arXiv subjects

Kitty Meeks

Publications and source records attributed to Kitty Meeks.

At least 37 records · Page 2Linked to original sources

arXiv2022

Making Life More Confusing for Firefighters

It is well known that fighting a fire is a hard task. The Firefighter problem asks how to optimally deploy firefighters to defend the vertices of a graph from a fire. This problem is NP-Complete on all but a few classes of graphs. Thankfully, firefighters do not have to work alone, and are often aided by the efforts of good natured civilians who slow the spread of a fire by maintaining firebreaks when they are able. We will show that this help, although well-intentioned, unfortunately makes the optimal deployment of firefighters an even harder problem. To model this scenario we introduce the Temporal Firefighter problem, an extension of Firefighter to temporal graphs. We show that Temporal Firefighter is also NP-Complete, and remains so on all but one of the underlying classes of graphs on which Firefighter is known to have polynomial time solutions. This motivates us to explore making use of the temporal structure of the graph in our search for tractability, and we conclude by presenting an FPT algorithm for Temporal Firefighter with respect to the temporal graph parameter vertex-interval-membership-width.

cs.DS↗

arXiv2022

Approximately counting and sampling small witnesses using a colourful decision oracle

In this paper, we design efficient algorithms to approximately count the number of edges of a given $k$-hypergraph, and to sample an approximately uniform random edge. The hypergraph is not given explicitly, and can be accessed only through its colourful independence oracle: The colourful independence oracle returns yes or no depending on whether a given subset of the vertices contains an edge that is colourful with respect to a given vertex-colouring. Our results extend and/or strengthen recent results in the graph oracle literature due to Beame et al. (ITCS 2018), Dell and Lapinskas (STOC 2018), and Bhattacharya et al. (ISAAC 2019). Our results have consequences for approximate counting/sampling: We can turn certain kinds of decision algorithms into approximate counting/sampling algorithms without causing much overhead in the running time. We apply this approximate-counting/sampling-to-decision reduction to key problems in fine-grained complexity (such as $k$-SUM, $k$-OV and weighted $k$-Clique) and parameterised complexity (such as induced subgraphs of size $k$ or weight-$k$ solutions to CSPs).

cs.DS↗

arXiv2021

Edge exploration of temporal graphs

We introduce a natural temporal analogue of Eulerian circuits and prove that, in contrast with the static case, it is NP-hard to determine whether a given temporal graph is temporally Eulerian even if strong restrictions are placed on the structure of the underlying graph and each edge is active at only three times. However, we do obtain an FPT-algorithm with respect to a new parameter called interval-membership-width which restricts the times assigned to different edges; we believe that this parameter will be of independent interest for other temporal graph problems. Our techniques also allow us to resolve two open question of Akrida, Mertzios and Spirakis [CIAC 2019] concerning a related problem of exploring temporal stars. Furthermore, we introduce a vertex-variant of interval-membership-width (which can be arbitrarily larger than its edge-counterpart) and use it to obtain an FPT-time algorithm for a natural vertex-exploration problem that remains hard even when interval-membership-width is bounded.

cs.CC↗

arXiv2021

Efficiently Enumerating Hitting Sets of Hypergraphs Arising in Data Profiling

The transversal hypergraph problem is the task of enumerating the minimal hitting sets of a hypergraph. It is a long-standing open question whether this can be done in output-polynomial time. For hypergraphs whose solutions have bounded size, Eiter and Gottlob [SICOMP 1995] gave an algorithm that runs in output-polynomial time, but whose space requirement also scales with the output size. We improve this to polynomial delay and polynomial space. More generally, we present an algorithm that on $n$-vertex, $m$-edge hypergraphs has delay $O(m^{k^*+1} n^2)$ and uses $O(mn)$ space, where $k^*$ is the maximum size of any minimal hitting set. Our algorithm is oblivious to $k^*$, a quantity that is hard to compute or even approximate. Central to our approach is the extension problem for minimal hitting sets, deciding for a set $X$ of vertices whether it is contained in any solution. With $|X|$ as parameter, we show that this is one of the first natural problems to be complete for the complexity class $W[3]$. We give an algorithm for the extension problem running in time $O(m^{|X|+1} n)$. We also prove a conditional lower bound under the Strong Exponential Time Hypothesis, showing that this is close to optimal. We apply our enumeration method to the discovery problem of minimal unique column combinations from data profiling. Our empirical evaluation suggests that the algorithm outperforms its worst-case guarantees on hypergraphs stemming from real-world databases.

cs.DS↗

arXiv2021

Deleting edges to restrict the size of an epidemic in temporal networks

Spreading processes on graphs are a natural model for a wide variety of real-world phenomena, including information spread over social networks and biological diseases spreading over contact networks. Often, the networks over which these processes spread are dynamic in nature, and can be modeled with temporal graphs. Here, we study the problem of deleting edges from a given temporal graph in order to reduce the number of vertices (temporally) reachable from a given starting point. This could be used to control the spread of a disease, rumour, etc. in a temporal graph. In particular, our aim is to find a temporal subgraph in which a process starting at any single vertex can be transferred to only a limited number of other vertices using a temporally-feasible path. We introduce a natural edge-deletion problem for temporal graphs and provide positive and negative results on its computational complexity and approximability.

cs.DS↗

arXiv2021

The Interactive Sum Choice Number of Graphs

We introduce a variant of the well-studied sum choice number of graphs, which we call the interactive sum choice number. In this variant, we request colours to be added to the vertices' colour-lists one at a time, and so we are able to make use of information about the colours assigned so far to determine our future choices. The interactive sum choice number cannot exceed the sum choice number and we conjecture that, except in the case of complete graphs, the interactive sum choice number is always strictly smaller than the sum choice number. In this paper we provide evidence in support of this conjecture, demonstrating that it holds for a number of graph classes, and indeed that in many cases the difference between the two quantities grows as a linear function of the number of vertices.

cs.DM↗

arXiv2020

Improved inference for areal unit count data using graph-based optimisation

Spatial correlation in areal unit count data is typically modelled by a set of random effects that are assigned a conditional autoregressive (CAR) prior distribution. The spatial correlation structure implied by this model depends on a binary neighbourhood matrix, where two random effects are assumed to be partially autocorrelated if their areal units share a common border, and are conditionally independent otherwise. This paper proposes a novel graph-based optimisation algorithm for estimating the neighbourhood matrix from the data, by viewing the areal units as the vertices of the graph and the neighbour relations as the set of edges. The superiority of our methodology compared to using the border sharing rule is comprehensively evidenced by simulation, before the method is applied to a new respiratory disease surveillance study in the Greater Glasgow and Clyde Health board in Scotland between 2011 and 2017.

stat.ME↗

arXiv2020

Assigning times to minimise reachability in temporal graphs

Temporal graphs (in which edges are active at specified times) are of particular relevance for spreading processes on graphs, e.g.~the spread of disease or dissemination of information. Motivated by real-world applications, modification of static graphs to control this spread has proven a rich topic for previous research. Here, we introduce a new type of modification for temporal graphs: the number of active times for each edge is fixed, but we can change the relative order in which (sets of) edges are active. We investigate the problem of determining an ordering of edges that minimises the maximum number of vertices reachable from any single starting vertex; epidemiologically, this corresponds to the worst-case number of vertices infected in a single disease outbreak. We study two versions of this problem, both of which we show to be $\NP$-hard, and identify cases in which the problem can be solved or approximated efficiently.

cs.CC↗

arXiv2020

Two dichotomies for model-checking in multi-layer structures

Multi-layer graphs can capture qualitatively different types of connection between entities, and networks of this kind are prevalent in biological and social systems: for example, a social contact network typically involves both virtual and face-to-face interactions between individuals. Since each layer is likely to exhibit stronger and/or more easily identifiable structurally properties than the overall system, it is natural to ask whether we can exploit the structural properties of individual layers to solve NP-hard problems efficiently on the overall network. In this paper we provide a complete characterisation of the structural properties required in each layer to guarantee the existence of an FPT algorithm to solve problems definable in either first-order or monadic second-order logic on the overall system, subject to the assumption that the structural properties are preserved under deletion of vertices and/or edges.

cs.DM↗

arXiv2019

The parameterised complexity of computing the maximum modularity of a graph

The maximum modularity of a graph is a parameter widely used to describe the level of clustering or community structure in a network. Determining the maximum modularity of a graph is known to be NP-complete in general, and in practice a range of heuristics are used to construct partitions of the vertex-set which give lower bounds on the maximum modularity but without any guarantee on how close these bounds are to the true maximum. In this paper we investigate the parameterised complexity of determining the maximum modularity with respect to various standard structural parameterisations of the input graph G. We show that the problem belongs to FPT when parameterised by the size of a minimum vertex cover for G, and is solvable in polynomial time whenever the treewidth or max leaf number of G is bounded by some fixed constant; we also obtain an FPT algorithm, parameterised by treewidth, to compute any constant-factor approximation to the maximum modularity. On the other hand we show that the problem is W[1]-hard (and hence unlikely to admit an FPT algorithm) when parameterised simultaneously by pathwidth and the size of a minimum feedback vertex set.

cs.DS↗

arXiv2019

Extremal properties of flood-filling games

The problem of determining the number of "flooding operations" required to make a given coloured graph monochromatic in the one-player combinatorial game Flood-It has been studied extensively from an algorithmic point of view, but basic questions about the maximum number of moves that might be required in the worst case remain unanswered. We begin a systematic investigation of such questions, with the goal of determining, for a given graph, the maximum number of moves that may be required, taken over all possible colourings. We give several upper and lower bounds on this quantity for arbitrary graphs and show that all of the bounds are tight for trees; we also investigate how much the upper bounds can be improved if we restrict our attention to graphs with higher edge-density.

math.CO↗

arXiv2018

Solving Hard Stable Matching Problems Involving Groups of Similar Agents

Many important stable matching problems are known to be NP-hard, even when strong restrictions are placed on the input. In this paper we seek to identify structural properties of instances of stable matching problems which will allow us to design efficient algorithms using elementary techniques. We focus on the setting in which all agents involved in some matching problem can be partitioned into k different types, where the type of an agent determines his or her preferences, and agents have preferences over types (which may be refined by more detailed preferences within a single type). This situation would arise in practice if agents form preferences solely based on some small collection of agents' attributes. We also consider a generalisation in which each agent may consider some small collection of other agents to be exceptional, and rank these in a way that is not consistent with their types; this could happen in practice if agents have prior contact with a small number of candidates. We show that (for the case without exceptions), several well-studied NP-hard stable matching problems including Max SMTI (that of finding the maximum cardinality stable matching in an instance of stable marriage with ties and incomplete lists) belong to the parameterised complexity class FPT when parameterised by the number of different types of agents needed to describe the instance. For Max SMTI this tractability result can be extended to the setting in which each agent promotes at most one `exceptional' candidate to the top of his/her list (when preferences within types are not refined), but the problem remains NP-hard if preference lists can contain two or more exceptions and the exceptional candidates can be placed anywhere in the preference lists, even if the number of types is bounded by a constant.

cs.GT↗

arXiv2018

Randomised enumeration of small witnesses using a decision oracle

Many combinatorial problems involve determining whether a universe of $n$ elements contains a witness consisting of $k$ elements which have some specified property. In this paper we investigate the relationship between the decision and enumeration versions of such problems: efficient methods are known for transforming a decision algorithm into a search procedure that finds a single witness, but even finding a second witness is not so straightforward in general. We show that, if the decision version of the problem can be solved in time $f(k) \cdot poly(n)$, there is a randomised algorithm which enumerates all witnesses in time $e^{k + o(k)} \cdot f(k) \cdot poly(n) \cdot N$, where $N$ is the total number of witnesses. If the decision version of the problem is solved by a randomised algorithm which may return false negatives, then the same method allows us to output a list of witnesses in which any given witness will be included with high probability. The enumeration algorithm also gives rise to an efficient algorithm to count the total number of witnesses when this number is small.

cs.DS↗

arXiv2017

Deleting edges to restrict the size of an epidemic

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 resulting graph avoids a set $\mathcal{F}$ of forbidden subgraphs; of particular interest is the problem of determining whether it is possible to delete at most $k$ edges so that the resulting graph has no connected component of more than $h$ vertices, as this bounds the worst-case size of an epidemic. While even this special case of the problem is NP-complete in general (even when $h=3$), 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 general problem in time \genruntime ~on an input graph having $n$ vertices and whose treewidth is bounded by a fixed constant $w$, if each of the subgraphs we wish to avoid has at most $r$ vertices. For the special case in which we wish only to ensure that no component has more than $h$ vertices, we improve on this to give an algorithm running in time $O((wh)^{2w}n)$, which we have implemented and tested on real datasets based on cattle movements.

cs.DS↗

arXiv2017

On the complexity of finding and counting solution-free sets of integers

Given a linear equation $\mathcal{L}$, a set $A$ of integers is $\mathcal{L}$-free if $A$ does not contain any `non-trivial' solutions to $\mathcal{L}$. This notion incorporates many central topics in combinatorial number theory such as sum-free and progression-free sets. In this paper we initiate the study of (parameterised) complexity questions involving $\mathcal{L}$-free sets of integers. The main questions we consider involve deciding whether a finite set of integers $A$ has an $\mathcal{L}$-free subset of a given size, and counting all such $\mathcal{L}$-free subsets. We also raise a number of open problems.

math.CO↗

arXiv2016

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).

cs.CC↗

arXiv2015

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 the problems of counting even and odd k-vertex induced subgraphs both admit an FPTRAS. These approximation schemes are based on a surprising structural result, which exploits ideas from Ramsey theory.

math.CO↗

arXiv2015

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.

cs.CC↗