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We present a new approximation algorithm for the minimum 2-edge-connected spanning subgraph problem. Its approximation ratio is 4 3 , which matches the current best ratio. The approximation ratio of the algorithm is 6 5 on subcubic graphs, which is an improvement upon the previous best ratio of 5 4. The algorithm is a novel extension of the primal-dual schema, which consists of two distinct phases. Both the algorithm and the analysis are much simpler than those of the previous approaches. [ABSTRACT FROM AUTHOR]

The squared metric k -facility location problem is a frequently encountered generalization of the k -means problem, where a specific cost should be paid for opening each facility. The current best approximation ratio for this problem is 44.473 + ϵ , which was obtained using a local search algorithm. We advance the state-of-the-art for the problem by devising a Lagrangian relaxation-based algorithm that achieves an improved approximation guarantee of 36.342 + ϵ. Our improvement comes from a new deterministic rounding approach, which exploits the properties of the squared metric. • We give a (36.342 + ϵ) -approximation algorithm for the squared metric k -facility location problem. • We obtain a well-behaved fractional solution to the squared metric k -facility location problem based on the framework of Lagrangian relaxation. • We give a new deterministic rounding approach that exploits the properties of the squared metric. [ABSTRACT FROM AUTHOR]

This work considers the Minimum Path-Collection Exact Cover (PCEC) and the Minimum k -Path Splitting Exact Cover (k-PSEC). Both problems receive a digraph G and a set P of paths in G , and their objective is to find a minimum cardinality set S of paths, whose elements are arc-disjoint and cover all arcs of G. Despite the similarities, the solutions for each problem must satisfy different properties. For k-PSEC , the set S must be obtained by splitting the paths in P in order to guarantee that no element of S has length greater than a given integer k. For PCEC , the paths in P cannot be split, and the elements of S are single arcs of G or complete paths of P. PCEC and k-PSEC have practical applications in network design and network monitoring systems, being also natural versions of graph covering problems. However, there are no theoretical studies on their complexity. This work not only presents NP-hardness results for the problems, but also proves that, unless P = NP , PCEC cannot be | P | O (1) -approximated in polynomial-time. Moreover, polynomial-time algorithms are presented for paths, cycles, and trees, and polynomial-time approximation algorithms are proposed for special cases of k-PSEC. [ABSTRACT FROM AUTHOR]

Abstract polymer models are systems of weighted objects, called polymers, equipped with an incompatibility relation. An important quantity associated with such models is the partition function, which is the weighted sum over all sets of compatible polymers. Various approximation problems reduce to approximating the partition function of a polymer model. Central to the existence of such approximation algorithms are weight conditions of the respective polymer model. Such conditions are derived either via complex analysis or via probabilistic arguments. We follow the latter path and establish a new condition—the clique dynamics condition—, which is less restrictive than the ones in the literature. We introduce a new Markov chain where the clique dynamics condition implies rapid mixing by utilizing cliques of incompatible polymers that naturally arise from the translation of algorithmic problems into polymer models. This leads to improved parameter ranges for several approximation algorithms, such as a factor of at least 2 1 / α for the hard-core model on bipartite α -expanders. [ABSTRACT FROM AUTHOR]

The traveling salesman problem (TSP) is one of the classic research topics in the field of operations research, graph theory and computer science. In this paper, we propose a generalized model of traveling salesman problem, denoted by generalized traveling salesman path problem. Let G = (V , E , c) be a weighted complete graph, in which c is a nonnegative metric cost function on edge set E , i.e., c : E → R +. The traveling salesman path problem aims to find a Hamiltonian path in G with minimum cost. Compared to the traveling salesman path problem, we are given extra vertex subset V ′ and edge subset E ′ in the problem proposed in this paper; its goal is to construct a path which traverses all the edges in E ′ while only needs to visit each vertex in V ′ exactly once. Based on integer programming, we give a mathematical model of the problem, and design a 1 + 5 2 -approximation algorithm for the problem by combining linear programming rounding strategy and a special graph structure. • In this paper, we consider a generalized model of traveling salesman problem. • Based on integer programming, we give a mathematical model of the problem. • Combining linear programming rounding strategy and a special graph structure, we propose a 1 + 5 2 -approximation algorithm. [ABSTRACT FROM AUTHOR]

We investigate two parallel dedicated machine scheduling with conflict constraints. The problem of minimizing the makespan has been shown to be NP-hard in the strong sense under the assumption that the processing sequence of jobs on one machine is given and fixed a priori. The problem without any fixed sequence was previously recognized as weakly NP-hard. In this paper, we first present a 9 5 -approximation algorithm for the problem with a fixed sequence. Then we show that the tight approximation ratios of the algorithm are 7 4 and 5 3 for two subproblems which remain strongly NP-hard. We also send an improved algorithm with approximation ratio 3 − 2 ≈ 1.586 for one subproblem. Finally, we prove that the problem without any fixed sequence is actually strongly NP-hard, and design a 5 3 -approximation algorithm. • Consider the problem of scheduling with conflict constraints on two parallel dedicated machines. • Present a 9 5 -approximation algorithm for the strongly NP-hard case where jobs on one machine have a fixed processing sequence, and give improvements to its hard subproblems. • Prove that the problem without any fixed sequence is strongly NP-hard too and propose a 5 3 -approximation algorithm. [ABSTRACT FROM AUTHOR]

Emerging wireless and mobile applications, such as border intrusion detection with station-based drones, brought a new barrier coverage problem of using sink-based mobile sensors to cover a given line barrier with minimum energy consumption. In this paper, we focus on the uniform sink-based line barrier coverage (SLBC) problem, in which we are given a line barrier and k sink stations distributed on the plane which can emit an infinite number of sensors with an identical sensing radius. The problem aims to find their final positions on the barrier for the sensors emitted by the stations, such that the total moving distance of the sensors is minimized and each point of the barrier is within the sensing area of at least one sensor. We first observe the geometric structure of an optimal solution that any optimal solution can be considered as a set of intersecting tangent (disk) segments, where a tangent (disk) segment is a sequence of tangent disks. Then, we devise an algorithm to calculate all possible tangent (disk) segments and another algorithm to calculate the near-optimal positions for each of such segments. After computing all tangent (disk) segments and their near-optimal positions, an algorithm is proposed to transform uniform SLBC into an instance of the shortest path problem. It is shown the whole algorithm deserves a runtime O (k 2 log ⁡ k r ε) and consumes at most ε more movement than an optimal solution, where ε is any given positive real number, and r and k are the sensor radius and the number of sink stations, respectively. • The algorithm consumes a runtime O (k 2 log ⁡ k r ε) and produces a solution with at most ε more movement than that of an optimum solution. • With the runtime and the performance guarantee, the proposed algorithm behaves in a manner similar to a polynomial-time exact algorithm. • We proved several geometric properties of an optimum solution, which might have possible applications in other variants of the problem. [ABSTRACT FROM AUTHOR]

The rooted Budgeted Cycle Cover (BCC) problem is a fundamental optimization problem arising in wireless sensor networks and vehicle routing. Given a metric space (V , w) with vertex set V consisting of two parts D (containing depots) and V ∖ D (containing nodes), and a budget B ≥ 0 , the rooted BCC problem asks to find a minimum number of cycles to cover all the nodes in V ∖ D , satisfying that each cycle has length at most B and each cycle must contain a depot in D. In this paper, we give new approximation algorithms for the rooted BCC problem. For the rooted BCC problem with single depot, we give an O (log ⁡ B μ) -approximation algorithm, where μ is the minimum difference of any two different distances between the vertices in V and the root. For the rooted BCC problem with multiple depots, we give an O (log ⁡ n) -approximation algorithm, where n is the number of vertices. We also test our algorithms on the randomly generated instances. The experimental results show that the algorithms have good performance in practice. • A log-factor approximation algorithm for the single depot budgeted cycle cover problem. • A log-factor approximation algorithm for the multi-depot budgeted cycle cover problem. • Experiments on randomly generated instances show good performance of the algorithms. [ABSTRACT FROM AUTHOR]

In this work, we study a range of constrained versions of the k-supplier and k-center problems. In the classical (unconstrained) k -supplier problem, we are given a set of clients C in a metric space X , with distance function d (. ,.). We are also given a set of feasible facility locations L ⊆ X. The goal is to open a set F of k facilities in L to minimize the maximum distance of any client to the closest open facility, i.e., minimize, cost (F , C) ≡ max j ∈ C ⁡ { d (F , j) } , where d (F , j) is the distance of client j to the closest facility in F. The k -center problem is a special case of the k -supplier problem where L = C. We study various constrained versions of the k -supplier problem such as: capacitated , fault-tolerant , ℓ -diversity, etc. These problems fall under a broad framework of constrained clustering. A unified framework for constrained clustering was proposed by Ding and Xu [Algorithmica 2020] in the context of the k -median and k -means objectives. We extend this framework to the k -supplier and k -center objectives in this work. This unified framework allows us to obtain results simultaneously for the following constrained versions of the k -supplier problem: r-gather, r-capacity, balanced, chromatic, fault-tolerant, strongly private, ℓ-diversity, and fair k-supplier problems, with and without outliers. We design Fixed-Parameter Tractable (FPT) algorithms for these problems. FPT algorithms have polynomial running time if the parameter under consideration is a constant. This may be relevant even to a practitioner since the parameter k is a small number in many real clustering scenarios. We obtain the following results: • We give 3 and 2 approximation algorithms for the constrained k -supplier and k -center problems, respectively, with FPT running time k O (k) ⋅ n O (1) , where n = | C ∪ L |. Moreover, these approximation guarantees are tight; that is, for any constant ε > 0 , no algorithm can achieve (3 − ε) and (2 − ε) approximation guarantees for the constrained k -supplier and k -center problems in FPT time, assuming FPT ≠ W [ 2 ]. • We study the constrained clustering problem with outliers. Our algorithm gives 3 and 2 approximation guarantees for the constrained outlier k -supplier and k -center problems, respectively, with FPT running time (k + m) O (k) ⋅ n O (1) , where n = | C ∪ L | and m is the number of outliers. • Our techniques generalise for distance function d (. ,.) z. That is, for any positive real number z , if the cost of a client is defined by d (. ,.) z instead of d (. ,.) , then our algorithm gives 3 z and 2 z approximation guarantees for the constrained k -supplier and k -center problems, respectively. • We design 3 approximation algorithm for a wide range of constrained k -supplier problems with FPT running time f (k). p o l y (n). • We design 2 approximation algorithm for a range of constrained k -center problems with FPT running time f (k). p o l y (n). • We extend the algorithm to outlier setting with FPT running time f (k , m). p o l y (n) , where m = number of outliers. • We show matching lower bounds for the constrained k -supplier and k -center problems in the outlier and non-outlier settings. [ABSTRACT FROM AUTHOR]

We study worst-case fairness in resource allocation and cooperative games with transferable utility, the stable division most dissimilar to a normative standard of fairness. Motivated by welfare economics, similarity is quantified using information-theoretic divergences. Worst-case fairness aims to parallel the spirit of the price of anarchy from noncooperative game theory, quantifying how much unfairness is compatible with coalitional rationality. Computing worst-case fairness is tractable in weighted voting games and classes of coalitional skill games, but NP-hard in highway allocation, induced-subgraph games and some task-count coalitional skill games. In these cases we show approximation algorithms that yield constant approximations. We also upper bound the performance of a Reverse Greedy algorithm on general convex games in terms of two game-specific constants. • We study worst-case fairness in resource allocation and cooperative games with transferable utility. • We consider the imputation in the core with the largest divergence with a fixed imputation, a "standard of fairness". • We study the problem for several classes of cooperative games with transferable utility. • In cases when worst-case fairness is intractable we give algorithms that produce an approximately worst-case fair imputation. [ABSTRACT FROM AUTHOR]

A mixed dominating set of a graph G = (V , E) is a mixed set D of vertices and edges, such that for every edge or vertex, if it is not in D , then it is adjacent or incident to at least one vertex or edge in D. The mixed domination problem is to find a mixed dominating set with a minimum cardinality. It has applications in system control and some other scenarios and it is NP -hard to compute an optimal solution. This paper studies approximation algorithms and hardness of the weighted mixed dominating set problem. The weighted version is a generalization of the unweighted version, where all vertices are assigned the same nonnegative weight w v and all edges are assigned the same nonnegative weight w e , and the question is to find a mixed dominating set with a minimum total weight. Although the mixed dominating set problem has a simple 2-approximation algorithm, few approximation results for the weighted version are known. The main contributions of this paper include: 1. for w e ≥ w v , a 2-approximation algorithm; 2. for w e ≥ 2 w v , inapproximability within ratio 1.3606 unless P = N P and within ratio 2 under UGC; 3. for 2 w v > w e ≥ w v , inapproximability within ratio 1.1803 unless P = N P and within ratio 1.5 under UGC; 4. for w e < w v , inapproximability within ratio (1 − ϵ) ln ⁡ | V | unless P = N P for any ϵ > 0. [ABSTRACT FROM AUTHOR]

Although the knapsack-constrained and k -system-constrained non-monotone adaptive submodular maximization have been well studied in the literature, it has only been settled given the additional assumption of pointwise submodularity. In this paper, we remove the common assumption on pointwise submodularity and propose the first approximation solutions for both knapsack and k -system constrained adaptive submodular maximization problems. Inspired by two recent studies on non-monotone adaptive submodular maximization, we develop a sampling-based randomized algorithm that achieves a 1 10 approximation ratio for the case of a knapsack constraint and that achieves a 1 2 k + 4 approximation ratio for the case of a k -system constraint. • We study the non-monotone adaptive submodular maximization problem. • We present a 1 10 approximation algorithm for knapsack constraints. • We present a 1 2 k + 4 approximation algorithm for k -system constraints. • Our results do not rely on pointwise submodularity. [ABSTRACT FROM AUTHOR]

Consistent query answering (CQA) is a widely accepted paradigm for querying inconsistent knowledge bases (KBs). A consistent answer to a query is a tuple that is an answer to the query over every repair of the KB, which is in turn a consistent KB whose extensional knowledge "minimally" differs from the original one's. This coarse-grained classification of answers into consistent and non-consistent ones lacks any information about their degree of consistency, i.e., how likely it is that a tuple is an answer to the query, when considering all the repaired KBs. To overcome this limitation, we consider a fine-grained notion of repair for KBs with equality-generating dependencies (EGDs), based on attribute-level updates, and exploit this notion to propose a probabilistic CQA approach, which associates a confidence to each answer, thereby providing more informative query answers. We first show that computing the query answer confidence is # P -hard. Then, in the light of this intractability result, we study the existence of efficient randomized, approximation schemes. In particular, we show that absolute error approximation schemes always exist in the general case, while more refined relative error approximation schemes, i.e., fully polynomial-time, randomized approximation schemes (FPRAS) exist when assuming that the constraints of the knowledge base are primary keys. Finally, we extend our framework to knowledge bases with tuple-generating dependencies (TGDs) and generalize our approximability results to the new setting, and prove additional inapproximability results. [ABSTRACT FROM AUTHOR]

Given a complete undirected graph G = (V , E) , where V is the vertex set partitioned into K clusters V 1 , V 2 , ... , V K and E is the edge set with edge weights satisfying triangle inequality, and a positive integer k , the min-max clustered k-traveling salesmen problem (min-max C k -TSP) asks to find a set of k tours to visit all vertices, such that each cluster is visited by exactly one tour and the vertices of each cluster are visited consecutively. The objective is to minimize the weight of the maximum weight tour. The problem is known to be NP-hard even when k = 1 and K = 1. In this paper, we consider two variants of the problem. The first one is all the k tours have a common predefined starting vertex, and the other one is no starting vertex of any tour is specified. For both the variants we propose the first constant-factor approximation algorithms with ratios 5.5 and 16, respectively. • We consider the min-max clustered k-traveling salesmen problem. • We propose a 5.5-approximation algorithm for the case in which all the k tours have a common predefined starting vertex. • We provide a 16-approximation algorithm for the case in which no starting vertex of any tour is specified. • Both algorithms are the first constant-factor approximation algorithms. [ABSTRACT FROM AUTHOR]

• We study the heterogeneous weighted delivery (HWD) problem, which was introduced in [Bärtschi et al., STACS'17]. • We efficiently 8-approximate HWD with k = O (log ⁡ n) agents, closing an open problem in [Bärtschi et al., ATMOS'17]. • We design a O (k) -approximation algorithm that runs in polynomial time regardless of the value of k. • We design a polynomial-time O ˜ (log 3 ⁡ n) -approximation algorithm. • We show that the HWD problem is 36-approximable in polynomial time when each agent has one of two possible consumption rates. We study the heterogeneous weighted delivery (HWD) problem introduced in [Bärtschi et al., STACS'17] where k heterogeneous mobile agents (e.g., robots, vehicles, etc.), initially positioned on vertices of an n -vertex edge-weighted graph G , have to deliver m messages. Each message is initially placed on a source vertex of G and needs to be delivered to a target vertex of G. Each agent can move along the edges of G and carry at most one message at any time. Each agent has a rate of energy consumption per unit of traveled distance and the goal is that of delivering all messages using the minimum overall amount of energy. This problem has been shown to be NP-hard even when k = 1 , and is 4 ρ -approximable where ρ is the ratio between the maximum and minimum energy consumption of the agents. In this paper, we provide approximation algorithms with approximation ratios independent of the energy consumption rates. First, we design a polynomial-time 8-approximation algorithm for k = O (log ⁡ n) , closing a problem left open in [Bärtschi et al., ATMOS'17]. This algorithm can be turned into an O (k) -approximation algorithm that always runs in polynomial-time, regardless of the values of k. Then, we show that HWD problem is 36-approximable in polynomial-time when each agent has one of two possible consumption rates. Finally, we design a polynomial-time O ˜ (log 3 ⁡ n) -approximation algorithm for the general case. [ABSTRACT FROM AUTHOR]

Consider an information diffusion process on a graph G that starts with k > 0 burnt vertices, and at each subsequent step, burns the neighbors of the currently burnt vertices, as well as k other unburnt vertices. The k-burning number of G is the minimum number of steps b k (G) such that all the vertices can be burned within b k (G) steps. Note that the last step may have smaller than k unburnt vertices available, where all of them are burned. The 1-burning number coincides with the well-known burning number problem, which was proposed to model the spread of social contagion. The generalization to k -burning number allows us to examine different worst-case contagion scenarios by varying the spread factor k. In this paper we prove that computing k -burning number is APX-hard, for any fixed constant k. We then give an O ((n + m) log ⁡ n) -time 3-approximation algorithm for computing k -burning number, for any k ≥ 1 , where n and m are the number of vertices and edges, respectively. Finally, we show that even if the burning sources are given as an input, computing a burning sequence itself is an NP-hard problem. [ABSTRACT FROM AUTHOR]

In the capacitated facility location problem, we are given a set F of potential facilities and a set D of clients, where each facility has a capacity and an open cost, and each client has a demand to be served by the facilities with service costs. The goal is to open some facilities in F and assign all clients in D to these open facilities such that the total cost is minimum. Based on the natural integer programming formulation, Levi et al. [8] presented an LP-based 5-approximation algorithm for this problem under the assumption that the facility costs are uniform. Based on the same integer programming formulation, we remove the uniformity assumption and present an (R + R 2 + 8 R 2 + 3) -approximation algorithm for the capacitated facility location problem, where R is the upper bound of the ratio between facility costs. Our result is a slight extension of the corresponding result in [8] , as when R = 1 the worst-case ratio of our algorithm is R + R 2 + 8 R 2 + 3 = 5. [ABSTRACT FROM AUTHOR]

In a minimum general partial dominating set problem (MinGPDS), given a graph G = (V , E) , a profit function p : V → R + and a threshold K , the goal is to find a minimum subset of vertices D ⊆ V such that the total profit of those vertices dominated by D is at least K (a vertex is dominated by D if it is either in D or has at least one neighbor in D). In a maximum general budgeted dominating set problem (MaxGBDS), given a budget B , the goal is to find a vertex set D with at most B vertices such that the total profit of those vertices dominated by D is as large as possible. We present the first parallel algorithms for MinGPDS and MaxGBDS in unit disk graphs. They both run in O (log ⁡ n) rounds on O (n) machines, and achieve constant approximation ratios. • First parallel algorithms for minimum partial dominating set and maximum budgeted dominating set in unit disk graphs. • Constant approximation ratios are obtained in logarithmic rounds. • Overcome the challenge brought by the ̀̀partial" consideration by implementing a greedy idea in a parallelized manner. [ABSTRACT FROM AUTHOR]

• We combine the distorted objective function, the threshold and the greedy algorithms. • This problem includes some previously studied problems as special cases, such as submodular+supermodular maximization, γ -weakly submodular function maximization. • The approximation ratio in Algorithm 1 is the minimum of 1 − k f e − 1 and 1 − (k g) 2 , improving the result obtained by Bai et al. • Algorithm 1 has an approximation ratio 1 − (1 − γ + γ α) e − γ , improving the result obtained by Bian et al. • Set g (S) = 0. Algorithm 2 has an approximation ratio γ γ + 1 , improving the approximate ratio 1 − 1 2 γ by Wang et al. The concept of submodularity finds wide applications in data science, artificial intelligence, and machine learning, providing a boost to the investigation of new ideas, innovative techniques, and creative algorithms to solve different submodular optimization problems arising from a diversity of applications. However pure submodular or supermodular problems only represent a small portion of the problems we are facing in real life applications. The main focus of this work is to consider a non-submodular function maximization problem subject to a cardinality constraint, where the objective function is the sum of a monotone γ -weakly submodular function and a supermodular function. This problem includes some previously studied problems as special cases, such as the submodular+supermodular maximization problem when γ =1, and the γ -weakly submodular function maximization problem when the supermodular function is void. We present greedy algorithms for this generalized problem under both offline and streaming models, improving existing results. [ABSTRACT FROM AUTHOR]

Consider a Gaussian downlink between an access point with power budget P > 0 , and a set of users specified by their effective noises and rate demands. In order to control the decoding complexity and error propagation, an integer-valued access limit M > 0 is imposed on the number of superimposed users. For each subset S of users, its (total) power demand p (S) is a strictly increasing and nonseparable function of the effective noises and rate demands of users in S. A subset S of users is feasible if | S | ≤ M and p (S) ≤ P. The goal is to select a feasible subset S of users whose total rate demand is maximized. In this paper, we show that this problem is NP-hard, and present a (1 − 1 / e) -approximation algorithm for this problem. In addition, we also give several other approximation algorithms with trade-offs between accuracy and efficiency. • The Gaussian downlink user selection problem studied in this paper is an NP-hard maximization problem. • The first 1 2 (1 − 1 / e) -approximation algorithm takes the relaxation/extraction approach. • A better algorithm utilizes partial enumeration for reducing the extraction loss, and achieves (1 − 1 / e) -approximation factor. • The paper is concluded with a summary of the general algorithmic framework and some open problems for further studies. [ABSTRACT FROM AUTHOR]

What approximation ratio can we achieve for the Facility Location problem if whenever a client u connects to a facility v , the opening cost of v is at most θ times the service cost of u ? We show that this and many other problems are a particular case of the Activation Edge-Cover problem. Here we are given a multigraph G = (V , E) , a set R ⊆ V of terminals, and thresholds { t u e , t v e } for each uv -edge e ∈ E. The goal is to find an assignment a = { a v : v ∈ V } to the nodes minimizing ∑ v ∈ V a v , such that the edge set E a = { e = u v : a u ≥ t u e , a v ≥ t v e } activated by a covers R. We obtain ratio 1 + max x ≥ 1 ⁡ ln ⁡ x 1 + x / θ ≈ ln ⁡ θ − ln ⁡ ln ⁡ θ for the problem, where θ ≤ max e ∈ E ⁡ max ⁡ { t u e , t v e } min ⁡ { t u e , t v e } is a problem parameter. This result is based on a simple generic algorithm for the problem of minimizing a sum of a decreasing and a sub-additive set functions, which is of independent interest. As an application, we get the same ratio for the above variant of Facility Location. If for each facility all service costs are identical then we show a better ratio 1 + max k ∈ N ⁡ H k − 1 1 + k / θ , where H k = ∑ i = 1 k 1 / i. For the Min-Power Edge-Cover problem we improve the ratio 1.406 of [4] (achieved by iterative randomized rounding) to 1.2785. For unit thresholds we improve the ratio 73 / 60 ≈ 1.217 of [4] to 1555 1347 ≈ 1.155. [ABSTRACT FROM AUTHOR]

A dominating set of a graph G = (V , E) is a subset of vertices S ⊆ V such that every vertex v ∈ V ∖ S has at least one neighbor in S. Finding a dominating set with the minimum cardinality in a connected graph G = (V , E) is known to be NP-hard. A polynomial-time approximation algorithm for this problem, described here, works in two stages. At the first stage a dominant set is generated by a greedy algorithm, and at the second stage this dominating set is purified (reduced). The reduction is achieved by the analysis of the flowchart of the algorithm of the first stage and a special kind of clustering of the dominating set generated at the first stage. The clustering of the dominating set naturally leads to a special kind of a spanning forest of graph G , which serves as a basis for the second purification stage. We expose some types of graphs for which the algorithm of the first stage already delivers an optimal solution and derive sufficient conditions when the overall algorithm constructs an optimal solution. We give three alternative approximation ratios for the algorithm of the first stage, two of which are expressed in terms of solely invariant problem instance parameters, and we also give one additional approximation ratio for the overall two-stage algorithm. The greedy algorithm of the first stage turned out to be essentially the same as the earlier known state-of-the-art algorithms for the set cover and dominating set problem Chvátal [6] and Parekh [31]. The second purification stage results in a significant reduction of the dominant set created at the first stage, in practice. The practical behavior of both stages was verified for randomly generated problem instances. The computational experiments emphasize the gap between a solution of Stage 1 and a solution of Stage 2. [ABSTRACT FROM AUTHOR]

• The smallest minimum color-spanning circle (S-MCSC) problem can be solved in O (n k log ⁡ n) time. • The largest minimum color-spanning circle (L-MCSC) problem is NP-Hard. • A 1 3 -factor (1 2 , if no two distinct color disks intersect) approximation to the L-MCSC problem can be computed in O (n k log ⁡ n) time. Let R be a set of n colored imprecise points, where each point is colored by one of k colors. Each imprecise point is specified by a unit disk in which the point lies. We study the problem of computing the smallest and the largest possible minimum color spanning circle, among all possible choices of points inside their corresponding disks. We present an O (n k log ⁡ n) time algorithm to compute a smallest minimum color spanning circle. Regarding the largest minimum color spanning circle, we show that the problem is NP - Hard and present a 1 3 -factor approximation algorithm. We improve the approximation factor to 1 2 for the case where no two disks of distinct color intersect. [ABSTRACT FROM AUTHOR]

For a graph G = (V , E) , a subset D of vertex set V , is a dominating set of G if every vertex not in D is adjacent to at least one vertex of D. A dominating set D of a graph G with no isolated vertices is called a paired dominating set (PD-set) , if the subgraph induced by D in G has a perfect matching. The Min-PD problem requires to compute a PD-set of minimum cardinality. The decision version of the Min-PD problem is NP-complete even when G belongs to restricted graph classes such as bipartite graphs, chordal graphs etc. On the other side, the problem is efficiently solvable for many graph classes including interval graphs, strongly chordal graphs, permutation graphs etc. In this paper, we study the complexity of the problem in AT-free graphs and planar graphs. The class of AT-free graphs contains cocomparability graphs, permutation graphs, trapezoid graphs, and interval graphs as subclasses. We propose a polynomial-time algorithm to compute a minimum PD-set in AT-free graphs. In addition, we also present a linear-time 2-approximation algorithm for the problem in AT-free graphs. Further, we prove that the decision version of the problem is NP-complete for planar graphs, which answers an open question asked by Lin et al. (2015) [19] and (2020) [20]. Alvarado et al. (2015) [1] conjectured that given a graph G = (V , E) , it is NP-hard to decide whether γ (G) = γ p r (G). In this paper we settle this conjecture affirmatively. [ABSTRACT FROM AUTHOR]

• Propose a new problem: prize-collecting sweep cover problem. • Give a 5-LMP algorithm for prize-collecting sweep cover assuming constant number of base-stations. • Give a 2-LMP algorithm for the rooted prize-collecting forest with k components problem. In a sweep cover problem, positions of interest (PoIs) are required to be visited periodically by mobile sensors. In this paper, we propose a new sweep cover problem: the prize-collecting sweep cover problem (PCSC), in which penalty is incurred by those PoIs which are not sweep-covered, and the goal is to minimize the covering cost plus the penalty. Assuming that every mobile sensor has to be linked to some base station, and the number of base stations is upper bounded by a constant, we present a 5-LMP (Lagrangian Multiplier Preserving) algorithm. As a step stone, we propose the prize-collecting forest with k components problem (PCF k), which might be interesting in its own sense, and presented a 2-LMP for rooted PCF k. [ABSTRACT FROM AUTHOR]

The influence maximization problem aims to find some seeds which can cause the maximum influence spread results in a social network. Most researches focus on the non-adaptive strategies, in which all seeds are selected at once. For the non-adaptive strategies, the seeds may influence other seeds in the influence spread process and make the waste of budget. This paper considers the adaptive strategies and studies the adaptive influence maximization and adaptive stochastic influence maximization in the general feedback model. These problems select seeds adaptively, and it completes each selection after the fixed observation time-step. In this paper, we utilize the adaptive greedy to solve these problems and propose a theoretical analysis by introducing a comparative factor. In addition, we present the feasible approximation algorithm using the reverse sampling technique. Finally, we carry out experiments on three networks to show the efficiency of adaptive strategies. • We study the adaptive influence maximization problem under a general model. • We propose a comparative factor and obtain the approximation ratio of an adaptive greedy algorithm. • We not only consider the problem with cardinality constraint but also the knapsack constraint. • We conduct several experiments on three real datasets to show the effectiveness of adaptive methods. [ABSTRACT FROM AUTHOR]

• We study the task utility maximization problem by scheduling mobile chargers and prove the NP-hardness of the problem. • We design three approximation algorithms for our problems, and prove the approximation ratio through theoretical analysis. • We validate the effectiveness of our algorithms by conducting extensive simulations. Replenishing energy to wireless sensor networks is always a crucial problem as the energy capacity of sensor nodes is very limited. Scheduling mobile chargers to charge sensor nodes has been widely studied due to its efficiency and flexibility. However, most existing works focus on maximizing the charging utility or charging efficiency, which ignores the task performing functions of sensor nodes. In this paper, we study the problem of scheduling a mobile charger to maximize the task utility achieved by sensor nodes. We consider two different scenarios where sensor nodes are deployed on a line road and a ring road, respectively. We prove the NP-Hardness of our problems and design two approximation algorithms with guaranteed performance. We also investigate how to jointly schedule two mobile chargers in the line road scenario, and design a corresponding approximation algorithm to address the problem. We prove the approximation ratio of our algorithms through theoretical analysis, and conduct extensive simulations to validate the performance of our algorithms. Simulation results show that our algorithms always outperform the baselines, which demonstrates the effectiveness of our algorithms. [ABSTRACT FROM AUTHOR]

We consider the single-machine time-dependent scheduling problem with linearly deteriorating jobs arriving over time. Each job i is associated with a release time r i and a processing time p i (s i) = α i + β i s i , where α i , β i > 0 are parameters and s i is the job's start time. In this setting, the approximability of both single-machine minimum makespan and total completion time problems remains open. We develop new bounds and approximation results for the special case of the problems with uniform deterioration, i.e. β i = β , for each i. The main contribution is a O (1 + 1 / β) -approximation algorithm for the makespan problem and a O (1 + 1 / β 2) approximation algorithm for the total completion time problem. Further, we propose greedy constant-factor approximation algorithms for instances with β = O (1 / n) and β = Ω (n) , where n is the number of jobs. Our analysis is based on an approach for comparing computed and optimal schedules via bounding pseudomatchings. [ABSTRACT FROM AUTHOR]

K-best enumeration , which asks to output k -best solutions without duplication, is a helpful tool in data analysis for many fields. In such fields, graphs typically represent data. Thus subgraph enumeration has been paid much attention to such fields. However, k -best enumeration tends to be intractable since, in many cases, finding one optimum solution is NP-hard. To overcome this difficulty, we combine k -best enumeration with a concept of enumeration algorithms called approximation enumeration algorithms. As a main result, we propose a 4-approximation algorithm for minimal connected edge dominating sets which outputs k minimal solutions with cardinality at most 4 ⋅ OPT ‾ , where OPT ‾ is the cardinality of a minimum solution which is not outputted by the algorithm. Our proposed algorithm runs in O (n m 2 Δ) delay, where n , m , Δ are the number of vertices, the number of edges, and the maximum degree of an input graph. [ABSTRACT FROM AUTHOR]

Damian and Pemmaraju (2002) [8] introduced leaf sector covers for circle graphs and presented an O (n 2) -time algorithm to find this data structure. They subsequently employed it as an algorithmic tool, leading to the development of an 8-approximation algorithm for the dominating set problem, a (2 + ε) -approximation scheme for the dominating set problem, and a (3 + ε) -approximation scheme for the total dominating set problem. In this paper, we have improved the time complexity from O (n 2) to O (n + m) for finding a leaf sector cover. Furthermore, we employed leaf sector covers as a powerful algorithmic tool to design a 10-approximation algorithm for the total dominating set problem and a 12-approximation algorithm for the paired-dominating set problem. Moreover, we also proposed a (2 + ε) -approximation scheme for the total dominating set problem and a (4 + ε) -approximation scheme for the paired-dominating set problem. The results presented above are currently the best algorithms for circle graphs. [ABSTRACT FROM AUTHOR]

Let T be a tessellation composed of equilateral triangular regions, where each region has an associated positive weight. We present two methods that discretize the space based on the placement of Steiner points in the cells of T. Using such a discretization, we can use Dijkstra's algorithm for computing the shortest path in the geometric graph obtained. This will lead us to two approximation algorithms for solving the Weighted Region Problem. For a given parameter ε ∈ (0 , 1 ] , the first discretization scheme provides an approximate path that is (1 + 0.428 ε) times better than the approximation given by Aleksandrov et al. (2005) [5]. The other discretization scheme uses at least ( ε + 2 ε + 4 ε + 4) log 2 ⁡ e fewer points per segment of the triangulation with the same approximation factor. [ABSTRACT FROM AUTHOR]

Given a weighted graph G = (V , E) with weight functions c : E → R + and π : V → R + , and a subset U ⊆ V , the normalized cut value for U is defined as the sum of the weights of edges exiting U divided by the weight of vertices in U. The mean isoperimetry problem , ISO 1 (G , k) , for a weighted graph G is a generalization of the classical uniform sparsest cut problem in which, given a parameter k , the objective is to find k disjoint nonempty subsets of V minimizing the average normalized cut value of the parts. The robust version of the problem seeks an optimizer where the number of vertices that fall out of the subpartition is bounded by some given integer 0 ≤ ρ ≤ | V |. The problem may also be considered as the normalized version of the classical k -multiway cut problem on edge-weighted graphs. Our main result states that ISO 1 (G , k) , as well as its robust version, CRISO 1 (G , k , ρ) , subjected to the condition that each part of the subpartition induces a connected subgraph, are solvable in time O (k 2 ρ 2 π (V (T)) 3) on any weighted tree T , in which π (V (T)) is the sum of the vertex-weights. This result implies that ISO 1 (G , k) is strongly polynomial-time solvable on weighted trees when the vertex-weights are polynomially bounded and may be compared to the fact that the problem is NP-hard for weighted trees in general. As far as applications are concerned, the connectivity requirement may be interpreted as an approach to model the practical consistency of the parts, which together with having control on the size of the outlier set and applying a smooth "mean" cost function (as opposed to, say, the "max" version), characterizes our solution to CRISO 1 (G , k , ρ) on trees as one of the most flexible and accurate procedures within the framework of isoperimetry-based clustering. Also, using this, we show that both mentioned problems, ISO 1 (G , k) and CRISO 1 (G , k , ρ) as well as the ordinary robust mean isoperimetry problem RISO 1 (G , k , ρ) , admit polynomial-time O (log 1.5 ⁡ | V | log ⁡ log ⁡ | V |) -approximation algorithms for weighted graphs with polynomially bounded weights, using the Räcke-Shah tree cut sparsifier. [ABSTRACT FROM AUTHOR]

In the Upper Dominating Set problem, the goal is to find a minimal dominating set of maximum size. We study the complexity of parameterized algorithms for Upper Dominating Set , as well as its sub-exponential approximation. First, we prove that, under the ETH, Upper Dominating Set cannot be solved in time O (n o (k)) (improving on O (n o (k))), and in the same time we show under the same complexity assumption that for any constant ratio r and any ε > 0 , there is no r -approximation algorithm running in time O (n k 1 − ε ). Then, we settle the problem's complexity parameterized by pathwidth by giving an algorithm running in time O ⁎ (6 p w) (improving the current best O ⁎ (7 p w)), and a lower bound showing that our algorithm is the best we can get under the SETH. Furthermore, we obtain a simple sub-exponential approximation algorithm for this problem: an algorithm that produces an r -approximation in time n O (n / r) , for any desired approximation ratio r < n. We finally show that this time-approximation trade-off is tight, up to an arbitrarily small constant in the second exponent: under the randomized ETH, and for any ratio r > 1 and ε > 0 , no algorithm can output an r -approximation in time n (n / r) 1 − ε . Hence, we completely characterize the approximability of the problem in sub-exponential time. • We show that under ETH, Upper Dominating Set cannot be solved in time n o (k). • We give a 6 p w algorithm parameterized by pathwidth and show it is optimal under SETH. • We obtain a tight time-approximation trade-off under the ETH. [ABSTRACT FROM AUTHOR]

k-median with service installation costs is a frequently encountered problem in applications involving multiple service requirements, which generalizes the standard k -median problem in that each client is associated with a specific service and can only be connected to the facilities where the service is installed. In this paper, we give a (13.349 + ϵ) -approximation algorithm for the k -median with service installation costs problem, improving upon the previous best approximation ratio of 18. We propose a new deterministic rounding approach to deal with the challenges caused by the service requirements, which is the crucial step in getting the improved ratio. • We give a (13.349 + ϵ) -approximation algorithm for the k -median with service installation costs problem. • We obtain a well-behaved fractional solution to the problem based on the framework of Lagrangian relaxation. • We give a new deterministic rounding approach to deal with the challenges caused by the service requirements. [ABSTRACT FROM AUTHOR]

• We introduce the minimum power cover problem with submodular and linear penalties. • We present a combinatorial primal-dual approximation algorithm for the minimum power cover problem with submodular penalties. • We present a polynomial time approximation scheme for the minimum power cover problem with linear penalties. In this paper, we introduce the minimum power cover problem with submodular and linear penalties. Suppose U is a set of users and S is a set of sensors in a d -dimensional space R d with d ≥ 2. Each sensor can adjust its power and the relationship between the power p (s) and the radius r (s) of the service area of sensor s satisfies p (s) = c ⋅ r (s) α , where c > 0 and α ≥ 1. Let p be the power assignment for each sensor and R be the set of users who are not covered by any sensor supported by p. The objective is to minimize the total power of p plus the rejected penalty of R. For a submodular penalty function that is normalized and nondecreasing, we present a combinatorial primal-dual (3 α + 1) -approximation algorithm. For the case in which the submodular penalty function is linear, we present a polynomial time approximation scheme based on a plane subdivision technique. [ABSTRACT FROM AUTHOR]

We give deterministic distributed (1 + ϵ) -approximation algorithms for Minimum Vertex Coloring and Maximum Independent Set on chordal graphs in the LOCAL model. Our coloring algorithm runs in O (1 ϵ log ⁡ n) rounds, and our independent set algorithm has a runtime of O (1 ϵ log ⁡ (1 ϵ) log ⁎ ⁡ n) rounds. For coloring, existing lower bounds imply that the dependencies on 1 ϵ and log ⁡ n are best possible. For independent set, we prove that Ω (1 ϵ) rounds are necessary. Both our algorithms make use of a tree decomposition of the input chordal graph. They iteratively peel off interval subgraphs, which are identified via the tree decomposition of the input graph, thereby partitioning the vertex set into O (log ⁡ n) layers. For coloring, each interval graph is colored independently, which results in various coloring conflicts between the layers. These conflicts are then resolved in a separate phase, using the particular structure of our partitioning. For independent set, only the first O (log ⁡ 1 ϵ) layers are required as they already contain a large enough independent set. We develop a (1 + ϵ) -approximation maximum independent set algorithm for interval graphs, which we then apply to those layers. While tree decompositions have only played a minor role in distributed computing, our work demonstrates their potential for designing efficient distributed algorithms. [ABSTRACT FROM AUTHOR]

Given a metric graph G = (V , E , w) , a specific vertex c ∈ V , and an integer p , let T be a depth-2 spanning tree of G rooted at c such that c is adjacent to p vertices called hubs and each of the remaining vertices is adjacent to a hub. The Star p -Hub Routing Cost Problem is to find a spanning tree T of G satisfying the conditions stated above such that the sum of distances between all pairs of vertices in T is minimized. In this paper, we prove that the Star p -Hub Routing Cost Problem is NP-hard. A 3-approximation algorithm running in time O (n 2) is given for solving the same problem where n is the number of vertices in the input graph. Moreover, we give an example to show that the analysis of the approximation ratio cannot be better than 2 − ϵ for any ϵ > 0. [ABSTRACT FROM AUTHOR]

• The realization for multi-attribute influence maximization problem (MIMP) is proposed. • A Sandwich Approximation Algorithm for MIMP is proposed, which can get a max ⁡ { f (S U) f ‾ (S U) , f _ (S L ⁎) f (S o ⁎) } (1 − 1 e) -approximation ratio. • A hypergraph is constructed and a Supermodular Degree Algorithm is proposed, its solution has an 1 / (d + 2) approximation ratio. The most valuable feature of social networks is that they can generate contents for users and spread them quickly on the network, which is a very important platform for viral marketing. Most of the related work on viral marketing focuses on the spread of single information, while a product may associate with multiple attributes in real life. Information about multiple attributes of a product propagates in the social networks simultaneously and independently. The attribute information that a user receives will determine whether he would purchase the product or not. We extend the traditional single information influence maximization problem to the multi-attribute based influence maximization problem. We also present the Multi-dimensional IC model (MIC model) for the proposed problem, then formulate the problem as the Multi-attribute based Influence Maximization Problem (MIMP). The objective function for MIMP is proved to be non-submodular, then we solve the problem with two different algorithms: the Sandwich Algorithm and the Supermodular Algorithm, whose solutions can get a m a x { f (S U) f ‾ (S U) , f _ (S L ⁎) f (S o ⁎) } (1 − 1 / e) approximation ratio and an 1 / (d + 2) approximation ratio to the optimal solution, respectively. Experiments based on the real world social network datasets verify the effectiveness and correctness of our proposed solutions. [ABSTRACT FROM AUTHOR]

• Research was motivated from recent research in cloud computing. • A new model for research on scheduling of multiple processors. • Computational complexity of the proposed scheduling problems. • Approximation algorithms for the proposed scheduling problems. • Relationship between the proposed scheduling problems and multiple knapsack packing. Recently, motivated by applications in cloud computing, scheduling multiple two-stage flowshops with the objective of minimizing the makespan has drawn increasing attention. Motivated by the same applications, the current paper studies the problem of scheduling multiple two-stage flowshops with a deadline, with the objective of maximizing the profit of the jobs that can be completed by the deadline. Algorithms and complexity of the problem are investigated. For the case where the number of two-stage flowshops is part of the input, we present an efficient approximation algorithm with a constant ratio. The approximation ratio is improved via a study of the relationship between the problem and the multiple knapsack problem, combined with the recent progresses in the research in approximation algorithms for the multiple knapsack problem. By integrating techniques in the study of the classical Knapsack problem and the classical Makespan problem on multiple processors, plus additional new techniques, a polynomial-time approximation algorithm with a further improved approximation ratio is developed for the case where the number of flowshops is a fixed constant. [ABSTRACT FROM AUTHOR]

• Show the complexity of approximating real Boolean Holant problems. • Present holographic transformations from the Holant problems to the Ising model. • Prove zero-freeness of the Holant problems. • Establish AP-reductions between the Holant problems and counting perfect matchings. Holant problems provide a novel framework to study the complexity of counting problems. It is a refinement to counting constraint satisfaction problems (#CSP) with a more explicit role for the constraint functions. Both graph homomorphisms and #CSP can be viewed as special cases of Holant problems. For approximation algorithms on Holant problems, the attention is focused on proving zero-freeness of the partition functions and establishing fully polynomial-time approximation schemes (FPTAS) using the Taylor expansion method. In this paper, we study the Holant problems defined by a real constraint function satisfying a generalized second-order recurrence. We present fully polynomial-time (deterministic or randomized) approximation schemes for the Holant problems except for a couple of cases. Our algorithms are established in two ways: 1) we construct holographic transformations from the Holant problems to the Ising model, and obtain the algorithms using the approaches with respect to the Ising model; 2) we prove the zero-freeness of the Holant problems, and present an algorithm based on the Taylor expansion method. In addition, for most of the other cases, there exist approximation-preserving reductions between the Holant problems and the problem of counting perfect matchings, which is a central open problem in approximate counting. [ABSTRACT FROM AUTHOR]

Online social network has deeply changed our lives, such as the style of communication and business, and hence promotes a lot of researches in social influence. The prior works in social influence mainly consider the influence from the view of individuals. However, in many cases, influencing the most of members of an important group such as the board of directors in a company can bring bigger profit than directly influencing the individuals of the company. We call such high profit group which obeys the vote rule as an union, different from existed targeted influence model, we consider such scenarios to make union acceptable and propose the union acceptable profit problem (UAPM) to choose seeds to maximize the union-acceptable profit, i.e., maximize the probability of the union being acceptable. The objective of profit in UAPM is #P-hard, and not submodularity or supmodularity. To solve the problem, we propose an efficient estimation method for the objective and design a heuristic algorithm and further a data-driven β (1 − 1 ϵ) -approximation algorithm where β is the data-driven parameter which is related to the input data. At last we evaluate the performance of the algorithms we proposed on effectiveness and efficiency by the experiments in real-world social network datasets. • We formulate a new problem of UAPM which proved to be hard to solve. • An efficient estimation method for the objective is proposed based on an optimized sampling algorithm. • We propose a heuristic algorithm and a data-driven approximation algorithm respectively to solve the problem. • Experiments with the real-world databases validate the effectiveness of our algorithms. [ABSTRACT FROM AUTHOR]

• The first TotP-complete problems under parsimonious reductions. • Hardness of approximation for these problems and some of their generalizations. • A family of hard-to-approximate instances for the complete problem Size-of-Subtree. • Exponential time hardness results for Size-of-Subtree under variants of the Exponential Time Hypothesis. • Approximation preserving reduction of the general #SAT to instances of it that are promised to be self-reducible with easy decision. Hard counting problems with easy decision version, like computing the permanent and counting independent sets, are of particular importance in counting complexity theory. TotP is the class of all self-reducible problems in # P with decision version in P. It has been a long standing open question to determine TotP -complete problems under strong reductions that preserve exactly the values of the functions (Karp), since Cook reductions blur structural differences between counting classes, specifically under Cook reductions TotP -complete problems are also # P -complete. In this paper we present the first such problems and we show some implications of these results to counting complexity. The problems that we prove to be TotP -complete under Karp reductions are: (a) #Tree-Monotone-Circuit-SAT , which further implies that it is hard to compute, both exactly and approximately, the number of satisfying assignments of monotone circuits, where the monotonicity is defined by a partial order which is given as part of the input. (b) Max-Lower-Set-Size , which implies that given a poset it is hard to compute and approximate the size of a principal upper set, and the size of the maximum lower set all elements of which share a given property P. (c) Size-of-Subtree , which directly implies that every problem in TotP admits a polynomial-time approximation algorithm the error of which depends on the amount of imbalance of the respective self-reducibility tree of the problem. We settle the worst case complexity of this problem, we provide a family of instances that are hard to approximate, and we show exponential time hardness results under variants of the exponential time hypothesis (ETH). (d) #Clustered-Monotone-SAT , which implies that approximating #SAT remains hard even on instances for which some kind of navigation among solutions is possible. Thus we narrow down any further research regarding the polynomial-time approximability of #SAT to instances with the aforementioned property. Among other implications, our results show that all these problems do not admit an FPRAS, unless NP = RP , something that we would not be able to conclude by showing them to be TotP -complete under Cook reductions.1 [ABSTRACT FROM AUTHOR]

We consider the rooted orienteering problem in Euclidean space: Given n points P in R d , a root point s ∈ P and a budget B > 0 , find a path that starts from s , has total length at most B , and visits as many points of P as possible. This problem is known to be NP-hard, hence we study (1 − δ) -approximation algorithms. The previous Polynomial-Time Approximation Scheme (PTAS) for this problem, due to Chen and Har-Peled (2008), runs in time n O (d d / δ) 2 (d / δ) O (d) , and improving on this time bound was left as an open problem. Our main contribution is a PTAS with a significantly improved time complexity of n O (1 / δ) 2 (d / δ) O (d) . A known technique for approximating the orienteering problem is to reduce it to solving 1 / δ correlated instances of rooted k -TSP (a k -TSP tour is one that visits at least k points). However, the k -TSP tours in this reduction must achieve a certain excess guarantee (namely, their length can surpass the optimum length only in proportion to a parameter of the optimum called excess) that is stronger than the usual (1 + δ) -approximation. Our main technical contribution is to improve the running time of these k -TSP variants, particularly in its dependence on the dimension d. Indeed, our running time is polynomial even for a moderately large dimension, roughly up to d = O (log ⁡ log ⁡ n) instead of d = O (1). [ABSTRACT FROM AUTHOR]

This paper studies the problem of programming a robotic panda gardener to keep a bamboo garden from obstructing the view of the lake by your house. The garden consists of n bamboo stalks with known daily growth rates and the gardener can cut at most one bamboo per day. As a computer scientist, you found out that this problem has already been formalized in [Gąsieniec et al., SOFSEM'17] as the Bamboo Garden Trimming (BGT) problem , where the goal is that of computing a perpetual schedule (i.e., the sequence of bamboos to cut) for the robotic gardener to follow in order to minimize the makespan , i.e., the maximum height ever reached by a bamboo. Two natural strategies are Reduce-Max and Reduce-Fastest(x). Reduce-Max trims the tallest bamboo of the day, while Reduce-Fastest(x) trims the fastest growing bamboo among the ones that are taller than x. It is known that Reduce-Max and Reduce-Fastest(x) achieve a makespan of O (log ⁡ n) and 4 for the best choice of x = 2 , respectively. We prove the first constant upper bound of 9 for Reduce-Max and improve the one for Reduce-Fastest(x) to 3 + 5 2 < 2.62 for x = 1 + 1 5 . Another critical aspect stems from the fact that your robotic gardener has a limited amount of processing power and memory. It is then important for the algorithm to be able to quickly determine the next bamboo to cut while requiring at most linear space. We formalize this aspect as the problem of designing a Trimming Oracle data structure, and we provide three efficient Trimming Oracles implementing different perpetual schedules, including those produced by Reduce-Max and Reduce-Fastest(x). • We prove the first constant upper bound on the makespan attained by Reduce-Max by showing that it is at most 9. • We improve the best known upper bound on the makespan of Reduce-Fastest(x) to 3 + 5 2 < 2.62 (for x = 1 + 1 5 ). • We show that Reduce-Fastest(2) (the former best choice for x), achieves a makespan of 19 6 (improved from 4). • We design compact data structures (Trimming Oracles) that are able to quickly report the next bamboo to cut. • Our Trimming Oracles implement Reduce-Max, Reduce-Fastest(x), and a novel trimming strategy achieving a makespan of 2. [ABSTRACT FROM AUTHOR]

We consider the multistage framework introduced in (Gupta et al., Eisenstat et al., both in ICALP 2014), where we are given a time horizon and a sequence of instances of a (static) combinatorial optimization problem (one for each time step), and the goal is to find a sequence of solutions (one for each time step) reaching a tradeoff between the quality of the solutions in each time step and the stability/similarity of the solutions in consecutive time steps. We first introduce a novel rounding scheme, tailored for multistage problems, that accounts for the moving cost (or stability revenue) of adjacent solutions. Using this rounding scheme, we propose improved approximation algorithms for the multistage variants of Prize-Collecting Steiner Tree and Prize-Collecting Traveling Salesman problems. Furthermore, we introduce a 2-approximation algorithm for multistage Multi-Cut on trees, and we also show that our general scheme can be extended to maximization problems, by providing a 0.75-approximation algorithm for the multistage variant of MaxSat. [ABSTRACT FROM AUTHOR]

The support of a flow x in a network is the subdigraph induced by the arcs uv for which x (u v) > 0. We discuss a number of results on flows in networks where we put certain restrictions on structure of the support of the flow. Many of these problems are NP-hard because they generalize linkage problems for digraphs. For example deciding whether a network N = (D , s , t , c) has a maximum flow x such that the maximum out-degree of the support D x of x is at most 2 is NP-complete as it contains the 2-linkage problem as a very special case. Another problem which is NP-complete for the same reason is that of computing the maximum flow we can send from s to t along p paths (called a maximum p -path-flow) in N. Baier et al. (2005) gave a polynomial time algorithm which finds a p -path-flow x whose value is at least 2 3 of the value of a optimum p -path-flow when p ∈ { 2 , 3 } , and at least 1 2 when p ≥ 4. When p = 2 , they show that this is best possible unless P=NP. We show for each p ≥ 2 that the value of a maximum p -path-flow cannot be approximated by any ratio larger than 9 11 , unless P=NP. We also consider a variant of the problem where the p paths must be disjoint. For this problem, we give an algorithm which gets within a factor 1 H (p) of the optimum solution, where H (p) is the p 'th harmonic number (H (p) ∼ ln ⁡ (p)). We show that in the case where the network is acyclic, we can find such a maximum p -path-flow in polynomial time for every p. We determine the complexity of a number of related problems concerning the structure of flows. For the special case of acyclic digraphs, some of the results we obtain are in some sense best possible. [ABSTRACT FROM AUTHOR]

Given an edge-weighted metric complete graph with n vertices, the maximum weight metric triangle packing problem is to find a set of n / 3 vertex-disjoint triangles with the total weight of all triangles in the packing maximized. Several simple methods can lead to a 2/3-approximation ratio. However, this barrier is not easy to break. Chen et al. proposed a randomized approximation algorithm with an expected ratio of (0.66768 − ε) for any constant ε > 0. In this paper, we improve the approximation ratio to (0.66835 − ε). Furthermore, we can derandomize our algorithm. [ABSTRACT FROM AUTHOR]

The fair k -center problem has been paid lots of attention recently. In the fair k -center problem, we are given a set X of points in a metric space and a parameter k ∈ Z + , where the points in X are divided into several groups, and each point is assigned a color to denote which group it is in. The goal is to partition X into k clusters such that the number of cluster centers with each color is equal to a given value, and the k -center problem objective is minimized. In this paper, we consider the fair k -center problem with outliers and capacity constraints, denoted as the fair k -center with outliers (F k CO) problem and the capacitated fair k -center (CF k C) problem, respectively. The outliers constraints allow up to z outliers to be discarded when computing the objective function, while the capacity constraints require that each cluster has size no more than L. In this paper, we design an Fixed-Parameter Tractability (FPT) approximation algorithm and a polynomial approximation algorithm for the above two problems. In particular, our algorithms give (1 + ϵ) -approximations with FPT time for the F k CO and CF k C problems in doubling metric space. Moreover, we also propose a 3-approximation algorithm in polynomial time for the F k CO problem with some reasonable assumptions. [ABSTRACT FROM AUTHOR]

We consider the product of determinantal point processes (DPPs), a point process whose probability mass is proportional to the product of principal minors of multiple matrices, as a natural, promising generalization of DPPs. We study the computational complexity of computing its normalizing constant , which is among the most essential probabilistic inference tasks. Our complexity-theoretic results (almost) rule out the existence of efficient algorithms for this task unless the input matrices are forced to have favorable structures. In particular, we prove the following: • Computing ∑ S det ⁡ (A S , S) p exactly for every (fixed) positive even integer p is UP -hard and Mod 3 P -hard, which gives a negative answer to an open question posed by Kulesza and Taskar [51]. • ∑ S det ⁡ (A S , S) det ⁡ (B S , S) det ⁡ (C S , S) is NP -hard to approximate within a factor of 2 O (| I | 1 − ε) or 2 O (n 1 / ε) for any ε > 0 , where | I | is the input size and n is the order of the input matrix. This result is stronger than the # P -hardness for the case of two matrices derived by Gillenwater [36]. • There exists a k O (k) n O (1) -time algorithm for computing ∑ S det ⁡ (A S , S) det ⁡ (B S , S) , where k is the maximum rank of A and B or the treewidth of the graph formed by nonzero entries of A and B. Such parameterized algorithms are said to be fixed-parameter tractable. These results can be extended to the fixed-size case. Further, we present two applications of fixed-parameter tractable algorithms given a matrix A of treewidth w : • We can compute a 2 n 2 p − 1 -approximation to ∑ S det ⁡ (A S , S) p for any fractional number p > 1 in w O (w p) n O (1) time. • We can find a 2 n -approximation to unconstrained maximum a posteriori inference in w O (w n) n O (1) time. [ABSTRACT FROM AUTHOR]

The coordination among drones and ground vehicles for last-mile delivery has gained significant interest in recent years. In this paper, we study multiple drone delivery scheduling problem (MDSP) [1] for last-mile delivery, where we have a set of drones with an identical battery budget and a set of delivery locations; along with profit for delivery, cost, and delivery time intervals. The objective of the MDSP is to find conflict-free schedules for each drone such that the total profit gained is maximum subject to the battery constraint of the drones. This paper proposes an optimal pseudo-polynomial algorithm and a fully polynomial time approximation scheme (FPTAS) for the single drone delivery scheduling problem (SDSP). Also, we propose two approximation algorithms of factor 1 3 for the MDSP with some constraints on the number of drones. We formulate the problem for the heterogeneous case, where the drones have non-identical battery capacities, and propose a 1 6 ρ -approximation algorithm, where ρ is the ratio between the minimum and maximum battery capacity of the drones. • An optimal pseudo-polynomial algorithm and FPTAS for the single drone delivery scheduling problem is proposed. • Two approximation algorithms for the multiple drone delivery scheduling problem are proposed. • An approximation algorithm for the multiple heterogeneous drone delivery scheduling problem is proposed. • We analyzed the approximation factors of all the aforementioned proposed algorithms and time complexities. [ABSTRACT FROM AUTHOR]