Erik Jan van Leeuwen

On the complexity of metric dimension

The metric dimension of a graph G is the size of a smallest subset L⊆V(G) such that for any x,y∈V(G) there is a z∈L such that the graph distance between x and z differs from the graph distance between y and z. Even though this notion has been part of the literature for almost 40 years, the computational complexity of determining the metric dimension of a graph is still very unclear. Essentially, we only know the problem to be NP-hard for general graphs, to be polynomial-time solvable on trees, and to have a logn-approximation algorithm for general graphs. In this paper, we show tight complexity boundaries for the Metric Dimension problem. We achieve this by giving two complementary results. First, we show that the Metric Dimension problem on bounded-degree planar graphs is NP-complete. Then, we give a polynomial-time algorithm for determining the metric dimension of outerplanar graphs.

PTAS for weighted set cover on unit squares

We study the planar version of Minimum-Weight Set Cover, where one has to cover a given set of points with a minimum-weight subset of a given set of planar objects. For the unit-weight case, one PTAS (on disks) is known. For arbitrary weights however, the problem appears much harder, and in particular no PTASs are known. We present the first PTAS for Weighted Geometric Set Cover on planar objects, namely on axis-parallel unit squares. By extending the algorithm, we also obtain a PTAS for Minimum-Weight Dominating Set on intersection graphs of unit squares and Geometric Budgeted Maximum Coverage on unit squares. The running time of the developed algorithms is optimal under the exponential time hypothesis. We also show inapproximability results for Geometric Set Cover on various object shapes that are more general than unit squares.

Parameterized complexity of firefighting revisited

The Firefighter problem is to place firefighters on the vertices of a graph to prevent a fire with known starting point from lighting up the entire graph. In each time step, a firefighter may be permanently placed on an unburned vertex and the fire spreads to its neighborhood in the graph in so far no firefighters are protecting those vertices. The goal is to let as few vertices burn as possible. This problem is known to be NP-complete, even when restricted to bipartite graphs or to trees of maximum degree three. Initial study showed the Firefighter problem to be fixed-parameter tractable on trees in various parameterizations. We complete these results by showing that the problem is in FPT on general graphs when parameterized by the number of burned vertices, but has no polynomial kernel on trees, resolving an open problem. Conversely, we show that the problem is W[1]-hard when parameterized by the number of unburned vertices, even on bipartite graphs. For both parameterizations, we additionally give refined algorithms on trees, improving on the running times of the known algorithms.

Subexponential-Time Parameterized Algorithm for Steiner Tree on Planar Graphs

The well-known bidimensionality theory provides a method for designing fast, subexponential-time parameterized algorithms for a vast number of NP-hard problems on sparse graph classes such as planar graphs, bounded genus graphs, or, more generally, graphs with a fixed excluded minor. However, in order to apply the bidimensionality framework the considered problem needs to fulfill a special density property. Some well-known problems do not have this property, unfortunately, with probably the most prominent and important example being the Steiner Tree problem. Hence the question whether a subexponential-time parameterized algorithm for Steiner Tree on planar graphs exists has remained open. In this paper, we answer this question positively and develop an algorithm running in O(2^{O((k log k)^{2/3})}n) time and polynomial space, where k is the size of the Steiner tree and n is the number of vertices of the graph. nOur algorithm does not rely on tools from bidimensionality theory or graph minors theory, apart from Bakers classical approach. Instead, we introduce new tools and concepts to the study of the parameterized complexity of problems on sparse graphs.

Domination when the stars are out

We algorithmize the recent structural characterization for claw-free graphs by Chudnovsky and Seymour. Building on this result, we show that Dominating Set on claw-free graphs is (i) fixed-parameter tractable and (ii) even possesses a polynomial kernel. To complement these results, we establish that Dominating Set is not fixed-parameter tractable on the slightly larger class of graphs that exclude K1,4 as an induced subgraph. Our results provide a dichotomy for Dominating Set in K1,l-free graphs and show that the problem is fixed-parameter tractable if and only if l ≤ 3. Finally, we show that our algorithmization can also be used to show that the related Connected Dominating Set problem is fixed-parameter tractable on claw-free graphs.

Reducing a target interval to a few exact queries

Many combinatorial problems involving weights can be formulated as a so-called ranged problem. That is, their input consists of a universe u, a (succinctly-represented) set family mathcal{f} subseteq 2^{u}f?2 u mathcal{f} subseteq 2^{u}, a weight function ?:u?{1,…,n}, and integers 0?=?l?=?u?=?8. Then the problem is to decide whether there is an x in mathcal{f}x?fx in mathcal{f} such that l?=? e?x ?(e)?=?u. Well-known examples of such problems include knapsack, subset sum, maximum matching, and traveling salesman. In this paper, we develop a generic method to transform a ranged problem into an exact problem (i.e. A ranged problem for which l?=?u). We show that our method has several intriguing applications in exact exponential algorithms and parameterized complexity, namely: , in exact exponential algorithms, we present new insight into whether subset sum and knapsack have efficient algorithms in both time and space. In particular, we show that the time and space complexity of subset sum and knapsack are equivalent up to a small polynomial factor in the input size. We also give an algorithm that solves sparse instances of knapsack efficiently in terms of space and time. In parameterized complexity, we present the first kernelization results on weighted variants of several well-known problems. In particular, we show that weighted variants of vertex cover and dominating set, traveling salesman, and knapsack all admit polynomial randomized turing kernels when parameterized by |u|. Curiously, our method relies on a technique more commonly found in approximation algorithms.

Faster algorithms on branch and clique decompositions

We combine two techniques recently introduced to obtain faster dynamic programming algorithms for optimization problems on graph decompositions. The unification of generalized fast subset convolution and fast matrix multiplication yields significant improvements to the running time of previous algorithms for several optimization problems. As an example, we give an O*(3ω/2k) time algorithm for Minimum Dominating Set on graphs of branchwidth k, improving on the previous O*(4k) algorithm. Here ω is the exponent in the running time of the best matrix multiplication algorithm (currently ω < 2.376). For graphs of cliquewidth k, we improve from O*(8k) to O*(4k). We also obtain an algorithm for counting the number of perfect matchings of a graph, given a branch decomposition of width k, that runs in time O*(2ω/2k). Generalizing these approaches, we obtain faster algorithms for all so-called [ρ, σ]-domination problems on branch decompositions if ρ and ρ are finite or cofinite. The algorithms presented in this paper either attain or are very close to natural lower bounds for these problems.

Making life easier for firefighters

Being a firefighter is a tough job, especially when tight city budgets do not allow enough firefighters to be on duty when a fire starts. This is formalized in the Firefighter problem, which aims to save as many vertices of a graph as possible from a fire that starts in a vertex and spreads through the graph. In every time step, a single additional firefighter may be placed on a vertex, and the fire advances to each vertex in its neighborhood that is not protected by a firefighter. The problem is notoriously hard: it is NP-hard even when the input graph is a bipartite graph or a tree of maximum degree 3, it is W[1]-hard when parameterized by the number of saved vertices, and it is NP-hard to approximate within n1−e for any e>0. We aim to simplify the task of a firefighter by providing algorithms that show him/her how to efficiently fight fires in certain types of networks. We show that Firefighter can be solved in polynomial time on various well-known graph classes, including interval graphs, split graphs, permutation graphs, and Pk-free graphs for fixed k. On the negative side, we show that the problem remains NP-hard on unit disk graphs.

Induced disjoint paths in AT-Free graphs

Paths P1,…,Pk in a graph G=(V,E) are said to be mutually induced if for any 1≤i<j≤k, Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to test whether a graph G with k pairs of specified vertices (si,ti) contains k mutually induced paths Pi such that Pi connects si and ti for i=1,…,k. This problem is known to be NP-complete already for k=2. We prove that it can be solved in polynomial time for AT-free graphs even when k is part of the input. As a consequence, the problem of testing whether a given AT-free graph contains some graph H as an induced topological minor admits a polynomial-time algorithm, as long as H is fixed; we show that such an algorithm is essentially optimal by proving that the problem is W[1]-hard, even on a subclass of AT-free graphs, namely cobipartite graphs, when parameterized by |VH|. We also show that the problems k-in-a-Path and k-in-a-Tree can be solved in polynomial time, even when k is part of the input. These problems are to test whether a graph contains an induced path or induced tree, respectively, spanning k given vertices.

Complexity results for the spanning tree congestion problem

We study the problem of determining the spanning tree congestion of a graph. We present some sharp contrasts in the complexity of this problem. First, we show that for every fixed k and d the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for graphs of degree at most d. In contrast, if we allow only one vertex of unbounded degree, the problem immediately becomes NP-complete for any fixed k ≥ 10. For very small values of k however, the problem becomes polynomially solvable. We also show that it is NP-hard to approximate the spanning tree congestion within a factor better than 11/10. On planar graphs, we prove the problem is NP-hard in general, but solvable in linear time for fixed k.