta-theorems for proving memership in FPT. First we consier the graph minor theorem an proof that it oes not hol for spanheight. Then in Chapter 8 we use Courcelle s theorem to fin FPT instances of oth prolems. Chapter 9 presents a linear time FPT algorithm for restricte-spanheight on graphs of oune restrictespanheight. Chapter 10 is aout a linear time FPT algorithm for spanheight on graphs of oune treeepth.

9 Chapter 2 Preliminaries 2.1 Definitions (Graphs). We will use G = (V, E) to efine an unirecte graph. We will use n to enote the size of the set of vertices V = n. The graph G is allowe to e isconnecte, meaning that it consist of multiple isconnecte components, unless it is specifically state that G is connecte. The set V contains all vertices of G. The set E contains all unirecte eges {v, w} E of G etween any pair of vertices v V an w V such that v w. (Ege an vertex sets). We will often use E(G) an V (G) to reference to the ege an vertex set of a graph G = (V, E). The notation is also use for paths an a trees. This limits the numer of variales we have to efine an the reaer has to rememer. (Path). A path P etween a vertex v an w is an orere sequence of vertices incluing the enpoints. The length of a path can e expresse as P an equals the numer of vertices on this path incluing the enpoints. When the path etween two points is exclusive, the enpoints are not on the path. (Trees an roote trees). Unroote trees are acyclic graphs an are efine y the tuple T = (V, F ), where V is a set of vertices, F is a set of tree-eges. In this thesis we will mostly use roote trees unless specifie otherwise. Let T = (V, F, r) e a roote tree, where r V is the root vertex. (Tree-Path). Let T = (V, F ) e a tree, a tree-path P is a path etween using vertices V (P ) V (T ) that uses only tree-eges E(P ) E(T ). (Height of a roote tree). The height of a vertex v in a roote tree is the maximum length of a path from v to a leaf vertex. Therefore, the height of a leaf equals 1. The height of a roote tree equals the height of the root vertex. ([property-] ecomposition). We use "ecomposition" to enote a structure create from a graph G, such this structure amits to the specifications of the given graph property. For example: A tree-ecomposition is a ecomposition of a graph into a tree structure of treewith at most tw. A path-ecomposition is a ecomposition of a graph into a path of pathwith at most tw. A anwith-ecomposition is a ecomposition of a graph into an interval graph of anwith at most. A spanheight-ecomposition is a ecomposition of a supergraph into a epth first search spanning tree of a spanheight at most k. A restricte-spanheight-ecomposition is a ecomposition of a graph into a epth first search spanning tree of a spanheight at most k. 5

10 Chapter 2. Preliminaries Depth first search spanning tree Let G = (V, E) e an unirecte an connecte graph where V is a set of n vertices an E is a set of unirecte eges. An example of such a graph G is given elow in Figure 2.1. A roote spanning tree of a graph G is a roote tree that contains all the vertices from the graph an a suset of the eges. Let T = (V, F ) e a roote spanning tree of the graph G using all vertices in V an let F E e a suset of the eges of the original graph G as isplaye in Figure 2.2. c a e Figure 2.1: An example of a graph G = (V, E). Figure 2.2: An example of a roote spanning tree of G. a e c Figure 2.3: The same spanning tree, with the non-tree eges ae in a grey color. a e c A DFS spanning tree is a special type of spanning tree in which we classify the non-tree eges, which are eges from the original graph G, that are not inclue in the spanning tree itself. We classify each non-tree ege as either a ack-ege or a cross-ege. In Figure 2.2 we have not isplaye the non-tree eges, ut in Figure 2.2 we ae all the non-tree eges of the graph G using grey colore eges. For the non-tree ege {a, }, the vertices a is a escenant of, we call this non-tree ege a ack-ege going ack up in the tree. The non-tree ege {c, e} is ifferent, the vertex c is not an ancestor of the vertex of e ecause it is containe in a ifferent ranch of the spanning tree. This type of non-tree ege is calle a cross-ege ecause it crosses etween tree-ranches. Of these two types of non-tree eges, a DFS spanning tree foris the existence of cross-eges. As a consequence all non-tree eges must e ack-eges. The spanning tree from Figure 2.2 an 2.3 is therefore not a DFS spanning tree. A DFS spanning tree of the graph G is isplaye in Figure 2.4. Every graph of n vertices amits at least n ifferent DFS spanning trees y picking ifferent roots. Computing a DFS spanning tree of a graph can e one in O( V + E ) time y the famous epth first search algorithm that enumerates all vertices in a pre-fix manner. a e c Figure 2.4: A DFS spanning tree of the graph from Figure 2.1. For a ack-ege etween two vertices a an c. Let P path etween a an c. We say that vertices P \{a, c} are spanne y the ack-ege. We will also reference to these vertices as elow the ack-ege. DFS spanning trees have many applications, e.g. solving puzzles, fining the connecte components of a graph, an fining a path etween two vertices in a graph. There are also many algorithms that use the epth first search algorithm to solve har prolems heuristically, e.g. heuristics for the constraint satisfaction prolem, an for jo scheuling prolems.

11 Chapter 2. Preliminaries Treewith The graph property calle treewith measures how much a connecte unirecte graph resemles a tree-structure. It efines a tree-ecomposition as a mapping of an aritrary connecte (cyclic) graph G = (V, E) onto a special type of unroote tree ({X i i I}, T = (I, F )). The tree T is uil with numere ags, with numers i I. The set of eges etween the numere ags are represente as the set F. Each ag i I of the T contains a suset of vertices X i V. A tree-ecomposition has the following properties: i I X i = V, every vertex is containe in the tree-ecomposition. For every ege {v, w} E of the original graph, there exist a ag i I containing oth v an w, or equivalently ( v, w V )(({v, w} E) ( i I)(v, w X i )). For every vertex v V, the set of ags containing it {X i v X i, i I}, form a connecte su-tree of T. An example of a tree-ecomposion is given in Figure 2.5. The treewith of a tree-ecomposition is the size of the iggest ag minus 1 or equivalently max i I X i 1. The treewith of a graph epens on the ensity/sparsity, a single vertex has treewith 0, an a tree has treewith 1. a,, a,c, e c j,e,g c,h,j f g h i f,e,g h,i,j Figure 2.5: A graph an its tree-ecomposition of treewith 2. Deciing if there exist a tree ecomposition with tree-with at most k of a graph is NP- Complete [2], ut the prolem is shown to e in FPT an solvale in linear time an linear space ut with a constant exponential factor. A lot of NP-Har prolems have a lower orer of complexity when the input graph G is a tree structure. However, an algorithm that only solves a prolem for trees is not very useful in practice. Tree-ecompositions are more practical than general trees ecause it allows us to exploit treelike properties of cyclic graphs. A path-ecomposition is a special case of a tree-ecomposition, where no join noes are allowe. As a consequence, it will always e a path Nice tree-ecomposition The operations on a tree-ecomposition naturally ivie into four separate operations: leaf, introuce, forget, an join. Instea of etecting which operations apply to each ag of the treeecomposition we transform the tree-ecomposition in what is calle a nice tree-ecomposition. In a nice tree-ecomposition the ags have the following properties. Every ag has zero, one or two chilren. Leaf ags contain only a single vertex. A join ag X, an its two chilren X an X contain the exact same vertices X = X = X. An introuce ag X introucing the vertex v has a single chil X, where X \ {v} = X. A forget ag X forgetting the vertex v has a single chil X, where X = X \ {v}.

12 Chapter 2. Preliminaries Treeepth The graph property calle treeepth measures for a graph G what the smallest height of any DFS spanning tree is when we are allowe to create new eges. The ecision prolem is NP- Complete on ipartite an coipartite graphs [8]. Treeepth Instance: An unirecte connecte graph G = (V, E) an a positive integer k. Question: Can we a eges to E an get the graph H = (V, E ). Does there exist a DFS spanning tree T = (V, F ) of H with height at most k? Graphs of low treeepth look like stars, with the root vertex as the center. It has a numer of practical applications that have generate interest in the prolem. It has een introuce multiple times using ifferent names an efinitions: as minimum elimination tree y Pothen in 1988; as orere coloring y Katchalksi et al. in 1995; as vertex ranking y Bolaener et al. in 1998; as treeepth y Nešetřil an Ossona e Menez in It has some interesting relations to the other prolems in this thesis, it upper ouns oth the spanheight an the treewith of a graph (Chapter 4). We will create an FPT algorithm for spanheight y parameterizing on the treeepth of a graph. The version of treeepth without ege creation oes not have any pulishe research to our knowlege. Therefore, we introuce it in Chapter 5 as part of an NP-Completeness proof. 2.5 Banwith Banwith Instance: An unirecte connecte graph G = (V, E) an a positive integer. Question: Does there exist a function f that maps vertices v V to the interval [1..n], such that for every ege {v, w} E the istance f(v) f(w) etween the vertices is at most?

13 Chapter 3 DFS Spanning Tree Reconfiguration In our search for an algorithm for restricte-spanheight we iscovere an DFS spanning tree reconfiguration operator. While not use in any of our algorithms, it is an interesting sie result. The transformation operator has the aility to make small local changes to a DFS spanning trees. Furthermore, the operation can e use to turn any tree T into a DFS spanning tree with small local changes to remove cross-eges, such that the main structure of T can e maintaine. We will show that this operator can e use to create any DFS spanning tree of a graph in a polynomial numer of steps. Applications of our reconfiguration operator are heuristic an local search algorithms for prolems on DFS spanning trees. Graph transformation A graph transformation transforms the structure of a graph G into a graph H. There exists a goo summary on the topic y W. Goar an H. Smart in "Distances etween graphs uner ege operations", 1997 [22]. While we are intereste DFS spanning trees instea of graphs we will use similar notations an proofs. Suppose ε enotes a symmetric nonreflexive inary relation on the graphs. Then we say that graph G can e transforme into graph H in k steps y ε if there exists a sequence G = ε G 0, G 1, G 2,..., G k = H of graphs such that G i Gi+1 for 0 i k 1. The istance δ ε (G, H) etween G an H is the minimum value of k such that G can e transforme into H in k steps y ε, if such k exists; otherwise the istance is efine to e. Not all pairs of graphs have finite istance k for every function ε. For example, G cannot e transforme into H when they have a ifferent numer of vertices/eges an ε oes not have the aility to introuce new vertices/eges. Geometric non-crossing spanning tree transformation In geometric graphs there exists the prolem of transforming a spanning tree T with non-crossing eges (planar) to T. Noncrossing enotes that no pair of eges of the spanning tree cross each other given the euler coorinates of the vertices. The transformation uner the planer restriction allows any ege to e create etween any pair of vertices, which again is a little ifferent from our goals, ut the analyses of their planar restriction is interesting. An upper oun is given in "A quaratic istance oun on sliing etween crossing-free spanning trees" (2006) y O. Aichholzer an K. Reinhart [1]. The proof uses the ege operation calle "slie triangle" ST which moves the eges along a triangle on 3 points. This satisfies the non-crossing property when the triangle has no interior vertices. They then show that every non-crossing spanning tree can e transforme to a x-monotone non-crossing spanning tree in O(N 2 ) steps using ST, an the symmetrical property of ST allows for transformations in the general case. For ST they prove that this upper oun is tight y giving an example case that requires O(N 2 ) steps. The previous transformations o not apply to DFS spanning trees ecause they require that an ege can e create etween any pair of vertices. An in DFS spanning trees we can only use eges from the original graph. Furthermore, the previous transformations only affect the new an ol enpoints of a single ege; while an operation on a DFS spanning tree can effect all other vertices of the graph as well. 9

14 Chapter 3. DFS Spanning Tree Reconfiguration 10 Definition (Tree-ege introuction σ ). Let σ enote a symmetric nonreflexive inary relation on DFS spanning trees. The function σ takes an ack-ege {x, y} that spans the path P in the DFS spanning tree T an introuces it as a tree-ege in the new DFS spanning tree T. This creates a cycle in T, which we reak y setting the parent of every internal vertex of P to their chil in P in the new tree T. As a result of the tree-ege introuce, the vertices from P are suject to cross-eges with other vertices in T. Iteratively, pick the highest cross-ege from T, an introuce it using σ, until all cross-eges are remove from the tree. This process is illustrate in Figure 3.1. The propose metho preserves the ol su-tree as much as possile. Lemma 3.1. A tree-ege introuction σ can e compute in O(n 3 ) time. Proof. Let T = (V, E, r) e a tree with n vertices. Let a tree-ege introuction σ e performe on the cross-ege {v, w} where epth(v) = i an epth(w) = j. Let a e their lowest common ancestor vertex. Aing {v, w} as a tree-ege of T creates a cycle through a. Assume w.l.o.g. that the vertex v has a greater epth than w. The cycle is roken y making every vertex on the path P etween v an a an escenant of w. We make two oservations: There exist a vertex u on the path P with the same epth as w, an as a consequence of the tree-ege introuction, its epth increases. We conclue that there are at most O(n) vertices at epth j in T, an the numer of vertices at epth j ecreases y 1, after each cross-ege removal incient to a vertex at epth j. Only the vertices in the path P are move at each cross-ege removal. Any new cross-eges create as a sie effect, must e incient to P. An the vertices incient to these new cross-eges must have een escenants v. We conclue that new cross-eges are create etween only vertices of greater epth than j The algorithm iteratively removes cross-eges starting at epth 1, until the lowest leaf noe with epth at most O(n) is reache. At each epth i, the algorithm introuces cross-eges using the σ operation, until one of the following cases is reache: There is only 1 vertex at epth i left, an it cannot have cross-eges to vertices at epth i or lower. There are multiple vertices at epth i left, an they have no cross-eges to vertices at epth i or lower. Every epth takes at most O(n) steps, an it takes O(n) time to process a step, giving a O(n 3 ) algorithm. Our efinition of the transformation σ uses a top-own orering of the cross-eges to avoi re-introucing cross-eges after they have een remove. This allows us to oun the numer of steps y O(n 2 ) with O(n) work per step. Does this upper oun also hols for an aritrary a a c e f a e c f a e f c e f c (a) efore () iteration 1 (c) iteration 2 () en Figure 3.1: Introucing {a, } as a tree-ege with σ. After the first step, this creates crosseges, which are remove in a top own approach.

15 Chapter 3. DFS Spanning Tree Reconfiguration 11 orering of the cross-eges? We elieve this to e true, ecause for every aritrary tree T we teste, even an aritrary orering seems to remove all cross-eges in at most O(n 2 ) steps. We have unale to fin a formal proof to answer this question. A graph transformation is of limite usefulness when it can only search a suset of the search space of DFS spanning trees. The following theorem shows that our transformation can reach the complete search space. Theorem 3.2. The istance etween a pair of aritrary DFS spanning trees is oune y O(n) steps of the tree-ege introuce σ operation. Proof. Let T = (V, E, r) e the DFS spanning tree that we transform into the DFS spanning tree T = (V, E, r ). Iterate the vertices of T in prefix orering. For every vertex v V, let w V e the parent, an introuce the ege {v, w} in T as tree-ege using σ. Once w is a parent of v in T, then this remains this way, ecause new cross-eges are create only elow v. After all vertices are iterate, T is transforme into T, ecause the parent-chil relations are equivalent.

16 Chapter 4 Properties of Spanheight The spanheight prolem allows eges to e ae to the graph in orer to get a new graph with lower spanheight. In Figure 4.1 we illustrate a DFS spanning tree with spanheight 2, which trivially is optimal. Once we allow the ege {a, c} to e create, the spanheight of the new graph is reuce to 1 as illustrate in Figure 4.2. a c Figure 4.1: An optimal DFS spanning tree without ege creation. a c Figure 4.2: A DFS spanning tree with the create ege etween a an c of the graph from Figure 4.1. Any solution for restricte-spanheight is also a solution to spanheight, an therefore the spanheight of a graph upper ouns a restricte-spanheight of the same graph. In the introuction it was also shown that treespan upper ouns the spanheight. A notale trivial case for spanheight is trees, ecause they o not contain ack-eges, an are alreay epth first search spanning trees. They have a spanheight zero. 4.1 Relation to anwith Spanheight irectly resemles the anwith prolem, ecause they oth restrict the istance etween ajacent vertices in their respective output structures. The ifference is that the output structure of anwith is a path, an a DFS spanning tree in the case of spanheight. The ifference in output structure makes us to elieve that there is not a strong relation etween the two prolems. Clearly anwith is an upper oun on the spanheight of a graph, ecause every anwith ecomposition of anwith is also a DFS spanning tree. The NP-Harness cases are also ifferent; anwith is NP-Complete on tree structures, while spanheight is easy on tree structures. The relation to anwith that we exploit in this thesis are the methos use to esign algorithms for anwith. 4.2 Relation to treewith Lemma 4.1. Let T = (V, H, r) e a DFS spanning tree roote at r of G with spanheight k, then there exist a tree-ecomposition ({X i i I}, T = (I, F )) of G with treewith k + 1. Proof. We transform T into a tree-ecomposition an prove that the three properties of a treeecomposition are satisfie. The first step is to create a ag i v I for every vertex v V. Then we copy the tree structure of T y creating an ege {i v, i w } F if an only if the corresponing ege {v, w} H exist in T. An for every vertex ag i v, let X iv e the set containing the first k + 2 vertices on the path from v to the root r, inclusive. Cleary every vertex from G is containe in T an therefore has an associate ag i v in the tree-ecomposition. The ags that can contain a vertex v form a connecte su-graph roote at 12

17 Chapter 4. Properties of Spanheight 13 the ag i v. We have left to show that every pair v, w V of ajacent vertices in G, are containe together in a ag of T. The vertices v, w must form an ancestor/escenant relationship in T an the path P connecting them has length at most k + 2 y efinition. Let v e the lowest vertex of the two, then the ag X iv contains every vertex on P incluing w y construction. This proofs the last property. A consequence of this lemma is that the spanheight of a graph upper-ouns the treewith of a graph. 4.3 Relation to treeepth The maximum istance etween any pair of vertices in a treeepth-ecomposition, is t. This is also a spanheight-ecomposition of spanheight t, an it follows that the treeepth is an upper oun on spanheight of a graph. Later in this thesis we will use the following property to create an FPT algorithm for spanheight. Lemma 4.2. Let T = (V, F, r) e DFS spanning tree of a connecte unirecte graph G = (V, E) with spanheight at most k an with ege creation F E. Let q e the maximum length of any path in G. The height of T is at most (q 1) k + q. Proof. Let v V e a leaf of T. Let P e a path from v to r in G. Oserve that any pair of ajacent vertices on P, have istance at most k + 1 etween each other in T, or the spanheight is larger than k. Using structural inuction, starting at the leaf v in T. We travel on T to reach the next vertex u on the path P, which has istance at most k + 1 from v. Continue this proceure until the root r of T is reache. At every step of this proceure, we move only up or own in the tree ecause there o not exist cross-eges in T. If we move only up, then we visit at most ( P 1) k + P vertices. This proves that the maximum istance etween any leaf v an the root r is a function of the spanheight in comination with the maximum length of any path in G. This upper oun can also e expresse in terms of the treeepth of a graph. Corollary 4.3. Let T = (V, F, r) e DFS spanning tree of a connecte unirecte graph G = (V, E) with spanheight at most k an with ege creation F E. Let t e the treeepth of G. The height of T is at most (2 t 1) k + 2 t t2 t. Proof. Nešetřil an Menez [24, formula 6.2] show that for graphs of oune treeepth t, the maximum length q of any path in G, is upper oune y 2 t. In the formula from Lemma 4.2 we sustitute q with 2 t. Recall that treeepth upper ouns the spanheight an sustitute k with t. Oserve that t k + 1, an we simplify (2 t 1) k + 2 t < 2 t (k + 1) 2 t t.

18 Chapter 5 NP-Completeness In this section we will present a NP-completeness proof for the spanheight prolem with a reuction from treeepth. But first we show that treeepth when restricte to eges from the original graph remains NP-har. This proof is analog to the NP-harness proof for the normal version of treeepth (Or equivalently vertex ranking) in [8]. Restricte-treeepth Instance: An unirecte connecte graph G = (V, E), an a positive integer k. Question: Does there exist a DFS spanning tree F = (V, E, r) of G with height lower or equal to k, such that V = V an E E? We will show that this prolem reuces to the alance complete ipartite supgraph prolem (areviate BCBS). The BCBS prolem is the NP-complete [21, GT24] an efine as follows: Balance complete ipartite supgraph Instance: A ipartite graph G = (V, E), an a positive integer k. Question: Are there two isjoint suset W 1, W 2 V such that W 1 = W 2 = k an such that u W 1, v W 2 implies that {u, v} E? BCBS on i- Theorem 5.1. Restricte-treeepth on coipartite graphs is reucile to partite graphs. Proof. Let G = (V 1, V 2, E) e a ipartite graph of n vertices an let k e a positive integer. Together they form an instance to the BCBS prolem. We claim that G contains a alance complete ipartite graph using 2 k vertices, if an only if the coipartite graph G create from G amits a DFS spanning tree restricte to eges from G of height (n k + 2). Let W 1 V 1, W 2 V 2 with W 1 = W 2 = k e a solution to the BCBS prolem on G. Let G = (V 1, V 2, E ) e the coipartite graph create from G such that V 1 = V 1, V 2 = V 2, an create eges e E e E. We create a DFS spanning tree for G as illustrate in Figure 5.1: Create an universal vertex a with two ranches/sutrees elow it. One ranch is a path using the vertices from W 1 an the other ranch is a path using the vertices from W 2. These paths can e create without creating new eges ecause y efinition of G the vertices containe in W 1 an W 2 form cliques in G. This also oes not create cross-eges ecause y efinition of G there oes not exist an ege from any vertex u W 1 to v W 2 in G. By efinition the vertices from V 1 \ W 1 from a clique an we create a path using these vertices aove the vertex a. Then we a a new universal vertex as root, an finally a a path using the vertices V 2 \ W 2 aove. The tree is a long path ening the vertex a, with two ranches elow it of length k, giving a tree height of (n k + 2). Let F e any DFS spanning tree of G, then F resemles the structure escrie aove. Let e the root of F with a as a chil of, an elow a two ranches. One ranch is a path using the vertices from V 1 an the other a path using the vertices from V 2. Let this e the left tree from Figure 5.2. Oserve that vertices containe in V 1 an V 2 form a clique in G, an therefore they can never create new su-ranches in F without creating cross-eges. Therefore, vertices can only move up an own while re-configuring the tree. As a consequence, any cross-eges 14

19 Chapter 5. NP-Completeness 15 q 1 q 2 q 5 z 1 z 2 a z 3 z 4 q 3 q 4 Figure 5.1: A DFS spanning tree of a coipartite graph with V 1 = {z 1, z 2, z 3, z 4 }, an V 2 = {q 1, q 2, q 3, q 4, q 5 }. The eges of the cliques V 1 an V 2 are represente y otte eges, the eges with an enpoint in V 1 an V 2 are represente y a soli grey ege, an the tree-eges are rawn using a soli lack color. The eges of the two universal vertices a an are not rawn for increase clarity. q 1 a q 2 z 1 z 1 z 2 z 3 z 4 q 1 q 2 q 3 q 4 q 5 z 2 a z 3 z 4 q 3 q 4 q 5 Figure 5.2: Two spanning trees of a coipartite graph G as efine in Figure 5.1. The vertices forming cross-eges in the left tree are move up into the tree on the right. etween the ranches elow a must e remove y moving a vertex from elow a to aove a. Let the right tree of Figure 5.2 e an example where all cross-eges are remove. We conclue that every DFS spanning tree of G consist of a path ening in some vertex x, with two ranches elow x. Let F e any aritrary DFS spanning tree of G with treeepth (n k + 2) with k 1, then as proven its structure consist of a path ening in some vertex x with two ranches elow it. As proven one sie of the ranch can only use vertices W 1 from V 1 an the other sie uses the vertices W 2 from V 2. Then W 1 an W 2 form a complete ipartite su-graph of G y efinition of G. If W 1 > W 2, then the treeepth is (n W 1 +2), an we can take any suset of W 1 W 1 such that W 1 = W 2 = k, an they form a alance complete ipartite su-graph in G. Corollary 5.2. Restricte-treeepth is NP-Complete on coipartite graphs. Proof. The NP-harness on coipartite graphs follows irectly from the reuction in Theorem 5.1 to the BCBS prolem which is known to e NP-Complete from [21]. To prove NP-Completeness we have left to show that there exist a polynomial time certificate testing the valiity of a solution. Let T = (V, F, r) e DFS spanning tree of trepepth k of the unirecte connecte graph G = (V, E). First compute for each vertex in T the height y iterating them in postfix orer. If the height of the root vertex is lower or higher than k then the treeepth is satisfie. To test if T is a vali DFS tree we must test whether T contains all vertices from G, is acyclic an connecte, ut also that is oes not contain cross-eges. These tests are trivially

20 Chapter 5. NP-Completeness 16 one in polynomial time. We conclue that there trivially exist such an certificate proving the corollary. Theorem 5.3. Restricte-spanheight is reucile to restricte-treeepth on aritrary graphs. Proof. Let G = (V, E) e a an unirecte connecte graph of n vertices an let k e a positive integer. Together they form an instance to the treeepth prolem. We claim that G amits a DFS spanning tree of height k, if an only if the graph G, constructe from G y aing a gaget, amits a DFS spanning tree of spanheight k 1. Let T = (V, F, r) e a DFS spanning tree of G with treeepth k, for an example see the left tree in Figure 5.3. A a set Z containing (2 k + 1) new vertices to G to get the new graph G. In aition a the ege-set ({z i, z j } z i, z j Z, 1 i < j Z, j i < k) an make z k+1 an universal vertex in G as illustrate in Figure 5.3. We claim that any DFS spanning tree T of G with spanheight k 1 must contain a DFS spanning tree T of G with treeepth k as a su-tree, with as a consequence the correctness of the theorem. This claim follows from the fact that for every vertex z i Z, the parent in T is z i+1, except the highest two z 2 k an z 2 k+1. The vertices from G are require to have istance at most k + 1 from the universal vertex z k+1. Since z k+1 alreay has a parent, the vertices from G are either ancestors or escenants of z k+1. When we o not allow ege creation, the vertices from G have no eges to the ancestors of z k+1, an are therefore escenants. Even if we allow the creation of eges, placing any vertex etween any pair of z i an z i+1 is impossile, ecause them eing a clique of size k requires them to form a path 7.1. Placing a vertex from G aove the vertex z 2 k will create a ack-ege with spanheight at least k, which is higher than the allowe k 1. We conclue that the vertices from G form a su-tree of T elow z k+1 with height at most k, such that the leafs have istance at most k + 1 to the universal vertex z k+1. By efinition this sutree is a DFS spanning tree T with treeepth k of G. We have left to show that for each vertex z i Z except z 2 k an z 2 k+1, the parent in T is require to e z i+1. The reverse orering is also possile w.l.o.g. The parent relation follows from the efinition of the cliques, there are a cliques on the vertices with inices (1 to k + 1), (2 to k + 2),..., an (k+1 to 2 k + 1). Each clique has size k + 1, an must form a path in the in the tree T. Let i e the inex of vertex z i, an j = i+1 of z j. Then z i an z j share a clique an must have an ancestor-escenant relationship. Furthermore, at least 1 of the following hols: z i is part of a clique (i k to i) with only vertices with inex < j, every vertex of this clique must e on one sie of j to e ale to form path. z j is part of a clique (j to j + k) using only vertices with inex > i, every vertex of this clique must e on one sie of i to e ale to form path. Therefore, there are only two orerings of the vertices with spanheight at most k. Orere y ecreasing inex an y increasing inex. The highest vertex z 2 k+1 is the only exception, ecause it is only part of one clique, it can move elow z k+1 ut this oes not affect the placement an spanheight of any vertices. Corollary 5.4. Restricte-spanheight is NP-Complete. Proof. The NP-harness follows irectly from the reuction in Theorem 5.3 to the restrictetreeepth prolem which is proven to e NP-Complete on coipartite graphs in Corollary 5.2. To prove NP-Completeness we have left to show that there exist a polynomial time certificate testing the valiity a solution. Let T = (V, F, r) e DFS spanning tree of spanheight k of the unirecte connecte graph G = (V, E). First compute for each vertex in T the height y iterating them in post-fix orer, an then test for each ege if the height ifference is lower or equal to k + 1. The DFS property is trivially teste. We conclue that there trivially exist a certificate proving the corollary.

21 Chapter 5. NP-Completeness 17 z 9 z 8 z 7 z 6 z 5 a z 4 a f e g z 3 z 2 z 1 f e g Figure 5.3: Left the DFS spanning tree of treeepth k = 4 of a graph G. On the right the DFS spanning tree of spanheight k 1 the graph of G create from G y aing the vertices z i for 1 i 2 k + 1, an the universal vertex z k+1. The lighter grey eges are ack-eges, not all ack-eges of the universal vertex are not shown for clarity. Corollary 5.5. Spanheight is reucile to treeepth on aritrary graphs. Proof. This follows irectly from Theorem 5.3. Let T e a DFS spanning tree of G with treeepth k. The reuction creates a graph G such that the DFS spanning tree T of G with spanheight k 1 contains the tree T as a su-tree. Allowing the tree T to use eges that o not exist in G will still allow us to use it as su-graph of T for the spanheight prolem. Furthermore, Theorem 5.3 shows that even with ege creation the gaget still works as placing vertices aove the universal vertex is impossile with respect to the spanheight. Corollary 5.6. Spanheight is NP-Complete. Proof. The NP-harness follows irectly from the reuction in Corollary 5.5 to the treeepth prolem which is proven to e NP-Complete on ipartite an coipartite graphs in [8]. To prove NP-Completeness we have left to show that there exist a polynomial time certificate testing the valiity a solution. Let T = (V, F, r) e a DFS spanning tree with spanheight k of the unirecte connecte graph G = (V, E). First compute for each vertex in T the height y iterating them in post-fix orer, an then test for each ege if the height ifference is lower or equal to k + 1. The other DFS properties are trivially teste. We conclue that there trivially exist a certificate proving the corollary.

22 Chapter 6 Exact Algorithms In this chapter we introuce exact algorithms for the ecision prolems spanheight an restrictespanheight. Both prolems can trivially e solve e performing an exhaustive search in the solution space. The size of the solution space for NP-Complete prolems is so large that such an enumeration is intractale. For a lot of cominatorial prolems there exist faster algorithms using ynamic programming techniques. These algorithms still require exponential time, ut are much more efficient than an exhaustive search. The running time of these "efficient" algorithms is in the orer of O (c n ), where c is a small constant, preferaly as low as possile. The fastest known exact algorithm for treeepth uses O( n ) time ue to Fomin, Giannopoulou, an Pilipczuk [19]. Another variant calle treeepth with capacity amits an O( n ) algorithm [27]. In this chapter we efine an single exponential time algorithm for oth spanheight an restricte-spanheight. We will raw inspiration from the existing exact algorithms for anwith. In [17] Feige an Killian introuce a ucket-assignment scheme to solve anwith in O (10 n ) time an polynomial space. This algorithm has een improve y Cygan an Pilipczuk in [13] to get a O (5 n ) time an O (2 n ) space algorithm. Finally using measure an conquer the same authors foun a O (4.83 n ) time an O (4 n ) space algorithm in [12]. 6.1 Spanheight algorithm Our algorithm will e an aaptation of the secon fastest algorithm y Cygan an Pilipczuk. It consists of two phases. First we assign vertices to segments. Definition (Segments an layers). Let T = (V, E, r) e roote tree. Vertices in this tree have a fixe height. A layer at height i contains all vertices of height i. The height of a layer is counte from the ottom of the tree to the top. The height of layer is use to reference to ajacent layers. In Figure 6.2 the layers are rawn using horizontal lines, an height is isplaye at the right sie. A segment represents a group of k + 2 layers. In Figure 6.2 the tree is partitione in 3 segments covering all layers of the tree. The size of a segment is chosen as k + 2 such that any ack-ege in the tree lies within a segment, or etween two neighoring segments. If not, then this ack-ege must span at least k + 2 vertices, which is not allowe in our solution space. The main concept of the algorithm is to first partition all vertices into segments. An then fin the position of these vertices within the segment in a secon phase of the algorithm. Partition the interval [1, 2n] in 2n k+2 segments. The segments form a line, with a parentchil relations ship etween every pair of consecutive segments. Assign a vertex v V to the mile segment. Iteratively, take a vertex v V that is not assigne to a segment, ut has at least one neighor u that is assigne to a segment. The placement of v must e in the in the same segment as u, or one of the two connecting segments. If not then the ack-ege etween them spans at least 1 segment with a path using k + 1 vertices, which violates the spanheight constraint. If there is no vali placement for v, then this partial segment assignment will never lea to a solution. For every vertex there are 3 options which will e enumerate in an exhaustive search. 18