Parameterized Algorithms on Width Parameters of Graphs

Masaryk University

Faculty of Informatics

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Doctoral Thesis Proposal

Robert Ganian

Supervisor: doc. RNDr. Petr Hlinˇen´y, Ph.D. Acknowledgement

I would like to thank doc. RNDr. Petr Hlinˇen´y, Ph.D. for his guidance and support during my studies. Contents

1 Preliminaries ...... 3 1.1 Introduction...... 3 1.2 Clique-width and Rank-width of undirected graphs ...... 4 1.3 Directed Graphs ...... 6 1.4 Aimofthethesis...... 7 2 ObtainedResults...... 9 2.1 Graphs of bounded rank-width ...... 9 2.2 Width measures of directed graphs ...... 13 3 Goalsofthethesis...... 16 3.1 Schedule and expected outcomes ...... 17 1 Preliminaries

1.1 Introduction

The design of graph algorithms has become a focal point of computer science research in the past decades for practical as well as theoretical reasons. Unfortunately, most of the interesting problems have been proved to be NP-complete in general. One way to deal with this obstacle is to design parameterized algorithms (see e.g. [14]) – the idea is that, although a problem might be NP-hard in general, restricting a certain parameter to constant size would make it possible to design a polynomial algorithm for solving the problem. Given an instance of a problem with input size x parameterized by par, a parameterized algorithm is in FPT if it runs in time O(p(x) ⋅ f(par)) for some polynomial p and function f. This is the ideal case, since the degree of the polynomial does not depend on the value of the parameter. A parameterized algorithm is in XP if it runs in time O(p(x)f(par)), i.e. polynomial time for any fixed value of par, but the degree of the polynomial grows with the parameter. XP algorithms are usually only practical for small values of the parameter, but in certain cases it is not possible to use FPT algorithms since the associated parameter could be unbounded for our input class of graphs. And of course it can also happen that a chosen parameter does not help solving the problem at all. Parameterized algorithms are especially useful when considering real- world applications. Usually, we do not need to solve a given problem on the class of all graphs. In most cases the input graphs are specific in one way or another, and this can be exploited by finding a suitable parameter which is low on this class of graphs and yet leads to efficient parameterized algorithms. But which graph parameters are actually useful? We have to be careful when choosing the right parameter for our algorithms. Obviously it shouldn’t be too restrictive, since otherwise they will be limited to a very small admissible input class. But at the same time it should be powerful enough to design efficient parameterized algorithms for a wide range of problems. One example of a very successful graph parameter is the tree-width of Robertson and Seymour [28]. Although fairly restrictive, it is possible to design FPT algorithms for most natural NP-hard problems on graphs of bounded tree-width. There also exists a language (actually a monadic second order logic MS2) associated with graph properties computable in FPT on graphs of bounded tree-width.

3 1.2 Clique-width and Rank-width of undirected graphs

1.2.1 Clique-width

Despite the power of the tree-width parameter, its applicability was limited by how restrictive it was. The tree-width of a graph reﬂects how “tree-like” that graph is – yet many graphs which seem to be well struc- tured are not very “tree-like”’ in the sense it is measured by tree-width. In light of these facts, Courcelle and Olariu introduced a new width parameter called clique-width [12].

A k-expression is an algebraic expression with the following operators and 1 ≤ i ≤ k:

1. ∙ i is a nullary operator which represents a graph with a single vertex labeled i.

2. ij for i ∕= j is a unary operator which adds edges between all vertices labeled i and all vertices labeled j.

3. i→jis a unary operator which changes the labels of all vertices labeled i to j. 4. ∪˙ is a binary operator which represents the disjoint union of two graphs.

The clique-width of a graph G is then the smallest k such that there exists a k-expression generating G. Notice that cliques have clique-width at most 2, and yet they can have arbitrarily high tree-width. In general it holds that bounded clique-width implies bounded tree-width, but the opposite is not true.

There have been numerous articles on parameterized algorithms running on clique-width (on k-expressions of graphs to be more precise), see e.g. [10,16]. It even holds that all problems expressible in MS1 logic, a weaker application of monadic second-order logic than the one associated with tree-width, are solvable in FPT on graphs of bounded clique-width when the k-expression is provided. Unfortunately it is unfeasible to compute the respective k-expressions of graphs even if they have bounded clique-width. In other words, all of the parameterized algorithms utilizing clique-width need the k-expressions of input graphs to either be provided as part of the input or by an oracle.

4 1.2.2 Rank-width

Rank-width is a relatively new width parameter related to clique-width which not only overcomes the aforementioned problem, but which we have also (independently of the parallel work of Courcelle and Kanté[9] and Telle et al. [8]) proved to lead to more efficient parameterized algorithms than clique-width [1,2,3]. The usual way of defining rank-width is as the branch-width of the cut-rank function.

Branch-width: A set function f : 2M → ℤ is called symmetric if f(X) = f(M ∖ X) for all X ⊆ M. A tree is subcubic if all its nodes have degree at most 3. For a symmetric function f : 2M → ℤ on a finite set M, the branch-width of f is defined as follows: A branch-decomposition of f is a pair (T,) of a subcubic tree T and a bijective function : M → {t : t is a leaf of T }. For an edge e of T , the connected components of T ∖ e induce a bipartition (X,Y ) of the set of leaves of T . The width of an edge e of a branch-decomposition (T,) is f(−1(X)). The width of (T,) is the maximum width over all edges of T . The branch-width of f is the minimum of the width of all branch- decompositions of f. (If ∣M∣ ≤ 1, then we define the branch-width of f as f(∅).)

One natural application of this deﬁnition is the branch-width of a graph, as introduced by Robertson and Seymour [28] along with better known tree-width, and its natural matroidal counterpart. In that case we use M = E(G), and f the connectivity function of G. For rank-width we use the vertex set V (G)= M of a graph G as the ground set.

Rank-width: For a graph G, let AG[U, W ] be the bipartite adjacency matrix of a bipartition (U, W ) of the vertex set V (G) deﬁned over the two-element ﬁeld GF(2) as follows: the entry au,w, u ∈ U and w ∈ W , of AG[U, W ] is 1 if and only if uw is an edge of G. The cut-rank function G(U) = G(W ) then equals the rank of AG[U, W ] over GF(2). A rank-decomposition and rank-width of a graph G is the branch-decomposition and branch-width of the cut-rank function G of G on M = V (G), respectively.

5 d d c × 1 0

0011 0 0 1 ⎛0 1⎞ 0110 × e × c 1 0 0 0 0 ⎝ ⎠ 1001 1100

1001 × × a b e a b

Figure 1. A rank-decomposition of the graph cycle C5.

Finally, to understand the usefulness of rank-width the reader must be aware of two related results. First, rank-width is closely related to clique-width, although the relationship is not immediate.

Theorem 1.1 ([28]). rwd(G) ≤ cwd(G) ≤ 2rwd(G)+1 − 1 for all graphs G.

This means that one can compare the eﬃciency of parameterized algorithms on clique-width and rank-width. Second, it is possible to eﬃciently compute rank-decompositions of graphs of bounded rank-width – recall that the equivalent is not possible for clique-width.

Theorem 1.2 ([21]). For every fixed t there is an O(n3)-time algorithm that, for a given n-vertex graph G, either finds a rank-decomposition of G of width at most t, or confirms that the rank-width of G is more than t.

1.3 Directed Graphs

Following the success of tree-width, there have been numerous attempts to find a similarly useful notion for directed graphs. The direct approach was to find a directed equivalent to tree-width. The list of considered candidates is extensive, including directed tree-width [23], DAG-width [25,7], Kelly-width [22], cycle-rank [15], directed path-width and others. All of these directed graph measures share the property of having low, fixed values on directed acyclic graphs (DAGs in short) and are closely related to each other.

Theorem 1.3. Let G be a digraph. Then

1/3(dtw(G) − 1) ≤[7] dagw(G) ≤ dpw(G) ≤[18] cr(G)

1/6(dtw(G) + 2) ≤[22] kellyw(G) ≤ dpw(G) ≤[18] cr(G)

Moreover, when DAG-width is bounded, so is Kelly-width [20].

6 Many of these directed width measures also share a resemblance to undirected tree-width – they are strongly based on some variant of the directed cops-and-robber game on a digraph: There are k cops and a robber. Each cop can either occupy a vertex, or move around in a helicopter, and the robber occupies a vertex. The robber can, however, see the helicopter landing, and can move at a great speed along a cop-free directed path to another vertex. The objective of the cops is to capture the robber by landing on the vertex currently occupied by him, the objective of the robber is to avoid capture. DAG-width is a great example of this approach. It is fully characterized by a variant of the cops-and-robber game where the cop strategy has to be monotone, i.e. , a cop cannot be placed on a previously vacated vertex. Note that monotone and non-monotone strategies are not equivalent on digraphs. Theorem 1.4 ([25,7]). For any graph G, there is a DAG-decomposition of G of width k if, and only if, the cop player has a monotone winning strategy in the k-cops-and-robber game on G. But how do these width measures fare as parameters for algorithms? The ﬁrst results seemed encouraging: The Hamiltonian path problem can be solved in polynomial time (XP) if the directed tree width, the DAG- width, or the Kelly-width are bounded by a constant [23]. More recently, it has been shown that parity games can be solved in polynomial time on digraphs of bounded DAG-width [7] and Kelly-width [22].

1.4 Aim of the thesis The topic of the thesis is the study of structural parameters (often called widths or width parameters) of graphs and parameterized algorithms utilizing these parameters. Research in this ﬁeld was initiated by a series of highly acclaimed articles by Robertson and Seymour (see e.g. [27] [28]), which have lead to the introduction of tree-width as a powerful parameter capable of solving many NP-hard problems in FPT time. Later on, Arnborg, Lagergren and Seese have proved that all graph problems which can be described by the MS2 logic can be solved on graphs of bounded tree-width in FPT time [6], and a powerful formalism for designing such algorithms has been developed. This formalism has also been generalized (through LinEMSO2) to also include optimization problems. However, the price tree-width pays for its considerable power is the way it restricts graphs of bounded tree-width. A lot of research has recently shifted towards new width parameters, which would allow more

7 freedom while still permitting the design of efficient parameterized algorithms. This has lead to the introduction of clique-width in 2000 [12] and subsequently rank-width in 2004 [26]. At first rank-width was merely considered as a tool for bounding and approximating clique-width, but our work with my advisor [2,3] (based on my master thesis [1]) together with the independent work of Courcelle, Kanté[9] and of Bui-Xuan, Telle and Vatshelle [8] has shown that rank-width possesses several features making it a better parameter for designing algorithms than clique-width. Another hot topic of research is the development of width parameters for directed graphs. Finding a directed counterpart to tree-width has lead to the introduction of a wide range of promising directed width parameters related to tree-width. Although there were some positive results at first, our joint article (Robert Ganian, Petr Hlinˇený, Jan Obdrˇzálek, Joachim Kneis, Alexander Langer and Peter Rossmanith [4]) has shown that almost all classical problems remain NP-hard even on DAGs, mean- ing that none of the directed parameters considered thus far could be used to solve them. On the other hand, the article points out that bi-rank- width, a directed counterpart to rank-width, remains just as powerful as rank-width. At this point my research can continue in several directions. Unless unforeseen circumstances occur, it is quite probable that the thesis will explore some of the following research directions:

∙ Development of new FPT and XP graph algorithms ∙ New hardness results for various combinations of problems and graph parameters ∙ Identiﬁcation of new promising parameters for graph algorithms ∙ Possible development of frameworks for parameterized algorithm design

8 2 Obtained Results

Our main results so far can be divided into results in the ﬁeld of undirected graphs and rank-width and results concerning width measures of directed graphs.

2.1 Graphs of bounded rank-width Rank-width has all the characteristics of a very useful and powerful width parameter. For “tree-like” graphs, tree-width is the optimal width parameter, however once the input class of graphs becomes less restricted, using parameterized algorithms on rank-width makes more sense than utilizing clique-width. Or at least that would be the case if we could actually design the parameterized algorithms we needed! The original definition of rank-width does not seem suitable for designing dynamic algorithms at all, and while a formalism exists for easily designing algorithms on graphs of bounded tree-width, no similar result was known for rank-width when I started writing my master thesis in 2007. In the master thesis [1], a formalism was developed for designing efficient FPT algorithms on graphs of bounded rank-width. The thesis was awarded first place in the SVOC 2008 competition, in the Theoretical Computer Science section. With my advisor, based on the results of the master thesis, we then wrote an article which was accepted at IWOCA 2008 [2]. The formalism can be used to easily develop efficient FPT algorithms for any decision problem which can be described by the MS1 logic and is easily extensible to optimization problems described by the LinEMSO1 logic (for more details, see e.g. [10]). Additionally the developed algorithms, while always being at least as efficient as their known counterparts on clique-width, for some problems resulted in exponentially faster running times than the best known algorithms on clique-width.

Labeling parse tree: Lt A t-labeling of a graph is a mapping lab : V (G) → 2 where Lt = {1, 2,...,t} is the set of labels (this notion is exactly equivalent to multiple- coloured graphs of [9]). Having a graph G with an (implicitly) associated t-labeling lab, we refer to the pair (G, lab) as to a t-labeled graph and use notation G¯. Notice that each vertex of a t-labeled graph may have zero, one or more labels. We will often view (cf. [9] again) a t-labeling of G equivalently as a mapping V (G) → GF(2)t to the binary vector space of dimension t, where GF(2) is the two-element ﬁnite ﬁeld.

9 1 2 Considering t-labeled graphs G¯1 = (G1, lab ) and G¯2 = (G2, lab ), a t-labeling join G¯1 ⊗ G¯2 is defined on the disjoint union of G1 and G2 by adding all edges (u, v) such that ∣lab1(u) ∩ lab2(v)∣ is odd, where u ∈ V (G1), v ∈ V (G2). The resulting graph is unlabeled. Lt A t-relabeling is a mapping f : Lt → 2 . In linear algebra terms, a t-relabeling f is in a natural one-to-one correspondence with a linear t t transformation f : GF(2) → GF(2) , i.e. a t ⋅ t binary matrix Af . For a t-labeled graph G¯ = (G, lab) we define f(G¯) as the same graph with a vertex t-labeling lab′ = f ∘ lab. Here f ∘ lab stands for the linear transformation f applied to the la- ′ beling lab, or equivalently lab = lab ⋅ Af as matrix multiplication. Infor- mally, f is applied separately to each label in lab(v) and the outcomes are summed up “modulo 2”; e.g. for lab(v)= {1, 2} and f(1) = {1, 3, 4}, f(2) = {1, 2, 3}, we get f ∘ lab(v)= {2, 4} = {1, 3, 4}△{1, 2, 3}. Let ⊙ be a nullary operator creating a single new graph vertex of la- Lt bel {1}. For t-relabelings f1,f2, g : Lt → 2 , let ⊗[g ∣ f1,f2] be a binary operator—called t-labeling composition (as bilinear product of [9])—over 1 2 pairs of t-labeled graphs G¯1 = (G1, lab ) and G¯2 = (G2, lab ) defined as follows: G¯ ⊗[g ∣ f ,f ] G¯ = H¯ = G¯ ⊗ g(G¯ ), lab 1 1 2 2 1 2 i where a new labeling is lab(v)= fi ∘ lab (v) for v ∈ V (Gi), i = 1, 2. A t-labeling parse tree T , see also [1, Definition 6.11], is a finite rooted ordered subcubic tree (with the root degree at most 2) such that ∙ all leaves of T contain the ⊙ symbol, and ∙ each internal node of T contains one of the t-labeling composition symbols. A parse tree T then generates (parses) the graph G which is obtained by successive leaves-to-root applications of the operators in the nodes of T . See Figure 2, 3.

⊗[id ∣ ∅, ∅] d × ⊗[id ∣ id, 1→∅] ⊗[id ∣1→2, id]

id id → × e × c ⊗[ ∣ , 1 2]

⊙ a

× × a b ⊙ b ⊙ c ⊙ d ⊙ e Figure 2. An example of a labeling parse tree which generates a 2-labeled cycle C5, with symbolic relabelings at the nodes (id denotes the relabeling preserving all labels, and ∅ is the relabeling “forgetting” all labels).

d {1} d {2}

× ×

× × ×

e {1} × c {1} e {1} c {2}

× × × → → b {1} a {1} b {1}

d {2} d

× ×

× × ×

e {1} × c {2} e c

× × × → × → a {1} b ∅ a b

Figure 3. “Bottom-up” generation of C5 by the parse tree from Figure 2.

Analogously to the work of Courcelle and Kantéwe get a crucial state- ment: Theorem 2.1 (Rank-width parsing theorem [9,?]). A graph G has rank-width at most t if and only if (some labeling of) G can be generated by a t-labeling parse tree. Furthermore, a width-t rank-decomposition of G can be transformed into a t-labeling parse tree on (∣V (G)∣) nodes in time O(t2 ⋅ ∣V (G)∣2). Regularity theorem for rank-width: Based on these results, we have successfully proved an equivalent to the classical Myhill–Nerode regularity tool of automata theory for graphs of bounded rank-width. Similar results have been achieved in other fields in the past (such as for graphs of bounded tree-width [14] and for matroids of bounded branch-width [19]), however there were obstacles to applying this approach to rank-width directly. We will need some notation and a definition before writing the main theorem itself. First, ℛt denotes the class of all unlabeled graphs of rank-width at most t and ℛt is the class of all t-labeled graphs generated by labeling parse trees using t labels. Also, ⊗ is an abbreviation of ⊗[id ∣ id, id] .

Deﬁnition 2.2 (Canonical Equivalence [2]). G¯1 ≈D,t G¯2 for any G¯1, G¯2 ∈ ℛt if and only if, for all H¯ ∈ ℛt,

G¯1 ⊗ H¯ ∈ Dt ⇐⇒ G¯2 ⊗ H¯ ∈ Dt . Theorem 2.3 (Rank-width regularity theorem [2]). Let t ≥ 1, D be a graph class, and Dt = D ∩ ℛt. The collection of all those t-labeling

11 parse trees which generate the members of Dt is accepted by a ﬁnite tree automaton if, and only if, the canonical equivalence ≈D,t of Dt over ℛt is of ﬁnite index.

In informal words, the classes of ≈D,t “capture” all information we need to know about a t-labeled subgraph G¯ ∈ ℛt to decide membership in D further on in our parse tree processing. So, if we need to decide whether a graph has some property (i.e. belongs to a certain graph class), we only need to describe an appropriate canonical equivalence with ﬁnite index. Very casually speaking, this equivalence works as follows: two labeled graphs are equivalent when, no matter what H we add to both of them, we always get the same results with respect to the property in question. A FPT algorithm for deciding the given property can then be obtained immediately from this equivalence.

To illustrate the power of this approach, we have shown that it can be used to develop efficient algorithms for deciding any MS1-expressible properties (see Theorem 2.4). Theorem 2.4. Let t ≥ 1 be fixed. Suppose is a formula in the language MS1, and is a partial equipment signature for . Then ≈,t has finite index in the universe of t-labeled -partially equipped graphs. The article also includes new sample FPT algorithms based on the framework. A prime example is our algorithm for solving the well-known NP-hard 3-colourability problem. It trivially extends to deciding c-colourability and, for any c, runs in time O ct3 ⋅ 2c t(t+1)/2 ⋅ ∣V (G)∣ – exponentially faster than any such algorithm on clique-width.

XP algorithms: Of course, not all problems can be solved in FPT time on graphs of bounded rank-width. In another article [3], we focus on extending the previous framework to solve problems which do not allow FPT algorithms (i.e. at least W[1]-hard problems). Generally, it is possible to convert any XP algorithm on clique-width to rank-width with the use of labeling parse trees. However, the main contribution of the article lies in showing how the canonical equivalence framework can be used to design formally “clean” and more eﬃcient XP algorithms. Several new algorithms include deciding hamiltonicity, computing the chromatic number and the chromatic polynomial of graphs. While all such newly designed algorithms are guaranteed to be at least as fast as their counterparts running on clique-width, in some cases it is

12 again possible to achieve exponentially better results. Such is the case for computing the chromatic number, which runs in time:

ℎ(t) t(t+1)/2 O ∣V (G)∣ where ℎ(t)= O(2 ) , exponentially faster than the best such known algorithm on clique-width. Computing the chromatic polynomial is done by an extension of this algorithm, and is also exponentially faster than on clique-width.

2.2 Width measures of directed graphs When I began studying the area of directed graphs, a large number of width parameters all seemed to be plausible candidates for the position of directed counterpart to tree-width. These widths are all linked to tree- width in one way or another (often via cops-and-robber games) and all share the property of having low, fixed values on DAGs. As for their usefulness for parameterized algorithms, it has been shown that Hamil- tonian path and parity games are solvable in polynomial time when any of these parameters are bounded. But do these parameters truly have the potential of being as powerful as tree-width has become on undirected graphs? The results of our large joint article [4] indicate that this is in fact not the case. The scope of the article is actually unusually wide. First, it shows that several directed variants of classical graph problems remain NP-hard on digraphs of bounded DAG-width and digraphs of bounded cycle-rank. Due to Theorem 1.3, this alone means that basically none of the considered “classical” directed width measures can be used to solve these problems. Second, it introduces K-width and DAG-depth, two new width parameters as very severe restrictions of the classical directed width measures. The idea here is that perhaps the principles behind the considered directed parameters could actually lead to efficient parameterized algorithms, while the negative results could have originated from the par- ticular definitions being too general and not restrictive enough. Definition 2.5 (DAG-depth [4]). The DAG-depth ddp(G) of a digraph G is inductively defined: If ∣V (G)∣ = 1, then ddp(G) = 1. If G has a single reachable fragment, then ddp(G) = 1+min{ ddp(G − v) : v ∈ V (G) }. Otherwise, ddp(G) equals the maximum over the DAG-depth of the reachable fragments of G. Definition 2.6 (K-width [4]). The K-width of a digraph G is the maximum number of distinct (not necessarily disjoint) simple s–t paths in G over all pairs of distinct vertices s,t ∈ V (G).

13 To understand how restrictive each of these width measures are, consider the following remarks. DAG-depth actually bounds the length of the longest path in a graph. The notion has a t-cops-and-robber game characterization similar to those of DAG-width and directed tree-width, however the cop player may never move a cop once he lands – in other words, the cop player must win in t turns. As for K-width, it is possible to efficiently list through all paths in any digraphs with bounded K-width. Consider the implications for e.g. Hamiltonian path: on a digraph G with K-width at most t, Hamiltonian path can be trivially solved in time O(V (G)2 ⋅ t) by just processing all paths and checking if any of them are Hamiltonian. But even with such restrictions, many classical problems – problems solvable in FPT on graphs of bounded tree-width – remain NP-hard when extended to directed graphs. This is strong evidence that the principles behind these classical directed width parameters are not suitable for designing parameterized algorithms. After all these negative results, the third contribution of the article offers some positive ones as well. While all the directed generalizations of tree-width seem to lose their algorithmic applicability, this is not true for rank-width. We show that bi-rank-width, one of two known directed counterparts of rank-width, retains good algorithmic properties. The findings of the article are best summarized by the following table:

∗ Table 1. Old and new (in boldface) complexity results on digraph measures ( -marked results assume a decomposition is given in advance; p-NPC is a shortcut for the complexity class para-NPC; and c and are ﬁxed parameters of the respective problems).

Problem K-width DAG-depth DAG-width Cycle-rank DAG Bi-rank-width ∗ ∗ HAM FPT FPT XP XP P XP W[2]-hard W[2]-hard ∗ ∗ c-Path FPT FPT XP XP P FPT k-Path p-NPC p-NPC NPC NPC NPC open DiDS p-NPC p-NPC NPC NPC NPC FPT DiSTP p-NPC p-NPC NPC NPC NPC FPT MaxDiCut p-NPC p-NPC NPC NPC NPC XP c-OCN p-NPC p-NPC NPC NPC NPC FPT DFVS open open p-NPC p-NPC P FPT Kernel p-NPC p-NPC p-NPC, p-NPC, P FPT MSO1mc p-NPH p-NPH NPH NPH NPH FPT LTLmc p-coNPH p-coNPH coNPH coNPH coNPC p-coNPH ∗ ∗ Parity XP XP XP XP P XP

14 In a follow-up article [5], we have studied the parameterized complexity of the oriented 4-colouring decision problem (OCN4 in short). Oriented colouring was introduced by Courcelle [11] and has been extensively studied by Sopena [29], Neˇsetˇril [24] and others. While most problems on digraphs can be reformulated on undirected graphs and have also been studied as undirected problems, oriented colouring is in fact a purely directed problem and can not be easily translated to undirected graphs. For a digraph to be properly orientedly coloured, not only can there be no arc between vertices of the same colour, but all arcs between two colours must always have the same “direction” – i.e. if there is an arc from blue to red, there can be no arc from red to blue anywhere in the digraph. OCN4 is the most interesting case since OCN3 can be solved in polynomial time. The article contains several new results regarding the 4-colourability of digraphs. It starts with a few positive results and provides two algorithms for deciding OCN4 on DAG-depth 2 and a special case of K-width 1. Both algorithms are fairly simple, but surprisingly we continue by proving that it is the best one can do. The main contribution of the article is a reduction from 3-SAT to OCN4 on a digraph of K-width 1 and DAG-depth 3. Even on digraphs with at most a single path between any two vertices and with paths of length at most 6, OCN4 still remains NP-hard. On the other hand, OCNk is decidable in FPT on digraphs of bounded bi-rank- width. The article also contains a proof that OCN4 is NP-hard on DAGs (greatly improving the original reduction of Culus and Demange [13]).

15 3 Goals of the thesis

The research conducted up to now has not only lead to new results, but has opened up new questions as well. Parameterized algorithms on graphs are a promising research topic with theoretical as well as practical implications. I intend to continue the work done until now by developing new results in the ﬁeld and publishing them on renowned international conferences. Possible results include but are not limited to:

∙ Development of new FPT and XP graph algorithms While not the usual case, sometimes developing a single parameterized algorithm can be a huge result all by itself. This is especially true if the problem has been open for some time, if the parameter did not seem useful for solving the problem or if the algorithm uses a new approach which could have further applications. Any new XP and FPT parameterized algorithm extends the knowledge of the scientific community and is certainly within the scope of this thesis. Developing parameterized algorithms for computing OCN is a good example of a subject for future research. ∙ New hardness results for various combinations of problems and graph parameters Closely related to the development of new algorithms, hardness results are crucial in determining which parameters are actually useful in the context of designing algorithms. Similarly to developing new algorithms, any new hardness result contributes to the scientific community and may lead to new breakthroughs and results. As such, new hardness results for problems on graphs with bounded parameters lie in the scope of this thesis. It is currently an open problem whether OCN can be computed on bi-rank-width; An eventual proof that it remains hard even on digraphs with bounded bi-rank-width is a good example of an interesting hardness result (the undirected chromatic number can be computed in polynomial time on graphs of bounded rank-width). ∙ Identification of new promising parameters for graph algorithms Introducing new useful width parameters also belongs into the scope of the thesis. New width parameters not only extend the theoretical knowledge of the mathematical and computer science community, but also have practical applications. In real life, it is very rare to have to solve problems on truly random graphs. Usually the input of one pro- cess is the output of another, and such outputs tend to have a certain

16 structure. The question is whether such a structure (a bounded width parameter) allows for eﬃcient computation of the tasked problem. It should be noted that parameterized algorithms need not run only on specialized width measures – for example Jan Kratochv´ıl, Jiˇr´ı Fiala and Petr Golovach [17] have achieved very good results when using the feedback vertex set and vertex cover numbers as parameters.

∙ Possible development of frameworks for parameterized algorithm design New frameworks for designing parameterized algorithms on graphs of bounded width parameters are similar to the one for tree-width [14] and the one we have developed for rank-width [1] would be a great ad- dition to the thesis. Such frameworks are often associated with logical systems capable of describing any problem which can be solved by the framework. However, most of the currently known width parameters are either already associated with frameworks for the development of parameterized algorithms or do not seem to be suitable for too many algorithms. The development of a new framework would likely follow the introduction of a new powerful width parameter.

3.1 Schedule and expected outcomes The current state of the art has a notable effect on the schedule. There are several possible directions my future research could follow, and breakthroughs in the field could also lead to new research topics while making some others obsolete. For the short-term horizon, I plan to focus on developing algorithms for problems on directed graphs. The main expected outcome of the thesis is a survey of results in the field of parameterized graph algorithms. I plan to defend the thesis proposal and pass the doctoral examination in the winter of 2009/2010 and finish my studies in accordance with the standard schedule after 8 terms, i.e. after the spring term of 2012.

17 Research I have worked on

1. Ganian, R.: Automata formalization for graphs of bounded rank-width. Master thesis. Faculty of Informatics of the Masaryk University, Brno, Czech republic (2008). 2. Ganian, R. and Hlinˇený, P.: On Parse Trees and Myhill–Nerode–type Tools for handling Graphs of Bounded Rank-width. Manuscript, to appear. Extended abstract in IWOCA08, LNCS. Springer, 2008. 3. R. Ganian and P. Hlinˇený: Better Polynomial Algorithms on Graphs of Bounded Rank-width. Manuscript, to appear. Extended abstract in IWOCA09, LNCS. Springer, 2009. 4. R. Ganian, P. Hlinˇený, J. Kneis, A. Langer, J. Obdrˇzálek and P. Rossmanith: On Digraph Width Measures in Parameterized Algorithmics. In IWPEC09, to appear. 5. R. Ganian and P. Hlinˇený: New results on the complexity of oriented colouring on restricted digraph classes. Manuscript, to appear.

References

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