We study algorithmic properties of the graph class \({\textsc {Chordal}}{-ke}\), that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of fill-in at most k. It appears that a number of fundamental intractable optimization problems being parameterized by k admit subexponential algorithms on graphs from \({\textsc {Chordal}}{-ke}\). More

Let P be a set of n colored points in the d-dimensional Euclidean space. Introduced by Hart (1968), a consistent subset of P, is a set \(S\subseteq P\) such that for every point p in \(P {\setminus } S\), the closest point of p in S has the same color as p. The consistent subset problem is to find a consistent subset of P with minimum cardinality. This problem is known to be NP-complete even for two-colored

We develop new approximation algorithms for classical graph and set problems in the RAM model under space constraints. As one of our main results, we devise an algorithm for \(d\text {-}\textsc {Hitting Set}{}\) that runs in time \(n^{{{\,\mathrm{O}\,}}{(d^2 + (d / \epsilon ))}}\), uses \({{\,\mathrm{O}\,}}{((d^2 + (d / \epsilon ))\log {n})}\) bits of space, and achieves an approximation ratio of \({{\

In this paper, we study the convex-straight-skeleton Voronoi diagrams of line segments and convex polygons. We explore the combinatorial complexity of these diagrams, and provide efficient algorithms for computing compact representations of them.

We consider the following control problem on fair allocation of indivisible goods. Given a set I of items and a set of agents, each having strict linear preferences over the items, we ask for a minimum subset of the items whose deletion guarantees the existence of a proportional allocation in the remaining instance; we call this problem Proportionality by Item Deletion (PID). Our main result is a polynomial-time

A graph is distance-hereditary if for any pair of vertices, their distance in every connected induced subgraph containing both vertices is the same as their distance in the original graph. The Distance-Hereditary Vertex Deletion problem asks, given a graph G on n vertices and an integer k, whether there is a set S of at most k vertices in G such that \(G-S\) is distance-hereditary. This problem is

The problem of parameterized range majority asks us to preprocess a string of length n such that, given the endpoints of a range, one can quickly find all the distinct elements whose relative frequencies in that range are more than a threshold \(\tau\). This is a more tractable version of the classical problem of finding the range mode, which is unlikely to be solvable in polylogarithmic time and linear

The rooted triplet distance measures the structural dissimilarity of two phylogenetic trees or phylogenetic networks by counting the number of rooted phylogenetic trees with exactly three leaf labels (called rooted triplets, or triplets for short) that occur as embedded subtrees in one, but not both, of them. Suppose that \(N_1 = (V_1, E_1)\) and \(N_2 = (V_2, E_2)\) are phylogenetic networks over

We improve the lower bound on the asymptotic competitive ratio of any online algorithm for bin packing to above 1.54278. We demonstrate for the first time the advantage of branching and the applicability of full adaptivity in the design of lower bounds for the classic online bin packing problem. We apply a new method for weight based analysis, which is usually applied only in proofs of upper bounds

Recent empirical evaluations of exact algorithms for Feedback Vertex Set have demonstrated the efficiency of a highest-degree branching algorithm with a degree-based pruning. In this paper, we prove that this empirically fast algorithm runs in \(O(3.460^k n)\) time, where k is the solution size. This improves the previous best \(O(3.619^k n)\)-time deterministic algorithm obtained by Kociumaka and

Clustering is a fundamental tool for analyzing large data sets. A rich body of work has been devoted to designing data-stream algorithms for the relevant optimization problems such as k-center, k-median, and k-means. Such algorithms need to be both time and and space efficient. In this paper, we address the problem of correlation clustering in the dynamic data stream model. The stream consists of updates

In the Cluster Deletion problem the goal is to remove the minimum number of edges of a given graph, such that every connected component of the resulting graph constitutes a clique. It is known that the decision version of Cluster Deletion is NP-complete on (\(P_5\)-free) chordal graphs, whereas Cluster Deletion is solved in polynomial time on split graphs. However, the existence of a polynomial-time

We study the parameterized complexity of the problem of counting graph homomorphisms with given partial injectivity constraints, i.e., inequalities between pairs of vertices, which subsumes counting of graph homomorphisms, subgraph counting and, more generally, counting of answers to equi-join queries with inequalities. Our main result presents an exhaustive complexity classification for the problem

A permutation is happy, if it can be transformed into the identity permutation using as many short swaps as one third times the number of inversions in the permutation. The complexity of the decision version of sorting a permutation by short swaps, is still open. We present an O(n) time algorithm to decide whether it is true for a permutation to be happy, where n is the number of elements in the permutation

For any fixed graph G, the subgraph isomorphism problem asks whether an n-vertex input graph has a subgraph isomorphic to G. A well-known algorithm of Alon et al. (J ACM 42(4):844–856, 1995. https://doi.org/10.1145/210332.210337) efficiently reduces this to the “colored” version of the problem, denoted \(G{\text{-}}{\mathsf {SUB}}\), and then solves \(G{\text{-}}{\mathsf {SUB}}\) in time \(O(n^{{\

Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this area is deciding whether a given permutation of length n contains a given pattern of length k. In this work we give two new algorithms for this well-studied problem

We study a continuous facility location problem on a graph where all edges have unit length and where the facilities may also be positioned in the interior of the edges. The goal is to position as many facilities as possible subject to the condition that any two facilities have at least distance \(\delta\) from each other. We investigate the complexity of this problem in terms of the rational parameter

A resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to

Given a k-node pattern graph H and an n-node host graph G, the subgraph counting problem asks to compute the number of copies of H in G. In this work we address the following question: can we count the copies of H faster if G is sparse? We answer in the affirmative by introducing a novel tree-like decomposition for directed acyclic graphs, inspired by the classic tree decomposition for undirected graphs

A matching cut is a partition of the vertex set of a graph into two sets A and B such that each vertex has at most one neighbor in the other side of the cut. The Matching Cut problem asks whether a graph has a matching cut, and has been intensively studied in the literature. Motivated by a question posed by Komusiewicz et al. [Discrete Applied Mathematics, 2020], we introduce a natural generalization

The knapsack problem is one of the classical problems in combinatorial optimization: Given a set of items, each specified by its size and profit, the goal is to find a maximum profit packing into a knapsack of bounded capacity. In the online setting, items are revealed one by one and the decision, if the current item is packed or discarded forever, must be done immediately and irrevocably upon arrival

A bicolored rectangular family BRF is the collection of all axis-parallel rectangles formed by selecting a bottom-left corner from a finite set of points A and an upper-right corner from a finite set of points B. We devise a combinatorial algorithm to compute the maximum independent set and the minimum hitting set of a BRF that runs in \(O(n^{2.5}\sqrt{\log n})\)-time, where \(n=|A |+|B |\). This result

This paper studies optimal matroid partitioning problems for various objective functions. In the problem, we are given k weighted-matroids on the same ground set. Our goal is to find a feasible partition that minimizes (maximizes) the value of an objective function. A typical objective is the maximum over all subsets of the total weights of the elements in a subset, which is extensively studied in

We investigate the effect of crossover in the context of parameterized complexity on a well-known fixed-parameter tractable combinatorial optimization problem known as the closest string problem. We prove that a multi-start (\(\mu\)+1) GA solves arbitrary length-n instances of closest string in \(2^{O(d^2 + d \log k)} \cdot t(n)\) steps in expectation. Here, k is the number of strings in the input

We study the Travelling Salesperson (TSP) and the Steiner Tree problem (STP) in graphs of low highway dimension. This graph parameter roughly measures how many central nodes are visited by all shortest paths of a certain length. It has been shown that transportation networks, on which TSP and STP naturally occur for various applications in logistics, typically have a small highway dimension. While

Given an n-vertex digraph D and a non-negative integer k, the Minimum Directed Bisection problem asks if the vertices of D can be partitioned into two parts, say L and R, such that \({\vert {L} \vert }\) and \({\vert {R} \vert }\) differ by at most 1 and the number of arcs from R to L is at most k. This problem is known to be NP-hard even when \(k = 0\). We investigate the parameterized complexity

In the eternal domination game played on graphs, an attacker attacks a vertex at each turn and a team of guards must move a guard to the attacked vertex to defend it. The guards may only move to adjacent vertices on their turn. The goal is to determine the eternal domination number \(\gamma ^{\infty }_{all}\) of a graph, which is the minimum number of guards required to defend against an infinite sequence

Our input is a complete graph G on n vertices where each vertex has a strict ranking of all other vertices in G. The goal is to construct a matching in G that is popular. A matching M is popular if M does not lose a head-to-head election against any matching \(M'\): here each vertex casts a vote for the matching in \(\{M,M'\}\) in which it gets a better assignment. Popular matchings need not exist

In this paper we revisit the classical edge disjoint paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or FPT) algorithms. As our first result, we answer

Apart from the principles and methodologies inherited from Economics and Game Theory, the studies in Algorithmic Mechanism Design typically employ the worst-case analysis and design of approximation schemes of Theoretical Computer Science. For instance, the approximation ratio, which is the canonical measure of evaluating how well an incentive-compatible mechanism approximately optimizes the objective

In this paper, we study online algorithms for the Canadian Traveller Problem defined by Papadimitriou and Yannakakis in 1991. This problem involves a traveller who knows the entire road network in advance, and wishes to travel as quickly as possible from a source vertex s to a destination vertex t, but discovers online that some roads are blocked (e.g., by snow) once reaching them. Achieving a bounded

The cut-set \(\partial (S)\) of a graph \(G=(V,E)\) is the set of edges that have one endpoint in \(S\subset V\) and the other endpoint in \(V\setminus S\), and whenever G[S] is connected, the cut \([S,V\setminus S]\) of G is called a connected cut. A bond of a graph G is an inclusion-wise minimal disconnecting set of G, i.e., bonds are cut-sets that determine cuts \([S,V\setminus S]\) of G such that

In this paper, we investigate algorithms for finding centers of a given collection \(\mathcal N\) of sets. In particular, we focus on metric rational set similarities, a broad class of similarity measures including Jaccard and Hamming. A rational set similarity S is called metric if \(D=1-S\) is a distance function. We study the 1-center problem on these metric spaces. The problem consists of finding

A tournament is a directed graph in which there is a single arc between every pair of distinct vertices. Given a tournament T on n vertices, we explore the classical and parameterized complexity of the problems of determining if T has a cycle packing (a set of pairwise arc-disjoint cycles) of size k and a triangle packing (a set of pairwise arc-disjoint triangles) of size k. We refer to these problems

We consider a game on a graph \(G=\langle V, E\rangle \) with two confronting classes of randomized players: \(\nu \) attackers, who choose vertices and seek to minimize the probability of getting caught, and a single defender, who chooses edges and seeks to maximize the expected number of attackers it catches. In a Nash equilibrium, no player has an incentive to unilaterally deviate from her randomized

Dujmović et al. (FOCS2019) recently proved that every planar graph G is a subgraph of \(H\boxtimes P\), where \(\boxtimes\) denotes the strong graph product, H is a graph of treewidth 8 and P is a path. This result has found numerous applications to linear graph layouts, graph colouring, and graph labelling. The proof given by Dujmović et al. is based on a similar decomposition of Pilipczuk and Siebertz

In this paper, we consider the problem of maximizing a monotone submodular function subject to a knapsack constraint in a streaming setting. In such a setting, elements arrive sequentially and at any point in time, and the algorithm can store only a small fraction of the elements that have arrived so far. For the special case that all elements have unit sizes (i.e., the cardinality-constraint case)

We present the first universal reconfiguration algorithm for transforming a modular robot between any two facet-connected square-grid configurations using pivot moves. More precisely, we show that five extra “helper” modules (“musketeers”) suffice to reconfigure the remaining n modules between any two given configurations. Our algorithm uses \(O(n^2)\) pivot moves, which is worst-case optimal. Previous

In a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be \({\mathsf {NP}}\)-complete. While Matching Cut is trivial for graphs with minimum degree at most one, it is \({\mathsf {NP}}\)-complete on graphs with minimum degree two. In this paper, we show that, for any given constant \(c>1\)

In the first and so far only mathematical runtime analysis of an estimation-of-distribution algorithm (EDA) on a multimodal problem, Hasenöhrl and Sutton (GECCO 2018) showed for any \(k = o(n)\) that the compact genetic algorithm (cGA) with any hypothetical population size \(\mu = \Omega (ne^{4k} + n^{3.5+\varepsilon })\) with high probability finds the optimum of the n-dimensional jump function with

Motivated by many practical applications, in this paper we study budget feasible mechanisms with the goal of procuring an independent set of a matroid. More specifically, we are given a matroid \({\mathcal {M}}=(E,{\mathcal {I}})\). Each element of the ground set E is controlled by a selfish agent and the cost of the element is private information of the agent itself. A budget limited buyer has additive

In the Directed Feedback Vertex Set (DFVS) problem, the input is a directed graph D and an integer k. The objective is to determine whether there exists a set of at most k vertices intersecting every directed cycle of D. DFVS was shown to be fixed-parameter tractable when parameterized by solution size by Chen et al. (J ACM 55(5):177–186, 2008); since then, the existence of a polynomial kernel for

The compact Genetic Algorithm (cGA) evolves a probability distribution favoring optimal solutions in the underlying search space by repeatedly sampling from the distribution and updating it according to promising samples. We study the intricate dynamics of the cGA on the test function OneMax, and how its performance depends on the hypothetical population size K, which determines how quickly decisions

The runtime analysis of evolutionary algorithms using crossover as search operator has recently produced remarkable results indicating benefits and drawbacks of crossover and illustrating its working principles. Virtually all these results are restricted to upper bounds on the running time of the crossover-based algorithms. This work addresses this lack of lower bounds and rigorously bounds the optimization

In the last decade remarkable progress has been made in development of suitable proof techniques for analysing randomised search heuristics. The theoretical investigation of these algorithms on classes of functions is essential to the understanding of the underlying stochastic process. Linear functions have been traditionally studied in this area resulting in tight bounds on the expected optimisation

This paper addresses the classic online Steiner tree problem on edge-weighted graphs. It is known that a greedy (nearest neighbor) online algorithm has a tight competitive ratio for wide classes of graphs, such as trees, rings, any class including series-parallel graphs, and unweighted graphs with bounded diameter. However, we do not know any greedy or non-greedy tight deterministic algorithm for other

We consider the inverse Voronoi diagram problem in trees: given a tree T with positive edge-lengths and a collection \(\mathbb {U}\) of subsets of vertices of V(T), decide whether \({\mathbb {U}}\) is a Voronoi diagram in T with respect to the shortest-path metric. We show that the problem can be solved in \(O(N+n \log ^2 n)\) time, where n is the number of vertices in T and \(N=n+\sum _{U\in {\mathbb

Agrawal et al. (ACM Trans Comput Theory 10(4):18:1–18:25, 2018. https://doi.org/10.1145/3265027) studied a simultaneous variant of the classic Feedback Vertex Set problem, called Simultaneous Feedback Vertex Set (Sim-FVS). Here, we consider the edge variant of the problem, namely, Simultaneous Feedback Edge Set (Sim-FES). In this problem, the input is an n-vertex graph G, a positive integer k, and

We study the two party problem of randomly selecting a common string among all the strings of length n. We want the protocol to have the property that the output distribution has high Shannon entropy or high min entropy, even when one of the two parties is dishonest and deviates from the protocol. We develop protocols that achieve high, close to n, Shannon entropy and simultaneously min entropy close

Obvious strategyproofness (OSP) is an appealing concept as it allows to maintain incentive compatibility even in the presence of agents that are not fully rational, i.e., those who struggle with contingent reasoning (Li in Am Econ Rev 107(11):3257–3287, 2017). However, it has been shown to impose some limitations, e.g., no OSP mechanism can return a stable matching (Ashlagi and Gonczarowski in J Econ

In this article, we study some fault-tolerant covering problems in metric spaces. In the metric multi-cover problem (MMC), we are given two point sets Y (servers) and X (clients) in an arbitrary metric space \((X \cup Y, d)\), a positive integer k that represents the coverage demand of each client, and a constant \(\alpha \ge 1\). Each server can host a single ball of arbitrary radius centered on it

There exist many variants of guarding an orthogonal polygon in an orthogonal fashion: sometimes a guard can see within a rectangle, along a staircase, or along an orthogonal path with at most k bends. In this paper, we study all these guarding models for the special case of orthogonal polygons that have bounded treewidth in some sense. As our main result, we show that the problem of finding the minimum

We revisit the complexity of deciding, given a bimatrix game, whether it has a Nash equilibrium with certain natural properties; such decision problems were early known to be \({{\mathcal{N}}{\mathcal{P}}}\)-hard (Gilboa and Zemel in Games Econ Behav 1(1):80–93, 1989). We show that \({{\mathcal{N}}{\mathcal{P}}}\)-hardness still holds under two significant restrictions in simultaneity: the game is

Let \(G\) be a random geometric graph formed by \(n\) nodes with adjacency distance \(r_n\) and let each edge of \(G\) be assigned an independent exponential passage time with mean that depends on the graph size \(n.\) We connect \(G\) to two nodes source \(s_A\) and destination \(s_B\) at deterministic locations spaced \(d_n\) apart in the unit square and find upper and lower bounds on the minimum

A graph G is prismatic if for every triangle T of G, every vertex of G not in T has a unique neighbour in T. The complement of a prismatic graph is called antiprismatic. The complexity of colouring antiprismatic graphs is still unknown. Equivalently, the complexity of the clique cover problem in prismatic graphs is not known. Chudnovsky and Seymour gave a full structural description of prismatic graphs

In the Tree Augmentation problem we are given a tree \(T=(V,F)\) and a set \(E \subseteq V \times V\) of edges with positive integer costs \(\{c_e:e \in E\}\). The goal is to augment T by a minimum cost edge set \(J \subseteq E\) such that \(T \cup J\) is 2-edge-connected. We obtain the following results. Recently, Adjiashvili [SODA 17] introduced a novel LP for the problem and used it to break the

Distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the distance between any two vertices u and v can be determined efficiently by merely inspecting the labels of u and v, without using any other information. Similarly, routing labeling schemes label the vertices of a graph in a such a way that given the labels of a source node and a destination

We contribute results for a set of fundamental problems in the context of programmable matter by presenting algorithmic methods for evaluating and manipulating a collective of particles by a finite automaton that can neither store significant amounts of data, nor perform complex computations, and is limited to a handful of possible physical operations. We provide a toolbox for carrying out fundamental

Low-treedepth colorings are an important tool for algorithms that exploit structure in classes of bounded expansion; they guarantee subgraphs that use few colors have bounded treedepth. These colorings have an implicit tradeoff between the total number of colors used and the treedepth bound, and prior empirical work suggests that the former dominates the run time of existing algorithms in practice