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Parameterized Algorithms on Width Parameters of Graphs

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Parameterized Algorithms on Width Parameters of Graphs Masaryk University Faculty of Informatics } Û¡¢£¤¥¦§¨ª«¬­Æ !"#°±²³´µ·¸¹º»¼½¾¿45<ÝA| Parameterized Algorithms on Width Parameters of Graphs 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 polyno- mial 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 algo- rithms. 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 lim- ited 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 pa- rameter 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 run- ning 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 com- pute 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 Kant´e[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 bet- ter 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 bi- partition (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 algo- rithms 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 can- didates 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].
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