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 ...

We give a kernel with \(O(k^7)\) vertices for Trivially Perfect Editing, the problem of adding or removing at most k edges in order to make a given graph trivially perfect. This answers in affirmative an open question posed by Nastos and Gao (Soc Netw 35(3):439–450, 2013), and by Liu et al. (Tsinghua Sci Technol 19(4):346–357, 2014). Our general technique implies also the existence ...

We study the following problem: preprocess a set \(\mathcal {O}\) of objects into a data structure that allows us to efficiently report all pairs of objects from \(\mathcal {O}\) that intersect inside an axis-aligned query range \({Q}\). We present data structures of size \(O(n\cdot {{\mathrm{polylog\,}}}n)\) and with query time \(O((k+1)\cdot {{\mathrm{polylog\,}}}n)\) time, where ...

The maximum cut problem in graphs and its generalizations are fundamental combinatorial problems. Several of these cut problems were recently shown to be fixed-parameter tractable and admit polynomial kernels when parameterized above the tight lower bound measured by the size and order of the graph. In this paper we continue this line of research and considerably improve several of ...

In the Token Swapping problem we are given a graph with a token placed on each vertex. Each token has exactly one destination vertex, and we try to move all the tokens to their destinations, using the minimum number of swaps, i.e., operations of exchanging the tokens on two adjacent vertices. As the main result of this paper, we show that Token Swapping is \(W[1]\)-hard ...

The authors regret the following error in their article “A connection between sports and matroids: How many teams can we beat?” (Algorithmica, doi: 10.1007/s00453-016-0256-2), considering the computational complexity of the problem MinStanding(S). In Theorem 3 of our paper [4], we erroneously claimed a \(\mathsf {W}[1]\)-hardness result to hold even for the case where the ...

Island models denote a distributed system of evolutionary algorithms which operate independently, but occasionally share their solutions with each other along the so-called migration topology. We investigate the impact of the migration topology by introducing a simplified island model with behavior similar to \(\lambda \) islands optimizing the so-called Maze fitness function ...

We consider two-player zero-sum stochastic mean payoff games with perfect information. We show that any such game, with a constant number of random positions and polynomially bounded positive transition probabilities, admits a polynomial time approximation scheme, both in the relative and absolute sense.

Escaping local optima is one of the major obstacles to function optimisation. Using the metaphor of a fitness landscape, local optima correspond to hills separated by fitness valleys that have to be overcome. We define a class of fitness valleys of tunable difficulty by considering their length, representing the Hamming path between the two optima and their depth, the drop in ...

The hybridization number problem requires us to embed a set of binary rooted phylogenetic trees into a binary rooted phylogenetic network such that the number of nodes with indegree two is minimized. However, from a biological point of view accurately inferring the root location in a phylogenetic tree is notoriously difficult and poor root placement can artificially inflate the ...

A secure set S in a graph is defined as a set of vertices such that for any \(X\subseteq S\) the majority of vertices in the neighborhood of X belongs to S. It is known that deciding whether a set S is secure in a graph is \(\mathrm {\text {co-}NP}\)-complete. However, it is still open how this result contributes to the actual complexity of deciding whether for a given graph G and ...

We study the fundamental problem of the exact and efficient generation of random values from a finite and discrete probability distribution. Suppose that we are given n distinct events with associated probabilities \(p_1, \dots , p_n\). First, we consider the problem of sampling from the distribution where the i-th event has probability proportional to \(p_i\). Second, we study the ...

The field of a priori optimization is an interesting subfield of stochastic combinatorial optimization that is well suited for routing problems. In this setting, there is a probability distribution over active sets, vertices that have to be visited. For a fixed tour, the solution on an active set is obtained by restricting the solution on the active set. In the well-studied a ...

In the Independent Set of Convex Polygons problem we are given a set of weighted convex polygons in the plane and we want to compute a maximum weight subset of non-overlapping polygons. This is a very natural and well-studied problem with applications in many different areas. Unfortunately, there is a very large gap between the known upper and lower bounds for this problem. The ...

For a graph G, a function \(\psi \) is called a bar visibility representation of G when for each vertex \(v \in V(G)\), \(\psi (v)\) is a horizontal line segment (bar) and \(uv \in E(G)\) if and only if there is an unobstructed, vertical, \(\varepsilon \)-wide line of sight between \(\psi (u)\) and \(\psi (v)\). Graphs admitting such representations are well understood (via simple ...

In the Edge Bipartization problem one is given an undirected graph G and an integer k, and the question is whether k edges can be deleted from G so that it becomes bipartite. Guo et al. (J Comput Syst Sci 72(8):1386–1396, 2006) proposed an algorithm solving this problem in time \(\mathcal {O}(2^k\cdot {m}^2)\); today, this algorithm is a textbook example of an application of the ...

Given two graphs G and H, we define \(\mathsf{v}\hbox {-}\mathsf{cover}_{H}(G)\) (resp. \(\mathsf{e}\hbox {-}\mathsf{cover}_{H}(G)\)) as the minimum number of vertices (resp. edges) whose removal from G produces a graph without any minor isomorphic to H. Also \(\mathsf{v}\hbox {-}\mathsf{pack}_{H}(G)\) (resp. \(\mathsf{e}\hbox {-}\mathsf{pack}_{H}(G)\)) is the maximum number of ...

We investigate the following greedy approach to attack linear programs of type \(\max \{1^{T} x\mid l\le Ax\le u\}\) where A has entries in \(\{-1,0,1\}\): The greedy algorithm starts with a feasible solution x and, iteratively, chooses an improving variable and raises it until some constraint becomes tight. In the special case, where A is the edge-path incidence matrix of some ...