Ton Kloks

Fixed Parameter Algorithms for DOMINATING SET and Related Problems on Planar Graphs

Abstract. We present an algorithm that constructively produces a solution to the k -DOMINATING SET problem for planar graphs in time O(c^ \sqrt k n) , where c=4^ 6\sqrt 34 . To obtain this result, we show that the treewidth of a planar graph with domination number γ (G) is O(\sqrt \rule 0pt 4pt \smash γ (G) ) , and that such a tree decomposition can be found in O(\sqrt \rule 0pt 4pt \smash γ (G) n) time. The same technique can be used to show that the k -FACE COVER problem (find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c1^ \sqrt k n) time, where c1=3^ 36\sqrt 34 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k -DOMINATING SET, e.g., k -INDEPENDENT DOMINATING SET and k -WEIGHTED DOMINATING SET.

Efficient and Constructive Algorithms for the Pathwidth and Treewidth of Graphs

In this paper we give, for all constantsk,l, explicit algorithms that, given a graphG=(V,E) with a tree-decomposition ofGwith treewidth at mostl, decide whether the treewidth (or pathwidth) ofGis at mostk, and, if so, find a tree-decomposition (or path-decomposition) ofGof width at mostk, and that useO(|V|) time. In contrast with previous solutions, our algorithms do not rely on non-constructive reasoning and are single exponential inkandl. This result can be combined with a result of B. Reed in“Proceedings of the 24th Annual Symposium on Theory of Computing,” pp. 221?228, 1992, yielding explicitO(nlogn) algorithms for the problem, given a graphG, to determine whether the treewidth (or pathwidth) ofGis at mostk, and, if so, to find a tree- (or path-) decomposition of width at mostk(kconstant). Also, H. L. Bodlaender in“Proceedings of the 25th Annual Symposium on Theory of Computing,” pp. 226?234, 1993 has used the result of this paper to obtain linear time algorithms for these problems. We also show that for all constantsk, there exists a polynomial time algorithm that, when given a graphG=(V,E) with treewidth ?k, computes the pathwidth ofGand a path-decomposition ofGof minimum width.

Approximating treewidth, pathwidth, frontsize, and shortest elimination tree

Abstract Various parameters of graphs connected to sparse matrix factorization and other applications can be approximated using an algorithm of Leighton et al . that finds vertex separators of graphs. The approximate values of the parameters, which include minimum front size, treewidth, pathwidth, and minimum elimination tree height, are no more than O (log n ) (minimum front size and treewidth) and O (log 2 n ) (pathwidth and minimum elimination tree height) times the optimal values. In addition, we show that unless P = NP there are no absolute approximation algorithms for any of the parameters.

Fixed parameter complexity of λ-labelings

A λ-labeling of a graph G is an assignment of labels from the set {0, . . . , λ} to the vertices of a graph G such that vertices at distance at most two get different labels and adjacent vertices get labels which are at least two apart. We study the minimum value λ = λ(G) such that G admits a λ-labeling. We show that for every fixed value k ≥ 4 it is NP-complete to determine whether λ(G) ≤ k. We further investigate this problem for sparse graphs (k-almost trees), extending the already known result for ordinary trees. In a generalization of this problem we wish to find a labeling such that vertices at distance two are assigned labels that differ by at least q and the labels of adjacent vertices differ by at least p (where p and q are given positive integers). We denote the minimum number of labels by L(G; p, q) (hence λ(G) = L(G; 2, 1)). We show several hardness results for L(G; p, q) including that for any p > q ≥ 1 there is a λ = λ(p, q) such that deciding if L(G; p, q) ≤ λ(p, q) is NP-complete.

Listing all Minimal Separators of a Graph

An efficient algorithm listing all minimal vertex separators of an undirected graph is given. The algorithm needs polynomial time per separator that is found.

Time‐varying shortest path problems with constraints

We study a new version of the shortest path problem. Let G = (V, E) be a directed graph. Each are e ∈ E has two numbers attached to it: a transit time b(e, u) and a cost c(e, u), which are functions of the departure time u at the beginning vertex of the arc. Moreover, postponement of departure (i.e., waiting) at a vertex may be allowed. The problem is to find the shortest path, i.e., the path with the least possible cost, subject to the constraint that the total traverse time is at most some number T. Three variants of the problem are examined. In the first one, we assume arbitrary waiting times, where waiting at a vertex without any restriction is allowed. In the second variant, we assume zero waiting times, namely, waiting at any vertex is strictly prohibited. Finally, we consider the general case whre there is a vertex-dependent upper bound on the waiting time at each vertex. Several algorithms with pseudopolynomial time complexity are proposed to optimally solve the problems. First, we assume that all transit times b(e, u) are positive integers. In the last section, we show how to include zero transit times.

Approximating Treewidth, Pathwidth, and Minimum Elimination Tree Height

We show how the value of various parameters of graphs connected to sparse matrix factorization and other applications can be approximated using an algorithm of Leighton et al. that finds vertex separators of graphs. The approximate values of the parameters, which include minimum front size, treewidth, pathwidth, and minimum elimination tree height, are no more than O(log n) (minimum front size and treewidth) and O(log2n) (pathwidth and minimum elimination tree height) times the optimal values. In addition we examine the existence of bounded approximation algorithms for the parameters, and show that unless P = NP, there are no absolute approximation algorithms for them.

Finding and counting small induced subgraphs efficiently

Abstract We give two algorithms for listing all simplicial vertices of a graph running in time O (n α ) and O (e 2α/(α+1) )= O (e 1.41 ) , respectively, where n and e denote the number of vertices and edges in the graph and O (n α ) is the time needed to perform a fast matrix multiplication. We present new algorithms for the recognition of diamond-free graphs ( O (n α +e 3/2 ) ), claw-free graphs ( O (e (α+1)/2 )= O (e 1.69 ) ), and K 4 -free graphs ( O (e (α+1)/2 )= O (e 1.69 ) ). Furthermore, we show that counting the number of K 4 s in a graph can be done in time O (e (α+1)/2 ) . For all other graphs on four vertices we can count within O (n α +e 1.69 ) time the number of occurrences as induced subgraph.

On treewidth and minimum fill-in of asteroidal triple-free graphs

We present O(n5R + n3R3) time algorithms to compute the treewidth, pathwidth, minimum fill-in and minimum interval graph completion of asteroidal triple-free graphs, where n is the number of vertices and R is the number of minimal separators of the input graph. This yields polynomial time algorithms for the four NP-complete graph problems on any subclass of the asteroidal triple-free graphs that has a polynomially bounded number of minimal separators, as e.g. cocomparability graphs of bounded dimension and d-trapezoid graphs for any fixed d ⩾ 1.

λ -Coloring of Graphs

A λ-coloring of a graph G is an assignment of colors from the set {0,....,λ} to the vertices of a graph G such that vertices at distance at most two get different colors and adjacent vertices get colors which are at least two apart. The problem of finding λ-colorings with small or optimal λ arises in the context of radio frequency assignment. We show that the problems of finding the minimum λ for planar graphs, bipartite graphs, chordal graphs and split graphs are NP-Complete. We then give approximation algorithms for λ-coloring and compute upperbounds of the best possible λ for outerplanar graphs, planar graphs, graphs of treewidth k, permutation and split graphs. With the exception of the split graphs, all the above bounds for λ are linear in Δ, the maximum degree of the graph. For split graphs, we give a bound of λ ≤1.5+2Δ+2 and show that there are split graphs with λ = Ω(Δ1.5). Similar results are also given for variations of the λ-coloring problem.