Kitty Meeks

The complexity of flood-filling games on graphs

We consider the complexity of problems related to the combinatorial game Free-Flood-It, in which players aim to make a coloured graph monochromatic with the minimum possible number of flooding operations. Although computing the minimum number of moves required to flood an arbitrary graph is known to be NP-hard, we demonstrate a polynomial time algorithm to compute the minimum number of moves required to link each pair of vertices. We apply this result to compute in polynomial time the minimum number of moves required to flood a path, and an additive approximation to this quantity for an arbitrary kxn board, coloured with a bounded number of colours, for any fixed k. On the other hand, we show that, for k>=3, determining the minimum number of moves required to flood a kxn board coloured with at least four colours remains NP-hard.

The complexity of Free-Flood-It on 2íxn boards

We consider the complexity of problems related to the combinatorial game Free-Flood-It, in which players aim to make a coloured graph monochromatic with the minimum possible number of flooding operations. Our main result is that computing the length of an optimal sequence is fixed parameter tractable (with the number of colours as a parameter) when restricted to rectangular 2xn boards. We also show that, when the number of colours is unbounded, the problem remains NP-hard on such boards. These results resolve a question of Clifford, Jalsenius, Montanaro and Sach.

The challenges of unbounded treewidth in parameterised subgraph counting problems

Parameterised subgraph counting problems are the most thoroughly studied topic in the theory of parameterised counting, and there has been significant recent progress in this area. Many of the existing tractability results for parameterised problems which involve finding or counting subgraphs with particular properties rely on bounding the treewidth of these subgraphs in some sense; here, we prove a number of hardness results for the situation in which this bounded treewidth condition does not hold, resulting in dichotomies for some special cases of the general subgraph counting problem. The paper also gives a thorough survey of known results on this subject and the methods used, as well as discussing the relationships both between multicolour and uncoloured versions of subgraph counting problems, and between exact counting, approximate counting and the corresponding decision problems.

The parameterised complexity of counting connected subgraphs and graph motifs

We introduce a class of parameterised counting problems on graphs, p-#Induced Subgraph With Property(\Phi), which generalises a number of problems which have previously been studied. This paper focusses on the case in which \Phi defines a family of graphs whose edge-minimal elements all have bounded treewidth; this includes the special case in which \Phi describes the property of being connected. We show that exactly counting the number of connected induced k-vertex subgraphs in an n-vertex graph is #W[1]-hard, but on the other hand there exists an FPTRAS for the problem; more generally, we show that there exists an FPTRAS for p-#Induced Subgraph With Property(\Phi) whenever \Phi is monotone and all the minimal graphs satisfying \Phi have bounded treewidth. We then apply these results to a counting version of the Graph Motif problem.

Spanning Trees and the Complexity of Flood-Filling Games

We consider problems related to the combinatorial game (Free-) Flood-It, in which players aim to make a coloured graph monochromatic with the minimum possible number of flooding operations. We show that the minimum number of moves required to flood any given graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T. This result is then applied to give two polynomial-time algorithms for flood-filling problems. Firstly, we can compute in polynomial time the minimum number of moves required to flood a graph with only a polynomial number of connected subgraphs. Secondly, given any coloured connected graph and a subset of the vertices of bounded size, the number of moves required to connect this subset can be computed in polynomial time.

Some Hard Families of Parameterized Counting Problems

We consider parameterized subgraph counting problems of the following form: given a graph G, how many k-tuples of its vertices induce a subgraph with a given property? A number of such problems are known to be #W[1]-complete; here, we substantially generalize some of these existing results by proving hardness for two large families of such problems. We demonstrate that it is #W[1]-hard to count the number of k-vertex subgraphs having any property where the number of distinct edge densities of labeled subgraphs that satisfy the property is o(k2). In the special case in which the property in question depends only on the number of edges in the subgraph, we give a strengthening of this result, which leads to our second family of hard problems.

The parameterised complexity of list problems on graphs of bounded treewidth

We consider the parameterised complexity of several list problems on graphs, with parameter treewidth or pathwidth. In particular, we show that List Edge Chromatic Number and List Total Chromatic Number are fixed parameter tractable, parameterised by treewidth, whereas List Hamilton Path is W[1]-hard, even parameterised by pathwidth. These results resolve two open questions of Fellows, Fomin, Lokshtanov, Rosamond, Saurabh, Szeider and Thomassen (2011).

Spanning trees and the complexity of flood-filling games

We consider problems related to the combinatorial game (Free-) Flood-It, in which players aim to make a coloured graph monochromatic with the minimum possible number of flooding operations. We show that the minimum number of moves required to flood any given graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T. This result is then applied to give two polynomial-time algorithms for flood-filling problems. Firstly, we can compute in polynomial time the minimum number of moves required to flood a graph with only a polynomial number of connected subgraphs. Secondly, given any coloured connected graph and a subset of the vertices of bounded size, the number of moves required to connect this subset can be computed in polynomial time.

The parameterised complexity of counting even and odd induced subgraphs

We consider the problem of counting, in a given graph, the number of induced k-vertex subgraphs which have an even number of edges, and also the complementary problem of counting the k-vertex induced subgraphs having an odd number of edges. We demonstrate that both problems are #W[1]-hard when parameterised by k, in fact proving a somewhat stronger result about counting subgraphs with a property that only holds for some subset of k-vertex subgraphs which have an even (respectively odd) number of edges. On the other hand, we show that each of the problems admits an FPTRAS. These approximation schemes are based on a surprising structural result, which exploits ideas from Ramsey theory.

Deleting Edges to Restrict the Size of an Epidemic: A New Application for Treewidth

Motivated by applications in network epidemiology, we consider the problem of determining whether it is possible to delete at most k edges from a given input graph of small treewidth so that the maximum component size in the resulting graph is at most h. While this problem is NP-complete in general, we provide evidence that many of the real-world networks of interest are likely to have small treewidth, and we describe an algorithm which solves the problem in time