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Temporal graphs are graphs whose topology is subject to discrete changes over time. Given a static underlying graph G , a temporal graph is represented by assigning a set of integer time-labels to every edge e of G , indicating the discrete time steps at which e is active. We introduce and study the complexity of a natural temporal extension of the classical graph problem Maximum Matching , taking into account the dynamic nature of temporal graphs. In our problem, Maximum Temporal Matching , we are looking for the largest possible number of time-labeled edges (simply time-edges) (e , t) such that no vertex is matched more than once within any time window of Δ consecutive time slots, where Δ ∈ N is given. We prove strong computational hardness results for Maximum Temporal Matching , even for elementary cases, as well as fixed-parameter algorithms with respect to natural parameters and polynomial-time approximation algorithms. [ABSTRACT FROM AUTHOR]

The problem of deletion of vertices to a hereditary graph class is a well-studied problem in parameterized complexity. Recently, a natural extension of the problem was initiated where we are given a finite set of hereditary graph classes and we determine whether k vertices can be deleted from a given graph so that the connected components of the resulting graph belong to one of the given hereditary graph classes. The problem is shown to be fixed parameter tractable (FPT) when the deletion problem to each of the given hereditary graph classes is fixed-parameter tractable, and the property of being in any of the graph classes is expressible in the counting monodic second order (CMSO) logic. This paper focuses on pairs of specific graph classes (Π 1 , Π 2) in which we would like the connected components of the resulting graph to belong to, and design simpler and more efficient FPT algorithms. [ABSTRACT FROM AUTHOR]

We generalize the tree doubling and Christofides algorithm to parameterized approximations for ATSP (constant factor approximations that invest more runtime with respect to a chosen parameter). The parameters we consider are upper bounded by the number of asymmetric distances , which yields algorithms to efficiently compute good approximations for moderately asymmetric TSP instances. As generalization of the Christofides algorithm, we derive a parameterized 2.5-approximation, with the size of a vertex cover for the subgraph induced by the edges with asymmetric distances as parameter. Our generalization of tree doubling gives a parameterized 3-approximation, where the parameter is the minimum number of asymmetric distances in a minimum spanning arborescence. Further, we combine these with a notion of symmetry relaxation which allows to trade approximation guarantee for runtime. Since the parameters we consider are theoretically incomparable, we present experimental results showing that generalized tree doubling frequently outperforms generalized Christofides with respect to parameter size. [ABSTRACT FROM AUTHOR]

Trees with many leaves have applications for broadcasting a message to all recipients simultaneously. Internal nodes of a broadcasting tree require more expensive technology to forward the messages received. We address a problem that captures the main goal: finding spanning trees with few internal nodes in a given network. The Maximum Leaf Spanning Arborescence problem consists of, given a directed graph D , finding a spanning arborescence of D , if one exists, with the maximum number of leaves. This problem is NP-hard in general and MaxSNP-hard in the rooted directed acyclic graphs class. This paper explores a relationship between Maximum Leaf Spanning Arborescence in rooted directed acyclic graphs and maximum weight set packing. The latter problem is related to independent sets on particular classes of intersection graphs. Exploiting this relation, we derive a 7/5-approximation for Maximum Leaf Spanning Arborescence on rooted directed acyclic graphs, improving on the previous 3/2-approximation. [ABSTRACT FROM AUTHOR]

We introduce and study two natural generalizations of the Connected Vertex Cover (VC) problem: the p -Edge-Connected and p -Vertex-Connected VC problem (where p ≥ 2 is a fixed integer). We obtain an 2 O (p k) n O (1) -time algorithm for p -Edge-Connected VC and an 2 O (k 2) n O (1) -time algorithm for p -Vertex-Connected VC. Thus, like Connected VC, both constrained VC problems are FPT. Furthermore, like Connected VC, neither problem admits a polynomial kernel unless NP ⊆ coNP/poly, which is highly unlikely. We prove however that both problems admit time efficient polynomial sized approximate kernelization schemes. Finally, we describe a 2 (p + 1) -approximation algorithm for the p -Edge-Connected VC. The proofs for the new VC problems require more sophisticated arguments than for Connected VC. In particular, for the approximation algorithm we use Gomory-Hu trees and for the approximate kernels a result on small-size spanning p -vertex/edge-connected subgraphs of a p -vertex/edge-connected graph by Nishizeki and Poljak (1994) [30] and Nagamochi and Ibaraki (1992) [27]. [ABSTRACT FROM AUTHOR]

We study two generalizations of classic clustering problems called dynamic ordered k-median and dynamic k-supplier , where the points that need clustering evolve over time, and we are allowed to move the cluster centers between consecutive time steps. In these dynamic clustering problems, the general goal is to minimize certain combinations of the service cost of points and the movement cost of centers, or to minimize one subject to some constraints on the other. We obtain a constant-factor approximation algorithm for dynamic ordered k -median under mild assumptions on the input. We give a 3-approximation for dynamic k -supplier and a multi-criteria approximation for its outlier version where some points can be discarded, when the number of time steps is two. We complement the algorithms with almost matching hardness results. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the shortest link scheduling (SLS) in wireless networks under the Signal-to-Interference-plus-Noise-Ratio (SINR) constraints. We introduce a consistent graph G ¯ = (V ¯ , E ¯) where V ¯ is the links set and each edge in E ¯ indicates that its two ends can be active simultaneously. We interpret SLS as the finding of a clique partition in G ¯ and propose a fast approximation algorithm, MCBS (Maximum Clique Based Scheduling), for SLS with a uniform power assignment and with an approximation ratio to the optimal solution of O (1). Theoretical analysis demonstrates the correctness and effectiveness of the proposed algorithm. Our simulation results show that the MCBS algorithm overcomes the best known algorithms in terms of execution time and the number of time slots. [ABSTRACT FROM AUTHOR]

We introduce a novel domination problem, which we call Dual Domination (DD). We assume that the nodes in a network are partitioned into two categories: Positive nodes (V +) and negative nodes (V −). We study the Maximum Bounded Dual Domination , where, given a bound k , the problem is to find a set D ⊆ V + , which maximizes the number of nodes dominated in V + , dominating at most k nodes in V −. We show that such problem is hard to approximate to a factor better than (1 − 1 / e). We give a polynomial time algorithm with approximation guaranteed (1 − e − 1 / Δ) , where Δ represents the maximum number of neighbors in V + of any node in V − , and an O (| V | k 2) time algorithm to solve the problem on trees. We also study two related problems named Maximum Dual Domination and minimum Negative Dual Domination. For both problems, we show hardness results and provide O (| V |) time algorithms on trees. [ABSTRACT FROM AUTHOR]

For several decades, much effort has been put into identifying classes of CNF formulas whose satisfiability can be decided in polynomial time. Classic results are the linear-time tractability of Horn formulas (Aspvall, Plass, and Tarjan, 1979) and Krom (i.e., 2CNF) formulas (Dowling and Gallier, 1984). Backdoors, introduced by Williams, Gomes and Selman (2003), gradually extend such a tractable class to all formulas of bounded distance to the class. Backdoor size provides a natural but rather crude distance measure between a formula and a tractable class. Backdoor depth, introduced by Mählmann, Siebertz, and Vigny (2021), is a more refined distance measure, which admits the utilization of different backdoor variables in parallel. We propose FPT approximation algorithms to compute backdoor depth into the classes Horn and Krom. This leads to a linear-time algorithm for deciding the satisfiability of formulas of bounded backdoor depth into these classes. [ABSTRACT FROM AUTHOR]

Fair clustering is a constrained clustering problem where we need to partition a set of colored points. The fraction of points of each color in every cluster should be more or less equal to the fraction of points of this color in the dataset. The problem was recently introduced by Chierichetti et al. (2017) [1]. We propose a new construction of coresets for fair clustering for Euclidean and general metrics based on random sampling. For the Euclidean space R d , we provide the first coreset whose size does not depend exponentially on the dimension d. The question of whether such constructions exist was asked by Schmidt et al. (2019) [2] and Huang et al. (2019) [5]. For general metrics, our construction provides the first coreset for fair clustering. New coresets appear to be a handy tool for designing better approximation and streaming algorithms for fair and other constrained clustering variants. [ABSTRACT FROM AUTHOR]

Many systems have to be maintained while the underlying constraints, costs and/or profits change over time. Although the state of a system may evolve during time, a non-negligible transition cost is incurred for transitioning from one state to another. In order to model such situations, we look at a recently introduced multistage model where the input is a sequence of instances (one for each time step), and the goal is to find a sequence of solutions (one for each time step) that are both (i) near optimal for each time step and (ii) as stable as possible. We propose a PTAS for the Multistage Knapsack problem. This is the first approximation scheme for a combinatorial optimization problem in the considered multistage setting, and its existence contrasts with the inapproximability results for other combinatorial optimization problems that are even polynomial-time solvable in the static case. [ABSTRACT FROM AUTHOR]

We consider the directed minimum latency problem (DirLat), wherein we seek a path P visiting all points (or clients) in a given asymmetric metric starting at a given root node r , so as to minimize the sum of the client waiting times along P. We give the first constant-factor approximation guarantee for DirLat , but in quasi-polynomial time. A key ingredient of our result, and our chief technical contribution, is an extension of a recent result of Köhne et al. (2019) [17] showing that the integrality gap of the natural Held-Karp relaxation for asymmetric TSP-Path (ATSPP) is at most a constant. We also give a better approximation guarantee for the minimum total-regret problem , where the goal is to find a path P that minimizes the total time that nodes spend in excess of their shortest-path distances from r , which can be cast as a special case of DirLat involving so-called regret metrics. [ABSTRACT FROM AUTHOR]

We study the problem of maximizing a monotone submodular function subject to a Multiple Knapsack constraint. The input is a set I of items, each has a non-negative weight, and a set of bins of arbitrary capacities. Also, we are given a submodular, monotone and non-negative function f over subsets of the items. The objective is to find a packing of a subset of items A ⊆ I in the bins such that f (A) is maximized. Our main result is an almost optimal polynomial time (1 − e − 1 − ε) -approximation algorithm for the problem, for any ε > 0. The algorithm relies on a structuring technique which converts a general multiple knapsack constraint to a constraint in which the bins are partitioned into groups of exponentially increasing cardinalities, each consisting of bins of uniform capacity. We derive the result by combining structuring with a refined analysis of techniques for submodular optimization subject to knapsack constraints. [ABSTRACT FROM AUTHOR]

We study the approximability of the NP-complete Maximum Minimal Feedback Vertex Set problem. Informally, this natural problem seems to lie in an intermediate space between two more well-studied problems of this type: Maximum Minimal Vertex Cover , for which the best achievable approximation ratio is n , and Upper Dominating Set , which does not admit any n 1 − ϵ approximation. We confirm and quantify this intuition by showing the first non-trivial polynomial time approximation for Maximum Minimal Feedback Vertex Set with a ratio of O (n 2 / 3) , as well as a matching hardness of approximation bound of n 2 / 3 − ϵ , improving the previously known hardness of n 1 / 2 − ϵ. Having settled the problem's approximability in polynomial time, we move to the context of super-polynomial time. We devise a generalization of our approximation algorithm which, for any desired approximation ratio r , produces an r -approximate solution in time n O (n / r 3 / 2). This time-approximation trade-off is essentially tight under the ETH. [ABSTRACT FROM AUTHOR]

We show that a constant factor approximation of the shortest and closest lattice vector problem in any ℓ p -norm can be computed in time 2 (0.802 + ε) n. This matches the currently fastest constant factor approximation algorithm for the shortest vector problem in the ℓ 2 norm. To obtain our result, we combine the latter algorithm for ℓ 2 with geometric insights related to coverings. [ABSTRACT FROM AUTHOR]

Given a set of n integer-valued coin types and a target value t , the well-known change-making problem asks for the minimum number of coins that sum to t , assuming an unlimited number of coins in each type. In the more general all-targets version of the problem, we want the minimum number of coins summing to j , for every j = 0 , ... , t. We obtain a number of new results on these problems, including (i) an algorithm for the all-targets problem with running time O ˜ (t 4 / 3) (improving a previous O ˜ (t 3 / 2) -time algorithm), (ii) a very simple O ˜ (u 2 + t) -time algorithm for the all-targets problem, where u denotes the maximum coin value, and (iii) an algorithm for the original (single-target) problem with running time O ˜ (u) (improving a previous O ˜ (t) -time algorithm by Chan and He (SOSA'20)). We also describe applications to the (single-capacity) unbounded knapsack problem and the minimum word break problem. [ABSTRACT FROM AUTHOR]

This work firstly studies the Steiner tree problem with bounded edge-length d in which d is the ratio of the maximum edge cost to the minimum edge cost. This work analyzes the algorithm of Byrka et al. [19] and shows that the approximation ratio of d ln ⁡ 4 d + ln ⁡ 4 − 1 + ϵ for general graphs and approximation ratio of 73 ⋅ d 60 ⋅ d + 13 + ϵ for quasi-bipartite graphs. The algorithm implies approximation ratio of 1.162 + ϵ for the problem on complete graphs with edge distances 1 and 2. This finding represents an improvement upon the previous best approximation ratio of 1.25. This work then presents a combinatorial two-phase heuristic for the general Steiner tree in greedy strategy that achieves an approximation ratio of 1.4295. [ABSTRACT FROM AUTHOR]

We obtain approximation ratio 4 + 2 ℓ ≈ 4 + 4 lg ⁡ k lg ⁡ n − lg ⁡ k for the (undirected) k -Connected Subgraph problem, where ℓ = ⌊ lg ⁡ n − lg ⁡ k + 1 2 lg ⁡ k + 1 ⌋ is the largest integer such that 2 ℓ − 1 k 2 ℓ + 1 ≤ n. For large values of n this improves the ratio 6 of Cheriyan and Végh [4] when n ≥ k 3 (the case ℓ = 1). Our result implies an fpt-approximation ratio 4 + ϵ that matches (up to the "+ ϵ " term) the best known ratio 4 for k = 6 , 7 for both the general and the easier augmentation versions of the problem. Similar results are shown for the problem of covering an arbitrary symmetric crossing supermodular biset function. [ABSTRACT FROM AUTHOR]

We study clustering problems such as k -Median, k -Means, and Facility Location in graphs of low highway dimension, which is a graph parameter modeling transportation networks. It was previously shown that approximation schemes for these problems exist, which either run in quasi-polynomial time (assuming constant highway dimension) (Feldmann et al., 2018) [8] or run in FPT time (parameterized by the number of clusters k , the highway dimension, and the approximation factor) (Becker et al., 2018; Braverman et al., 2021) [9,10]. In this paper we show that a polynomial-time approximation scheme (PTAS) exists (assuming constant highway dimension). We also show that the considered problems are NP-hard on graphs of highway dimension 1. [ABSTRACT FROM AUTHOR]

Graph coloring is one of the most famous computational problems with applications in a wide range of areas such as planning and scheduling, resource allocation, and pattern matching. So far coloring problems are mostly studied on static graphs, which often stand in contrast to practice where data is inherently dynamic. A temporal graph has an edge set that changes over time. We present a natural temporal extension of the classical graph coloring problem. Given a temporal graph and integers k and Δ, we ask for a coloring sequence with at most k colors for each vertex such that in every time window of Δ consecutive time steps, in which an edge is present, this edge is properly colored at least once. We thoroughly investigate the computational complexity of this temporal coloring problem. More specifically, we prove strong computational hardness results, complemented by efficient exact and approximation algorithms. [ABSTRACT FROM AUTHOR]

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 modelled 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. [ABSTRACT FROM AUTHOR]

In this paper, we study the Maximum Internal Spanning Tree Problem (MIST). Given an undirected simple graph G , the task for the Maximum Internal Spanning Tree problem is to find a spanning tree of G with maximum number of internal vertices. We present an approximation algorithm with performance ratio 4 3 , which improves upon the best known performance ratio 3 2. Our algorithm benefits from a new observation for bounding the number of internal vertices of a spanning tree. We can also give an example to show that the performance ratio 4 3 is actually tight for this algorithm. Finally, we show that MIST is Max-SNP-hard. [ABSTRACT FROM AUTHOR]

The modularity is the best known and widely used quality function for community detection in graphs. We investigate the approximability of the modularity maximization problem and some related problems. We first design a polynomial-time 0.4209-additive approximation algorithm for the modularity maximization problem, which improves the current best additive approximation error of 0.4672. Our theoretical analysis also demonstrates that the proposed algorithm obtains a nearly-optimal solution for any instance with a high modularity value. We next design a polynomial-time 0.1660-additive approximation algorithm for the maximum modularity cut problem. Finally, we extend our algorithm to some related problems. [ABSTRACT FROM AUTHOR]

Maximum parsimony distance is a measure used to quantify the dissimilarity of two unrooted phylogenetic trees. It is NP-hard to compute, and very few positive algorithmic results are known due to its complex combinatorial structure. Here we address this shortcoming by showing that the problem is fixed parameter tractable. We do this by establishing a linear kernel i.e., that after applying certain reduction rules the resulting instance has size that is bounded by a linear function of the distance. As powerful corollaries to this result we prove that the problem permits a polynomial-time constant-factor approximation algorithm; that the treewidth of a natural auxiliary graph structure encountered in phylogenetics is bounded by a function of the distance; and that the distance is within a constant factor of the size of a maximum agreement forest of the two trees, a well studied object in phylogenetics. [ABSTRACT FROM AUTHOR]

We consider Internet of Things (IoT) sensors deployed inside an area to be monitored. Drones can be used to collect the data from the sensors, but they are constrained in energy and storage. Therefore, all drones need to select a subset of sensors whose data are the most relevant to be acquired, modeled by assigning a reward. We present an optimization problem called Multiple-drone Data-collection Maximization Problem (MDMP) whose objective is to plan a set of drones' missions aimed at maximizing the overall reward from the collected data, and such that each individual drone's mission energy cost and total collected data are within the energy and storage limits, respectively. We optimally solve MDMP by proposing an Integer Linear Programming based algorithm. Since MDMP is NP -hard, we devise suboptimal algorithms for single- and multiple-drone scenarios. Finally, we thoroughly evaluate our algorithms on the basis of random generated synthetic data. [ABSTRACT FROM AUTHOR]

A garden is populated by n bamboos, each with its own daily growth rate. The Bamboo Garden Trimming Problem (BGT) is to design for a robotic gardener a perpetual schedule of cutting bamboos to keep the elevation of the garden as low as possible. The frequency of cutting is constrained by the time needed to move from one bamboo to the next, which is one day in Discrete BGT and is defined by the distance between the two bamboos in Continuous BGT. The bamboo garden is a metaphor for a collection of machines which have to be serviced, with different frequencies, by a robot which can service only one machine at a time. For Discrete BGT, we show tighter approximation algorithms, with one of them settling a long-standing conjecture about the related Pinwheel problem. For Continuous BGT, we propose approximation algorithms which achieve logarithmic approximation ratios. [ABSTRACT FROM AUTHOR]

Translocation has been long learned as a basic operation to rearrange the structure of a genome. Translocation sorting asks to find a shortest sequence of translocations that transforms one genome into another, which has attracted attention of many scientists in algorithm design. Signed translocation sorting can be solved in polynomial time. Unsigned translocation sorting turns out to be NP-Hard and Max-SNP-Hard. The best known approximation algorithm by now for unsigned translocation sorting can achieve a performance ratio 1.408. In this paper, we propose a new approximation algorithm for unsigned translocation sorting which can achieve a asymptotic performance ratio 1.375. [ABSTRACT FROM AUTHOR]

A graph G = (V , E) is δ -hyperbolic if for any four vertices u , v , w , x , the two larger of the three distance sums d (u , v) + d (w , x) , d (u , w) + d (v , x) , d (u , x) + d (v , w) differ by at most 2 δ ≥ 0. This paper describes the eccentricity terrain of a δ -hyperbolic graph. The eccentricity function e G (v) = max ⁡ { d (v , u) : u ∈ V } partitions vertices of G into eccentricity layers C k (G) = { v ∈ V : e G (v) = r a d (G) + k } , k ∈ N , where r a d (G) = min ⁡ { e G (v) : v ∈ V } is the radius of G. The paper studies the eccentricity layers of vertices along shortest paths, identifying such terrain features as hills, plains, valleys, terraces, and plateaus. It introduces the notion of β -pseudoconvexity, which implies Gromov's ϵ -quasiconvexity, and illustrates the abundance of pseudoconvex sets in δ -hyperbolic graphs. It shows that all sets C ≤ k (G) = { v ∈ V : e G (v) ≤ r a d (G) + k } , k ∈ N , are (2 δ − 1) -pseudoconvex. Several bounds on the eccentricity of a vertex are obtained which yield a few approaches to efficiently approximating all eccentricities. [ABSTRACT FROM AUTHOR]

Truth Discovery is an important learning problem arising in data analytics related fields. It concerns about finding the most trustworthy information from a dataset acquired from a number of unreliable sources. The problem has been extensively studied and a number of techniques have already been proposed. However, all of them are of heuristic nature and do not have any quality guarantee. In this paper, we formulate the problem as a high dimensional geometric optimization problem, called Entropy based Geometric Variance. Relying on a number of novel geometric techniques, we further discover new insights to this problem. We show, for the first time, that the truth discovery problem can be solved with guaranteed quality of solution. Particularly, it is possible to achieve a (1 + ϵ) -approximation within nearly linear time under some reasonable assumptions. We expect that our algorithm will be useful for other data related applications. [ABSTRACT FROM AUTHOR]

We present the first study of non-deterministic weighted automata under probabilistic semantics. In this semantics words are random events, generated by a Markov chain, and functions computed by weighted automata are random variables. We consider the probabilistic questions of computing the expected value and the cumulative distribution for such random variables. The exact answers to the probabilistic questions for non-deterministic automata can be irrational and are uncomputable in general. To overcome this limitation, we propose approximation algorithms for the probabilistic questions, which work in exponential time in the size of the automaton and polynomial time in the size of the Markov chain and the given precision. We apply this result to show that non-deterministic automata can be effectively determinised with respect to the standard deviation metric. [ABSTRACT FROM AUTHOR]

Modern, inherently dynamic systems are usually characterized by a network structure which is subject to discrete changes over time. Given a static underlying graph, a temporal graph can be represented via an assignment of a set of integer time-labels to every edge, indicating the discrete time steps when this edge is active. While most of the recent theoretical research on temporal graphs focused on temporal paths and other "path-related" temporal notions, only few attempts have been made to investigate "non-path" temporal problems. In this paper we introduce and study two natural temporal extensions of the classical problem VERTEX COVER. We present a thorough investigation of the computational complexity and approximability of these two temporal covering problems. We provide strong hardness results, complemented by approximation and exact algorithms. Some of our algorithms are polynomial-time, while others are asymptotically almost optimal under the Exponential Time Hypothesis (ETH) and other plausible complexity assumptions. [ABSTRACT FROM AUTHOR]

Abstract In sum coloring, it is required to find a proper coloring of the vertices of a graph using positive integers, such that the sum of colors of the vertices is minimized. First Fit is the natural coloring algorithm that processes vertices one by one (in some order), and colors the current vertex with the minimum color that was not used for any of its already considered neighbors. Here, we study the approximation ratio of First Fit for sum coloring for the class of claw-free graphs, its most natural subclasses, and its generalizations, and get tight bounds for it. Highlights • A thorough study of the approximation ratio for the natural algorithm First Fit. • A study of sum coloring of graphs, of vertices or edges. • A tight analysis of (k + 1)-claw free graphs, with new types of examples. • A tight analysis for important subclasses of via linear programs. • A tight analysis for edge coloring and a tight analysis for unit interval graphs. [ABSTRACT FROM AUTHOR]

Abstract An HV-graph is a planar graph with vertex-degree at most four such that each edge is labeled either H (horizontal) or V (vertical). The HV-planarity testing problem asks whether an HV-graph admits an HV-drawing , that is, a planar drawing such that each edge with label H is drawn as a horizontal segment and each edge with label V is drawn as a vertical segment. We prove that the HV-planarity testing problem is NP-complete even for graphs with vertex-degree at most three, which answers an open question posed by both Manuch et al. [30] and Durocher et al. [17]. On the positive side, we prove that the HV-planarity testing problem can be solved in polynomial-time for series-parallel graphs. This result significantly extends a previous result by Durocher et al. about HV-planarity testing of biconnected outerplanar graphs of maximum vertex-degree three. Highlights • HV-planarity testing is NP-complete, even for graphs with vertex-degree at most 3. • Polynomial-time HV-planarity testing for 2-connected series-parallel graphs. • Polynomial-time HV-planarity testing for partial 2-trees. [ABSTRACT FROM AUTHOR]

Abstract Let R be a finite set of terminals in a convex metric space (M , d). We give approximation algorithms for problems of finding a minimum size set S ⊆ M of additional points such that the unit-disc graph G [ R ∪ S ] of R ∪ S satisfies some connectivity properties. Let Δ be the maximum number of independent points in a unit ball. For the Steiner Tree with Minimum Number of Steiner Points problem we obtain approximation ratio 1 + ln ⁡ (Δ − 1) + ϵ , which in R 2 reduces to 1 + ln ⁡ 4 + ϵ < 2.3863 + ϵ ; this improves the ratios ⌊ (Δ + 1) / 2 ⌋ + 1 + ϵ of [19] and 2.5 + ϵ of [7] , respectively. For the Steiner Forest with Minimum Number of Steiner Points problem we give a simple Δ-approximation algorithm, improving the ratio 2 (Δ − 1) of [21]. We also simplify the Δ-approximation of [4] , when G [ R ∪ S ] should be 2-connected. Highlights • We consider connectivity problems on unit-disc graphs in a convex metric space. • Let Δ be the maximum number of independent points in a unit ball. • For Steiner Tree with Minimum Number of Steiner Points we improve the ratio (Δ + 1) / 2 to 1 + ln ⁡ (Δ − 1). • For Steiner Forest with Minimum Number of Steiner Points we improve the ratio 2 (Δ − 1) to Δ. [ABSTRACT FROM AUTHOR]

We study the approximability of (Finite-)Valued Constraint Satisfaction Problems (VCSPs) with a fixed finite constraint language Γ consisting of finitary functions on a fixed finite domain. Ene et al. have shown that, under a mild technical condition, the basic LP relaxation is optimal for constant-factor approximation for VCSP ( Γ ) unless the Unique Games Conjecture fails. Using the algebraic approach to the CSP, we give new natural algebraic conditions for the finiteness of the integrality gap for the basic LP relaxation of VCSP ( Γ ) and show how this leads to efficient constant-factor approximation algorithms for several examples that cover all previously known cases that are NP-hard to solve to optimality but admit constant-factor approximation. Finally, we show that the absence of another algebraic condition leads to NP-hardness of constant-factor approximation. Thus, our results indicate where the boundary of constant-factor approximability for VCSPs lies. [ABSTRACT FROM AUTHOR]

We consider the classical scheduling problem on parallel identical machines to minimize the makespan. There is a long history of study on this problem, focusing on exact and approximation algorithms. It is thus natural to consider whether these algorithms can be further improved in terms of running times. We provide strong lower bounds on the running times of exact and approximation schemes for the classical scheduling problem based on Exponential Time Hypothesis, showing that the best known approximation and exact algorithms are almost the best possible in terms of the running time. [ABSTRACT FROM AUTHOR]

In bin packing with cardinality constraints (BPCC), there is an upper bound k ≥ 2 on the number of items that can be packed into each bin, additionally to the standard constraint on the total size of items. We study the algorithm First Fit (FF), acting on a list of items, packing each item into the minimum indexed bin that contains at most k − 1 items and has sufficient space for the item. We present a complete analysis of its asymptotic approximation ratio for all values of k . Many years after FF for BPCC was introduced, its tight asymptotic approximation ratio is finally found. [ABSTRACT FROM AUTHOR]

A power assignment is an assignment of transmission power to each of the wireless nodes of a wireless network, so that the induced graph satisfies some desired properties. The cost of a power assignment is the sum of the assigned powers. In this paper, we consider the dual power assignment problem, in which each wireless node is assigned a high- or low-power level, so that the induced graph is strongly connected and the cost of the assignment is minimized. We improve the best known approximation ratio from π 2 6 − 1 36 + ϵ ≈ 1.617 to 11 7 ≈ 1.571 . Moreover, we show that the algorithm of Khuller et al. [11] for the strongly connected spanning subgraph problem, which achieves an approximation ratio of 1.617, is 1.522-approximation algorithm for symmetric directed graphs. The innovation of this paper is in achieving these results by using interesting conditions for the existence of a second Hamiltonian cycle. [ABSTRACT FROM AUTHOR]

We motivate and describe a new parameterized approximation paradigm which studies the interaction between approximation ratio and running time for any parametrization of a given optimization problem. As a key tool, we introduce the concept of an α - shrinking transformation , for α ≥ 1 . Applying such transformation to a parameterized problem instance decreases the parameter value, while preserving the approximation ratio of α (or α-fidelity ). Moving even beyond the approximation ratio, we call for a new type of approximative kernelization race . Our α -shrinking transformations can be used to obtain approximative kernels which are smaller than the best known for a given problem. The smaller “ α -fidelity” kernels allow us to obtain an exact solution for the reduced instance more efficiently, while obtaining an approximate solution for the original instance. We show that such fidelity preserving transformations exist for several fundamental problems, including Vertex Cover , d-Hitting Set , Connected Vertex Cover and Steiner Tree . [ABSTRACT FROM AUTHOR]

In this paper we focus on problems inapproximable in polynomial time and explore how quickly their approximability improves as the allowed running time is gradually increased from polynomial to (sub-)exponential. We tackle a number of problems: For Min Independent Dominating Set , Max Induced Path , Forest and Tree , for any r ( n ) , a simple, known scheme gives an approximation ratio of r in time roughly r n / r . We show that, if this running time could be significantly improved, the ETH would fail. For Max Minimal Vertex Cover we give a r -approximation in time 2 n / r . We match this with a similarly tight result. We also give a log ⁡ r -approximation for Min ATSP in time 2 n / r and an r -approximation for Max Grundy Coloring in time r n / r . Finally, we investigate the approximability of Min Set Cover , when measuring the running time as a function of the number of sets in the input. [ABSTRACT FROM AUTHOR]

A complete weighted graph G = ( V , E , w ) is called Δ β -metric, for some β ≥ 1 / 2 , if G satisfies the β -triangle inequality, i.e., w ( u , v ) ≤ β ⋅ ( w ( u , x ) + w ( x , v ) ) for all vertices u , v , x ∈ V . Given a Δ β -metric graph G = ( V , E , w ) and a center c ∈ V , and an integer p , the Δ β -Star p -Hub Center problem ( Δ β -S p HCP) is to find a depth-2 spanning tree T of G rooted at c such that c has exactly p children (also called hubs) and the diameter of T is minimized. In this paper, we study Δ β -S p HCP for all β ≥ 1 2 . We show that for any ϵ > 0 , to approximate the Δ β -S p HCP to a ratio g ( β ) − ϵ is NP-hard and give r ( β ) -approximation algorithms for the same problem where g ( β ) and r ( β ) are functions of β . A subclass of metric graphs is identified that Δ β -S p HCP is polynomial-time solvable. Moreover, some r ( β ) -approximation algorithms given in this paper meet approximation lower bounds. [ABSTRACT FROM AUTHOR]

The model checking problem of continuous-time Markov chains with respect to continuous-time stochastic logic was introduced and shown to be decidable by Aziz et al. [1,2] in 1996. Unfortunately, their proof is only constructive, but highly unpractical. Later in 2000, an efficient approximate algorithm was proposed by Baier et al. [3,5] for a sublogic with binary until. In this paper, we apply transcendental number theory and classical linear algebra to bridge the gap between the precise but unpractical algorithm, and the imprecise but efficient approximate algorithm. We prove that the approximate algorithm in [3,5] can be used as an off-the-shell tool for a precise model checking algorithm for binary until formulas. Further, we discuss extensions of our results to nested until and continuous-time Markov decision processes. [ABSTRACT FROM AUTHOR]

Recent research in algorithmic robotics considers combinatorial filters , which concisely capture the discrete structure underlying many reasoning problems for robots. An important recent result is that the filter minimization problem —Given a filter, find the smallest equivalent filter—is NP-hard. This paper extends that result along several dimensions, including hardness proofs for some natural special cases and for approximation, and new results analyzing the only known algorithm for this problem. We show that this problem is not fixed-parameter tractable for any of the obvious parameters, but it is fixed-parameter tractable for a certain combination of new parameters. [ABSTRACT FROM AUTHOR]

We study the Weighted t -Uniform Sparsest Cut (Weighted t-USC) and other related problems. In an instance of the Weighted t -USC problem, a parameter t and an undirected graph G = ( V , E ) with edge-weights w : E → R ≥ 0 and vertex-weights η : V → R + are given. The goal is to find a vertex set S ⊆ V with | S | ≤ t while minimizing w ( S , V \ S ) / η ( S ) , where w ( S , V \ S ) is the total weight of the edges with exactly one endpoint in S and η ( S ) = ∑ v ∈ S η ( v ) . For this problem, we present a ( O ( log ⁡ t ) , 1 + ϵ ) factor bicriteria approximation algorithm. Our algorithm outperforms the current best algorithm when t = n o ( 1 ) . We also present better approximation algorithms for Weighted ρ -Unbalanced Cut and Min–Max k -Partitioning problems. [ABSTRACT FROM AUTHOR]

We present new results on Boolean matrix factorization and a new algorithm based on these results. The results emphasize the significance of factorizations that provide from-below approximations of the input matrix. While the previously proposed algorithms do not consider the possibly different significance of different matrix entries, our results help measure such significance and suggest where to focus when computing factors. An experimental evaluation of the new algorithm on both synthetic and real data demonstrates its good performance in terms of good coverage by the first few factors as well as a small number of factors needed for an almost exact decomposition and indicates that the algorithm outperforms the available ones in these terms. We also propose future research topics. [ABSTRACT FROM AUTHOR]

The network verification problem is that of establishing the accuracy of a high-level description of its physical topology, by making as few measurements as possible on its nodes. This task can be formalized as an optimization problem that, given a graph and a query model specifying the information returned by a query at a node, asks for finding a minimum-size subset of nodes to be queried so as to univocally identify the graph. This problem has been studied with respect to different query models, assuming that a node had some global knowledge about the network. Here, we propose a new query model based on the local knowledge a node instead usually has. Quite naturally, we assume that a query at a given node returns the associated routing table, i.e., a set of entries which provides, for each destination node, a corresponding (set of) first-hop node(s) along an underlying shortest path. [ABSTRACT FROM AUTHOR]

Abstract: We study several network design problems with degree constraints. For the minimum-cost Degree-Constrained 2-Node-Connected Subgraph problem, we obtain the first non-trivial bicriteria approximation algorithm, with violation for the degrees and a 4-approximation for the cost. This improves upon the logarithmic degree violation and no cost guarantee obtained by Feder, Motwani, and Zhu (2006). Then we consider the problem of finding an arborescence with at least k terminals and with minimum maximum outdegree. We show that the natural LP-relaxation has a gap of or with respect to the multiplicative degree bound violation. We overcome this hurdle by a combinatorial -approximation algorithm, where denotes the maximum degree in the optimum solution. We also give an lower bound on approximating this problem. Then we consider the undirected version of this problem, however, with an extra diameter constraint, and give an lower bound on the approximability of this version. We also consider a closely related Prize-Collecting Degree-Constrained Edge-Connectivity Survivable Network problem, and obtain several results in this direction by reducing the prize-collecting variant to the regular one. Finally, we consider the Terminal Steiner Tree problem, which is a simple variant of the Degree-Constrained Steiner Tree problem, when some terminals are required to be leaves. We show that this seemingly simple problem is equivalent to the Group Steiner Tree problem. [Copyright &y& Elsevier]

Abstract: Given a graph of interactions, a module (also called a community or cluster) is a subset of nodes whose fitness is a function of the statistical significance of the pairwise interactions of nodes in the module. The topic of this paper is a model-based community finding approach, commonly referred to as modularity clustering, that was originally proposed by Newman (Leicht and Newman, 2008 [25]) and has subsequently been extremely popular in practice (e.g., see Agarwal and Kempe, 2008 [1], Guimer‘a et al., 2007 [20], Newman, 2006 [28], Newman and Girvan, 2004 [30], Ravasz et al., 2002 [32]). Various heuristic methods are currently employed for finding the optimal solution. However, as observed in Agarwal and Kempe (2008) [1], the exact computational complexity of this approach is still largely unknown. To this end, we initiate a systematic study of the computational complexity of modularity clustering. Due to the specific quadratic nature of the modularity function, it is necessary to study its value on sparse graphs and dense graphs separately. Our main results include a -inapproximability for dense graphs and a logarithmic approximation for sparse graphs. We make use of several combinatorial properties of modularity to get these results. These are the first non-trivial approximability results beyond the NP-hardness results in Brandes et al. (2007) [10]. [Copyright &y& Elsevier]

Abstract: A cut-and-paste operation can be a reversal, a transposition, or a transreversal on circular or linear permutations. There are several approximation algorithms for sorting signed permutations by combinations of these operations. For sorting unsigned permutations, we only know an algorithm with performance ratio 3 and its improved version with performance ratio allowing reversals and transpositions. In this paper, we present a 2.25-approximation algorithm for sorting unsigned circular permutations by cut-and-paste operations. A structure called tie is proposed to represent the alternating path of length 5. We can classify the ties into 6 types and find ways to remove the breakpoints for each type of ties, so that every cut-and-paste operation can reduce at least breakpoints averagely. Our algorithm can be used to sort unsigned linear permutations and achieve the performance ratio 2.25 if another operation named revrev is allowed. [Copyright &y& Elsevier]

Abstract: The problem Constrained Longest Common Subsequence is a natural extension to the classical problem Longest Common Subsequence, and has important applications to bioinformatics. Given k input sequences and l constraint sequences , is the problem of finding a longest common subsequence of that is also a common supersequence of . Gotthilf et al. gave a polynomial-time algorithm that approximates within a factor , where is the length of the shortest input sequence and is the alphabet size. They asked whether there are better approximation algorithms and whether there exists a lower bound. In this paper, we answer their questions by showing that their approximation factor is in fact already very close to optimal although a small improvement is still possible: [1.] For any computable function f and any , there is no polynomial-time algorithm that approximates within a factor unless . Moreover, this holds even if the constraint sequence is unary. [2.] There is a polynomial-time randomized algorithm that approximates within a factor with high probability, where OPT is the length of the optimal solution, . For the problem over an alphabet of arbitrary size, we show that [3.] For any , there is no polynomial-time algorithm that approximates within a factor unless . [4.] There is a polynomial-time algorithm that approximates within a factor . We also present some complementary results on exact and parameterized algorithms for . [Copyright &y& Elsevier]