05301 Abstracts Collection -- Exact Algorithms and Fixed-Parameter Tractability

From 24.07.05 to 29.07.05, the Dagstuhl Seminar 05301 ``Exact Algorithms and Fixed-Parameter Tractability\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. This is a collection of abstracts of the presentations given during the seminar

Colouring on hereditary graph classes

The graph colouring problems ask if one can assign a colour from a palette of colour to every vertex of a graph so that any two adjacent vertices receive different colours. We call the resulting problem k-Colourability if the palette is of fixed size k, and Chromatic Number if the goal is to minimize the size of the palette. One of the earliest NP-completeness results states that 3-Colourability is NP-complete. Thereafter, numerous studies have been devoted to the graph colouring problems on special graph classes. For a fixed set of graphs H we denote F orb(H) by the set of graphs that exclude any graph H ∈ H as an induced subgraph. In this thesis, we explore the computational complexity of graph colouring problems on F orb(H) for different sets of H.In the first part of this thesis, we study k-Colourability on classes F orb(H) when H contains at most two graphs. We show that 4-Colourability and 5-Colourability are NPcomplete on F orb({P7}) and F orb({P6}), respectively, where Pt denotes a path of order t. These results leave open, for k ≥ 4, only the complexity of k-Colourability on F orb({Pt}) for k = 4 and t = 6. Secondly, we refine our NP-completeness results on k-Colourability to classes F orb({Cs, Pt}), where Cs denotes a cycle of length s. We prove new NP-completeness results for different combinations of values of k, s and t. Furthermore, we consider two common variants of the k-colouring problem, namely the list k-colouring problem and the pre-colouring extension of k-colouring problem. We show that in most cases these problems are also NP-complete on the class F orb({Cs, Pt}). Thirdly, we prove that the set of forbidden induced subgraph that characterizes the class of k-colourable (C4, P6)-free graphs is of finite size. For k ∈ {3, 4}, we obtain an explicit list of forbidden induced subgraphs and the first polynomial certifying algorithms for k-Colourability on F orb({C4, P6}).We also discuss one particular class F orb(H) when the size of H is infinite. We consider the intersection class of F orb({C4, C6, . . .}) and F orb(caps), where a cap is a graph obtained from an induced cycle by adding an additional vertex and making it adjacent to two adjacent vertices on the cycle. Our main result is a polynomial time 3/2-approximation algorithm for Chromatic Number on this class

New Graph Model for Channel Assignment in Ad Hoc Wireless Networks

The channel assignment problem in ad hoc wireless networks is investigated. The problem is to assign channels to hosts in such a way that interference among hosts is eliminated and the total number of channels is minimised. Interference is caused by direct collisions from hosts that can hear each other or indirect collisions from hosts that cannot hear each other, but simultaneously transmit to the same destination. A new class of disk graphs (FDD: interFerence Double Disk graphs) is proposed that include both kinds of interference edges. Channel assignment in wireless networks is a vertex colouring problem in FDD graphs. It is shown that vertex colouring in FDD graphs is NP-complete and the chromatic number of an FDD graph is bounded by its clique number times a constant. A polynomial time approximation algorithm is presented for channel assignment and an upper bound 14 on its performance ratio is obtained. Results from a simulation study reveal that the new graph model can provide a more accurate estimation of the number of channels required for collision avoidance than previous models

Deciding Relaxed Two-Colourability: A Hardness Jump

We study relaxations of proper two-colourings, such that the order of the induced monochromatic components in one (or both) of the colour classes is bounded by a constant. A colouring of a graph G is called (C1, C2)-relaxed if every monochromatic component induced by vertices of the first (second) colour is of order at most C1 (C2, resp.). We prove that the decision problem ‘Is there a (1, C)-relaxed colouring of a given graph G of maximum degree 3?' exhibits a hardness jump in the component order C. In other words, there exists an integer f(3) such that the decision problem is NP-hard for every 2 ≤ C < f(3), while every graph of maximum degree 3 is (1, f(3))-relaxed colourable. We also show f(3) ≤ 22 by way of a quasilinear time algorithm, which finds a (1, 22)-relaxed colouring of any graph of maximum degree 3. Both the bound on f(3) and the running time greatly improve earlier results. We also study the symmetric version, that is, when C1 = C2, of the relaxed colouring problem and make the first steps towards establishing a similar hardness jum

Structural solutions to maximum independent set and related problems

In this thesis, we study some fundamental problems in algorithmic graph theory. Most natural problems in this area are hard from a computational point of view. However, many applications demand that we do solve such problems, even if they are intractable. There are a number of methods in which we can try to do this: 1) We may use an approximation algorithm if we do not necessarily require the best possible solution to a problem. 2) Heuristics can be applied and work well enough to be useful for many applications. 3) We can construct randomised algorithms for which the probability of failure is very small. 4) We may parameterize the problem in some way which limits its complexity. In other cases, we may also have some information about the structure of the instances of the problem we are trying to solve. If we are lucky, we may and that we can exploit this extra structure to find efficient ways to solve our problem. The question which arises is - How far must we restrict the structure of our graph to be able to solve our problem efficiently? In this thesis we study a number of problems, such as Maximum Indepen- dent Set, Maximum Induced Matching, Stable-II, Efficient Edge Domina- tion, Vertex Colouring and Dynamic Edge-Choosability. We try to solve problems on various hereditary classes of graphs and analyse the complexity of the resulting problem, both from a classical and parameterized point of view

Star Colouring of Bounded Degree Graphs and Regular Graphs

A $k$-star colouring of a graph $G$ is a function $f:V(G)\to\{0,1,\dots,k-1\}$ such that $f(u)\neq f(v)$ for every edge $uv$ of $G$, and every bicoloured connected subgraph of $G$ is a star. The star chromatic number of $G$, $\chi_s(G)$, is the least integer $k$ such that $G$ is $k$-star colourable. We prove that $\chi_s(G)\geq \lceil (d+4)/2\rceil$ for every $d$-regular graph $G$ with $d\geq 3$. We reveal the structure and properties of even-degree regular graphs $G$ that attain this lower bound. The structure of such graphs $G$ is linked with a certain type of Eulerian orientations of $G$. Moreover, this structure can be expressed in the LC-VSP framework of Telle and Proskurowski (SIDMA, 1997), and hence can be tested by an FPT algorithm with the parameter either treewidth, cliquewidth, or rankwidth. We prove that for $p\geq 2$, a $2p$-regular graph $G$ is $(p+2)$-star colourable only if $n:=|V(G)|$ is divisible by $(p+1)(p+2)$. For each $p\geq 2$ and $n$ divisible by $(p+1)(p+2)$, we construct a $2p$-regular Hamiltonian graph on $n$ vertices which is $(p+2)$-star colourable. The problem $k$-STAR COLOURABILITY takes a graph $G$ as input and asks whether $G$ is $k$-star colourable. We prove that 3-STAR COLOURABILITY is NP-complete for planar bipartite graphs of maximum degree three and arbitrarily large girth. Besides, it is coNP-hard to test whether a bipartite graph of maximum degree eight has a unique 3-star colouring up to colour swaps. For $k\geq 3$, $k$-STAR COLOURABILITY of bipartite graphs of maximum degree $k$ is NP-complete, and does not even admit a $2^{o(n)}$-time algorithm unless ETH fails