On computing the smallest four-coloring of planar graphs and non-self-reducible sets in P

We show that computing the lexicographically first four-coloring for planar graphs is -hard. This result optimally improves upon a result of Khuller and Vazirani who prove this problem NP-hard, and conclude that it is not self-reducible in the sense of Schnorr, assuming P≠NP. We discuss this application to non-self-reducibility and provide a general related result. We also discuss when raising a problem's NP-hardness lower bound to -hardness can be valuable.     

The complexity for partitioning graphs by monochromatic trees,cycles and paths

Let G be an edge-coloured graph. We show in this paper that it is NP-hard to find the minimum number of vertex disjoint monochromatic trees which cover the vertices of the graph G. We also show that there is no constant factor approximation algorithm for the problem unless P?=?NP. The same results hold for the problem of finding the minimum number of vertex disjoint monochromatic cycles (paths, respectively) which cover the vertices of the graph.     

Vertex rankings of chordal graphs and weighted trees

In this paper we consider the vertex ranking problem of weighted trees. We show that this problem is strongly NP-hard. We also give a polynomial-time reduction from the problem of vertex ranking of weighted trees to the vertex ranking of (simple) chordal graphs, which proves that the latter problem is NP-hard. In this way we solve an open problem of Aspvall and Heggernes. We use this reduction and the algorithm of Bodlaender et al.'s for vertex ranking of partial k-trees to give an exact polynomial-time algorithm for vertex ranking of a tree with bounded and integer valued weight functions. This algorithm serves as a procedure in designing a PTAS for weighted vertex ranking problem of trees with bounded weight functions.     

APX-hardness of domination problems in circle graphs

We show that the problem of finding a minimum dominating set in a circle graph is APX-hard: there is a constant δ>0 such that there is no (1+δ)-approximation algorithm for the minimum dominating set problem on circle graphs unless P=NP. Hence a PTAS for this problem seems unlikely. This hardness result complements the (2+?)-approximation algorithm for the problem [M. Damian, S.V. Pemmaraju, A (2+?)-approximation scheme for minimum domination on circle graphs, J. Algorithms 42 (2) (2002) 255-276].     

Priority algorithms for graph optimization problems

We continue the study of priority or “greedy-like” algorithms as initiated in Borodin et al. (2003) [10] and as extended to graph theoretic problems in Davis and Impagliazzo (2009) [12]. Graph theoretic problems pose some modeling problems that did not exist in the original applications of Borodin et al. and Angelopoulos and Borodin (2002) [3]. Following the work of Davis and Impagliazzo, we further clarify these concepts. In the graph theoretic setting, there are several natural input formulations for a given problem and we show that priority algorithm bounds in general depend on the input formulation. We study a variety of graph problems in the context of arbitrary and restricted priority models corresponding to known “greedy algorithms”.     

Refined memorization for vertex cover

Memorization is a technique which allows to speed up exponential recursive algorithms at the cost of an exponential space complexity. This technique already leads to the currently fastest algorithm for fixed-parameter vertex cover, whose time complexity is O(k^1.2832k^1.5+kn), where n is the number of nodes and k is the size of the vertex cover. Via a refined use of memorization, we obtain an O(k^1.2759k^1.5+kn) algorithm for the same problem. We moreover show how to further reduce the complexity to O(k^1.2745k⁴+kn).     

Parameterized complexity of the induced subgraph problem in directed graphs

In this Letter, we consider the parameterized complexity of the following problem: Given a hereditary property P on digraphs, an input digraph D and a positive integer k, does D have an induced subdigraph on k vertices with property P? We completely characterize hereditary properties for which this induced subgraph problem is W[1]-complete for two classes of directed graphs: general directed graphs and oriented graphs. We also characterize those properties for which the induced subgraph problem is W[1]-complete for general directed graphs but fixed parameter tractable for oriented graphs. These results are among the very few parameterized complexity results on directed graphs.     

On the parameterized complexity of some optimization problems related to multiple-interval graphs

We show that for any constant t≥2, k-Independent Set and k-Dominating Set in t-track interval graphs are W[1]-hard. This settles an open question recently raised by Fellows, Hermelin, Rosamond, and Vialette. We also give an FPT algorithm for k-Clique in t-interval graphs, parameterized by both k and t, with running time , where n is the number of vertices in the graph. This slightly improves the previous FPT algorithm by Fellows, Hermelin, Rosamond, and Vialette. Finally, we use the W[1]-hardness of k-Independent Set in t-track interval graphs to obtain the first parameterized intractability result for a recent bioinformatics problem called Maximal Strip Recovery (MSR). We show that MSR-d is W[1]-hard for any constant d≥4 when the parameter is either the total length of the strips, or the total number of adjacencies in the strips, or the number of strips in the solution.     

Between 2- and 3-colorability

The recognition of 3-colorable graphs is an NP-complete problem, while 2-colorable (i.e., bipartite) graphs can be recognized in polynomial time. To make the complexity gap more precise, we study intermediate graph classes and respective problems. This note proposes a conjecture that separates difficult instances of the problem from polynomially solvable ones and proves the “polynomial” part of the conjecture.     

Complexity of homomorphisms to direct products of graphs

For a graph G, OALG asks whether or not an input graph H together with a partial map g:S→G, S⊆V(H), admits a homomorphism f:H→G such that f|_S=g. We show that for connected graphs G₁, G₂, OAL G₁×G₂ is in P if G₁ and G₂ are trees and NP-complete otherwise.     

Well-balanced orientations of mixed graphs

We show that deciding if a mixed graph has a well-balanced orientation is NP-complete.     

Reachability on prefix-recognizable graphs

We prove that on prefix-recognizable graphs reachability is complete for deterministic exponential time matching the complexity of alternating reachability.     

The complexity of determining the rainbow vertex-connection of a graph

A vertex-colored graph is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The rainbow vertex-connection of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. In this paper, we study the complexity of determining the rainbow vertex-connection of a graph and prove that computing rvc(G) is NP-Hard. Moreover, we show that it is already NP-Complete to decide whether rvc(G)=2. We also prove that the following problem is NP-Complete: given a vertex-colored graph G, check whether the given coloring makes G rainbow vertex-connected.     

Complexity analysis of a decentralised graph colouring algorithm

Colouring a graph with its chromatic number of colours is known to be NP-hard. Identifying an algorithm in which decisions are made locally with no information about the graph's global structure is particularly challenging. In this article we analyse the complexity of a decentralised colouring algorithm that has recently been proposed for channel selection in wireless computer networks.     

Parameterized graph separation problems

We consider parameterized problems where some separation property has to be achieved by deleting as few vertices as possible. The following five problems are studied: delete k   vertices such that (a) each of the given ?$?$ terminals is separated from the others, (b) each of the given ?$?$ pairs of terminals is separated, (c) exactly ?$?$ vertices are cut away from the graph, (d) exactly ?$?$ connected vertices are cut away from the graph, (e) the graph is separated into at least ?$?$ components. We show that if both k   and ?$?$ are parameters, then (a), (b) and (d) are fixed-parameter tractable, while (c) and (e) are W[1]-hard.