We resolve the computational complexity of Graph Isomorphism for classes of graphs characterized by two forbidden induced subgraphs \( H_{1} \) and \(H_2\) for all but six pairs \((H_1,H_2)\). Schweitzer had previously shown that the number of open cases was finite, but without specifying the open cases. Grohe and Schweitzer proved that Graph Isomorphism is polynomial-time solvable on graph classes

We provide a simple and novel algorithmic design technique, for which we call iterative partial rounding, that gives a tight rounding-based approximation for vertex cover with hard capacities (VC-HC). In particular, we obtain an f-approximation for VC-HC on hypergraphs, improving over a previous results of Cheung et al. (In: SODA’14, 2014) to the tight extent. This also closes the gap of approximation

We study the problem of enumerating the k-arc-connected orientations of a graph G, i.e., generating each exactly once. A first algorithm using submodular flow optimization is easy to state, but intricate to implement. In a second approach we present a simple algorithm with \(O(knm^2)\) time delay and amortized time \(O(m^2)\), which improves over the analysis of the submodular flow algorithm. As ingredients

A private learner is an algorithm that given a sample of labeled individual examples outputs a generalizing hypothesis while preserving the privacy of each individual. In 2008, Kasiviswanathan et al. (FOCS 2008) gave a generic construction of private learners, in which the sample complexity is (generally) higher than what is needed for non-private learners. This gap in the sample complexity was then

Let \({\mathcal {C}}\) and \({\mathcal {D}}\) be hereditary graph classes. Consider the following problem: given a graph \(G\in {\mathcal {D}}\), find a largest, in terms of the number of vertices, induced subgraph of G that belongs to \({\mathcal {C}}\). We prove that it can be solved in \(2^{o(n)}\) time, where n is the number of vertices of G, if the following conditions are satisfied: the graphs

For an integer t, we let \(P_t\) denote the t-vertex path. We write \(H+G\) for the disjoint union of two graphs H and G, and for an integer r and a graph H, we write rH for the disjoint union of r copies of H. We say that a graph G is H-free if no induced subgraph of G is isomorphic to the graph H. In this paper, we study the complexity of k-coloring, for a fixed integer k, when restricted to the

We address the problem of speeding up group computations in cryptography using a single untrusted computational resource. We analyze the security of two efficient protocols for securely outsourcing (multi-)exponentiations. We show that the schemes do not achieve the claimed security guarantees and we present practical polynomial-time attacks on the delegation protocols which allow the untrusted helper

After publication of the article [1] the authors have noticed that the funding information are not published in online and print version of the article. The omitted funding acknowledgement is given below.

This paper addresses the problem of designing a \(\beta \)-additive fault-tolerant approximate BFS (or FT-ABFS for short) structure, namely, a subgraph H of the network G such that subsequent to the failure of a single edge e, the surviving part of H still contains an approximate BFS spanning tree for (the surviving part of) G, whose distances satisfy \(\mathrm{dist}(s,v,H{\setminus } \{e\}) \le \mathrm{dist}(s

Flip graphs are a ubiquitous class of graphs, which encode relations on a set of combinatorial objects by elementary, local changes. Skeletons of associahedra, for instance, are the graphs induced by quadrilateral flips in triangulations of a convex polygon. For some definition of a flip graph, a natural computational problem to consider is the flip distance: Given two objects, what is the minimum

A robot modeled as a deterministic finite automaton has to build a structure from material available to it. The robot navigates in the infinite oriented grid \({\mathbb {Z}} \times {\mathbb {Z}}\). Some cells of the grid are full (contain a brick) and others are empty. The subgraph of the grid induced by full cells, called the shape, is initially connected. The (Manhattan) distance between the furthest

We introduce a dynamic version of the NP-hard graph modification problem Cluster Editing. The essential point here is to take into account dynamically evolving input graphs: having a cluster graph (that is, a disjoint union of cliques) constituting a solution for a first input graph, can we cost-efficiently transform it into a “similar” cluster graph that is a solution for a second (“subsequent”) input

In the article Reoptimization Time Analysis of Evolutionary Algorithms on Linear Functions Under Dynamic Uniform Constraints, we claimed a worst-case runtime of and for the Multi-Objective Evolutionary Algorithm and the Multi-Objective Genetic Algorithm, respectively, on linear profit functions under dynamic uniform constraint, where denotes the difference between the original constraint bound B and

Applications designed for data-parallel computation frameworks such as MapReduce usually alternate between computation and communication stages. Coflow scheduling is a recent popular networking abstraction introduced to capture such application-level communication patterns in datacenters. In this framework, a datacenter is modeled as a single non-blocking switch with m input ports and m output ports

It is generally accepted that populations are useful for the global exploration of multi-modal optimisation problems. Indeed, several theoretical results are available showing such advantages over single-trajectory search heuristics. In this paper we provide evidence that evolving populations via crossover and mutation may also benefit the optimisation time for hillclimbing unimodal functions. In particular

Given two strings S and T, each of length at most n, the longest common substring (LCS) problem is to find a longest substring common to S and T. This is a classical problem in computer science with an \(\mathcal {O}(n)\)-time solution. In the fully dynamic setting, edit operations are allowed in either of the two strings, and the problem is to find an LCS after each edit. We present the first solution

We introduce a novel adversarial model for scheduling with explorable uncertainty. In this model, the processing time of a job can potentially be reduced (by an a priori unknown amount) by testing the job. Testing a job j takes one unit of time and may reduce its processing time from the given upper limit \(\bar{p}_j\) (which is the time taken to execute the job if it is not tested) to any value between

Finding maximum-cardinality matchings in undirected graphs is arguably one of the most central graph primitives. For m-edge and n-vertex graphs, it is well-known to be solvable in \(O(m\sqrt{n})\) time; however, for several applications this running time is still too slow. We investigate how linear-time (and almost linear-time) data reduction (used as preprocessing) can alleviate the situation. More

We study Subgraph Isomorphism on graph classes defined by a fixed forbidden graph. Although there are several ways for forbidding a graph, we observe that it is reasonable to focus on the minor relation since other well-known relations lead to either trivial or equivalent problems. When the forbidden minor is connected, we present a near dichotomy of the complexity of Subgraph Isomorphism with respect

Despite significant progress in the theory of evolutionary algorithms, the theoretical understanding of evolutionary algorithms which use non-trivial populations remains challenging and only few rigorous results exist. Already for the most basic problem, the determination of the asymptotic runtime of the \((\mu +\lambda )\) evolutionary algorithm on the simple OneMax benchmark function, only the special

In this work, we first study a natural generalisation of the Min-Cut problem, where a graph is augmented by a superadditive set function defined on its vertex subsets. The goal is to select a vertex subset such that the weight of the induced cut plus the set function value are minimised. In addition, a lower and upper bound is imposed on the solution size. We present a polynomial-time algorithm for

Searching for all occurrences of a pattern in a text is a fundamental problem in computer science with applications in many other fields, like natural language processing, information retrieval and computational biology. Sampled string matching is an efficient approach recently introduced in order to overcome the prohibitive space requirements of an index construction, on the one hand, and drastically

We derandomize Valiant’s (J ACM 62, Article 13, 2015) subquadratic-time algorithm for finding outlier correlations in binary data. This demonstrates that it is possible to perform a deterministic subquadratic-time similarity join of high dimensionality. Our derandomized algorithm gives deterministic subquadratic scaling essentially for the same parameter range as Valiant’s randomized algorithm, but

Large real-world networks typically follow a power-law degree distribution. To study such networks, numerous random graph models have been proposed. However, real-world networks are not drawn at random. Therefore, Brach et al. (27th symposium on discrete algorithms (SODA), pp 1306–1325, 2016) introduced two natural deterministic conditions: (1) a power-law upper bound on the degree distribution (PLB-U)

The bin covering problem asks for covering a maximum number of bins with an online sequence of n items of different sizes in the range (0, 1]; a bin is said to be covered if it receives items of total size at least 1. We study this problem in the advice setting and provide asymptotically tight bounds of \(\Theta (n \log {\textsc {Opt}})\) on the size of advice required to achieve optimal solutions

We study a scheduling problem arising in demand response management in smart grid. Consumers send in power requests with a flexible feasible time interval during which their requests can be served. The grid controller, upon receiving power requests, schedules each request within the specified interval. The electricity cost is measured by a convex function of the load in each timeslot. The objective

In this paper, we investigate the complexity of Maximum Independent Set (MIS) in the class of H-free graphs, that is, graphs excluding a fixed graph as an induced subgraph. Given that the problem remains NP-hard for most graphs H, we study its fixed-parameter tractability and make progress towards a dichotomy between FPT and W[1]-hard cases. We first show that MIS remains W[1]-hard in graphs forbidding

We propose and analyze a self-adaptive version of the \((1,\lambda )\) evolutionary algorithm in which the current mutation rate is encoded within the individual and thus also subject to mutation. A rigorous runtime analysis on the OneMax benchmark function reveals that a simple local mutation scheme for the rate leads to an expected optimization time (number of fitness evaluations) of \(O(n\lambda

A (possibly degenerate) drawing of a graph G in the plane is approximable by an embedding if it can be turned into an embedding by an arbitrarily small perturbation. We show that testing whether a piece-wise linear drawing of a planar graph G in the plane is approximable by an embedding can be carried out in polynomial time, if a desired embedding of G belongs to a fixed isotopy class. In other words

A space bounded \( O(d/\log d)\)-competitive hypercube packing algorithm with one active bin only is presented. As a starting point we give a simple 1-space bounded hypercube packing algorithm with competitive ratio \( (3/2)^{d}+O((21/16)^d)\), for \(d\ge 3.\)

We introduce the inverse Voronoi diagram problem in graphs: given a graph G with positive edge-lengths and a collection \({\mathbb {U}}\) of subsets of vertices of V(G), decide whether \({\mathbb {U}}\) is a Voronoi diagram in G with respect to the shortest-path metric. We show that the problem is NP-hard, even for planar graphs where all the edges have unit length. We also study the parameterized

The problem of truth discovery arises in many areas such as database, data mining, data crowdsourcing and machine learning. It seeks trustworthy information from possibly conflicting data provided by multiple sources. Due to its practical importance, the problem has been studied extensively in recent years. Two competing models were proposed for truth discovery, weight-based model and probabilistic

A graph G is called a pairwise compatibility graph (PCG, for short) if it admits a tuple \((T,w, d_{\min },d_{\max })\) of a tree T whose leaf set is equal to the vertex set of G, a non-negative edge weight w, and two non-negative reals \(d_{\min }\le d_{\max }\) such that G has an edge between two vertices \(u,v\in V\) if and only if the distance between the two leaves u and v in the weighted tree

We prove that, given two topologically-equivalent upward planar straight-line drawings of an n-vertex directed graph G, there always exists a morph between them such that all the intermediate drawings of the morph are upward planar and straight-line. Such a morph consists of O(1) morphing steps if G is a reduced planar st-graph, O(n) morphing steps if G is a planar st-graph, O(n) morphing steps if

In this work we study the problem of collecting protected data in ad-hoc sensor network using a mobile entity called MULE. The objective is to increase information survivability in the network. Sensors from all over the network, route their sensing data through a data gathering tree, towards a particular node, called the sink. In case of a failed sensor, all the aggregated data from the sensor and

This paper is dedicated to the counting problem of writing an integer number z as a sum of an ordered sequence of n integers from n given intervals, i.e., counting the number of configurations \((z_1,\ldots ,z_n)\) with \(z = z_1 + \cdots + z_n\) for \(z_i \in [x_i, y_i]\) with integers \(x_i\) and \(y_i\) and \(1 \le i \le n\). We show an algorithm computing this number in \( \mathop {}\mathopen {}{\mathcal

The problem of merging sorted lists in the least number of pairwise comparisons has been solved completely only for a few special cases. Graham and Karp (Sorting Search 3:197–207, 1999) independently discovered that the tape merge algorithm is optimal in the worst case when the two lists have the same size. In their seminal papers, Stockmeyer and Yao (SIAM J Comput 9(1):85–90, 1980), Murphy and Paull

In the range \(\alpha\)-majority query problem, we are given a sequence \(S[1\ldots n]\) and a fixed threshold \(\alpha \in (0, 1)\), and are asked to preprocess S such that, given a query range \([i\ldots j]\), we can efficiently report the symbols that occur more than \(\alpha (j-i+1)\) times in \(S[i\ldots j]\), which are called the range \(\alpha\)-majorities. In this article we describe the first

The similarity of two one-dimensional sequences is usually measured by the longest common subsequence (LCS) algorithms. However, these algorithms cannot be directly extended to solve the two or higher dimensional data. Thus, for the two-dimensional data, computing the similarity with an LCS-like approach remains worthy of investigation. In this paper, we utilize a systematic way to give the generalized

Alamdari and Shmoys introduced the following variant of the k-median problem. In this variant, we are given an instance of the k-median problem and a threshold value. Then this variant is the same as the k-median problem except that if the distance between a client i and a facility j is more than the threshold value, then i is not allowed to be connected to j. In this paper, we consider a matroid generalization

An input to a conflict-free variant of a classical problem \(\Gamma \), called Conflict-Free\(\Gamma \), consists of an instance I of \(\Gamma \) coupled with a graph H, called the conflict graph. A solution to Conflict-Free\(\Gamma \) in (I, H) is a solution to I in \(\Gamma \), which is also an independent set in H. In this paper, we study conflict-free variants of Maximum Matching and Shortest Path

Motivated by the role of triadic closures in social networks, and the importance of finding a maximum subgraph avoiding a fixed pattern, we introduce and initiate the parameterized study of the StrongF-closure problem, where F is a fixed graph. This is a generalization of Strong Triadic Closure, whereas it is a relaxation of F-free Edge Deletion. In StrongF-closure, we want to select a maximum number

In this paper we study the hardness of the \(k\)-Center problem on inputs that model transportation networks. For the problem, a graph \(G=(V,E)\) with edge lengths and an integer k are given and a center set \(C\subseteq V\) needs to be chosen such that \(|C|\le k\). The aim is to minimize the maximum distance of any vertex in the graph to the closest center. This problem arises in many applications

A road map can be interpreted as a graph embedded in the plane, in which each vertex corresponds to a road junction and each edge to a particular road section. In this paper, we consider the computational cartographic problem to place non-overlapping road labels along the edges so that as many road sections as possible are identified by their name, i.e., covered by a label. We show that this is NP-hard

A subgraph complement of the graph G is a graph obtained from G by complementing all the edges in one of its induced subgraphs. We study the following algorithmic question: for a given graph G and graph class \({\mathscr {G}}\), is there a subgraph complement of G which is in \({\mathscr {G}}\)? We show that this problem can be solved in polynomial time for various choices of the graphs class \({\mathscr

In reoptimization, one is given an optimal solution to a problem instance and a (locally) modified instance. The goal is to obtain a solution for the modified instance. We aim to use information obtained from the given solution in order to obtain a better solution for the new instance than we are able to compute from scratch. In this paper, we consider Steiner tree reoptimization and address the optimality

A graph is d-orientable if its edges can be oriented so that the maximum in-degree of the resulting digraph is at most d. d-orientability is a well-studied concept with close connections to fundamental graph-theoretic notions and applications as a load balancing problem. In this paper we consider the \(d\)-Orientable Deletion problem: given a graph \(G=(V,E)\), delete the minimum number of vertices

The k-Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a fixed integer k such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list \(L(u)\subseteq \{1,\ldots ,k\},\) then we obtain the List k-Colouring problem. A graph G is H-free if G does not contain H as an induced subgraph. We continue

Anomaly detection problems (also called change-point detection problems) have been studied in data mining, statistics and computer science over the last several decades (mostly in non-network context) in applications such as medical condition monitoring, weather change detection and speech recognition. In recent days, however, anomaly detection problems have become increasing more relevant in the context

We initiate the study of a fundamental combinatorial problem: Given a capacitated graph \(G=(V,E)\), find a shortest walk (“route”) from a source \({s\in V}\) to a destination \(t\in V\) that includes all vertices specified by a set \(WP \subseteq V\): the waypoints. This Waypoint Routing Problem finds immediate applications in the context of modern networked systems. Our main contribution is an exact

We study the problem of finding shortest paths in the plane among h convex obstacles, where the path is allowed to pass through (violate) up to k obstacles, for \(k \le h\). Equivalently, the problem is to find shortest paths that become obstacle-free if k obstacles are removed from the input. Given a fixed source point s, we show how to construct a map, called a shortest k-path map, so that all destinations

The problem of pollution control has been mainly studied in the environmental economics literature where the methodology of game theory is applied for the pollution control. To the best of our knowledge this is the first time this problem is studied from the computational point of view. We introduce a new network model for pollution control and present two applications of this model. On a high level

A secure set S in a graph is defined as a set of vertices such that for any X⊆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 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 integer k, a non-empty secure set for G of size at

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 fitness. For this function

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 hybridization number

Real-world optimisation problems are often dynamic. Previously good solutions must be updated or replaced due to changes in objectives and constraints. It is often claimed that evolutionary algorithms are particularly suitable for dynamic optimisation because a large population can contain different solutions that may be useful in the future. However, rigorous theoretical demonstrations for how populations

We study sorting algorithms based on randomized round-robin comparisons. Specifically, we study Spin-the-bottle sort, where comparisons are unrestricted, and Annealing sort, where comparisons are restricted to a distance bounded by a temperature parameter. Both algorithms are simple, randomized, data-oblivious sorting algorithms, which are useful in privacy-preserving computations, but, as we show

We study a family of problems where the goal is to make a graph Eulerian, i.e., connected and with all the vertices having even degrees, by a minimum number of deletions. We completely classify the parameterized complexity of various versions: undirected or directed graphs, vertex or edge deletions, with or without the requirement of connectivity, etc. The collection of results shows an interesting

In this paper, we develop new tools and connections for exponential time approximation. In this setting, we are given a problem instance and an integer r > 1 , and the goal is to design an approximation algorithm with the fastest possible running time. We give randomized algorithms that establish an approximation ratio ofr for maximum independent set in O ∗ ( exp ( O ~ ( n / r log 2 r + r log 2 r )

In this paper we study graph problems in the dynamic streaming model, where the input is defined by a sequence of edge insertions and deletions. As many natural problems require Ω ( n ) space, where n is the number of vertices, existing works mainly focused on designing O ( n · poly log n ) space algorithms. Although sublinear in the number of edges for dense graphs, it could still be too large for