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The length of a tree-decomposition of a graph is the maximum distance (in the graph) between two vertices of a same bag of the decomposition. The treelength of a graph is the minimum length among its tree-decompositions. Treelength of graphs has been studied for its algorithmic applications in classical metric problems such as Traveling Salesman Problem or metric dimension of graphs and also, in compact routing in the context of distributed computing. Deciding whether the treelength of a general graph is at most 2 is NP-complete (graphs of treelength one are precisely the chordal graphs), and it is known that the treelength of a graph cannot be approximated up to a factor less than 3 2 (the best known approximation algorithm for treelength has an approximation ratio of 3). However, nothing is known on the computational complexity of treelength in planar graphs, except that the treelength of any outerplanar graph is equal to the third of the size of a largest isometric cycle. This work initiates the study of treelength in planar graphs by considering the next natural subclass of planar graphs, namely the one of series–parallel graphs. We first fully describe the treelength of melon graphs (set of pairwise internally disjoint paths linking two vertices), showing that, even in such a restricted graph class, the expression of the treelength is not trivial. Then, we show that treelength can be approximated up to a factor 3 2 in series–parallel graphs. Our main result is a quadratic-time algorithm for deciding whether a series–parallel graph has treelength at most 2. Our latter result relies on a characterization of series–parallel graphs with treelength 2 in terms of an infinite family of forbidden isometric subgraphs. [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]

• 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]

• 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 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]

Deterministic time–frequency representations are commonly used in signal processing, particularly in audio processing. Whilst presenting many potential advantages, their probabilistic counterparts are not widely used, essentially because of the computational load and the lack of clear interpretability of the different underlying models. However, using state space models, they have been shown recently to be equivalent to Spectral Mixtures Gaussian processes (SM-GP). This pioneer work unlocks this problem and opens the path for the development of tractable and interpretable probabilistic time–frequency analysis. In this paper, we propose a relatively simple yet a significant improvement of that work in terms of computational complexity, flexibility and practical application. To do so, we use a recent approach for covariance approximation to develop an algorithm for faster inference of SM-GP, while opting for a frequency-domain approach to hyperparameter learning. We illustrate the practical potential of our method using voiced speech data. We first show that key speech features can be accurately learned from the data. Second, we show that our method can yield better performances in denoising. [ABSTRACT FROM AUTHOR]

We study a constrained version of the Geometric Hitting Set problem where we are given a set of points, partitioned into pairwise disjoint subsets, and a set of intervals. The objective is to hit all the intervals with a minimum number of points such that if we select a point from a subset, we must select all the points from that subset. We consider two special cases of the problem where each subset can have at most 2 and 3 points. If each subset contains at most 2 points and the intervals are disjoint, we show that the problem admits a polynomial-time algorithm. On the contrary, if each subset contains at most t points, where t ≥ 2 , and the intervals are overlapping, we show that the problem becomes NP - Hard. Further, when each subset contains at most t points where t ≥ 3 , and the intervals are disjoint, we prove that the problem is NP - Hard , and we provide two constant factor approximation algorithms for this problem. We also study the problem from the parameterized complexity perspective. If the intervals are disjoint, then we prove that the problem is in FPT when parameterized by the size of the solution. We also complement this result by giving a lower bound in the size of the kernel for disjoint intervals, and we also provide a polynomial kernel when the size of all subsets is bounded by a constant. [ABSTRACT FROM AUTHOR]

Discrete random variables are essential ingredients in various artificial intelligence problems. These include the estimation of the probability of missing the deadline in a series-parallel schedule and the assignment of suppliers to tasks in a project in a manner that maximizes the probability of meeting the overall project deadline. The solving of such problems involves repetitive operations, such as summation, over random variables. However, these computations are NP-hard. Therefore, we explore techniques and methods for approximating random variables with a given support size and minimal Kolmogorov distance. We examine both the general problem of approximating a random variable and a one-sided version in which over-approximation is allowed but not under-approximation. We propose several algorithms and evaluate their performance through computational complexity analysis and empirical evaluation. All the presented algorithms are optimal in the sense that given an input random variable and a requested support size, they return a new approximated random variable with the requested support size and minimal Kolmogorov distance from the input random variable. Our approximation algorithms offer useful estimations of probabilities in situations where exact computations are not feasible due to NP-hardness complexity. [ABSTRACT FROM AUTHOR]

• We study a k-wise generalization of the Kemeny rule. • We study the properties of this distance, and of the induced aggregation rule. • We introduce a k-wise counterpart of the majority graph. • We provide computational complexity results for the aggregation problem. • A polytime 2-approximation algorithm is provided for the aggregation problem. In this paper, we advocate the use of setwise contests for aggregating a set of input rankings into an output ranking. We propose a generalization of the Kemeny rule where one minimizes the number of k -wise disagreements instead of pairwise disagreements (one counts 1 disagreement each time the top choice in a subset of alternatives of cardinality at most k differs between an input ranking and the output ranking). After an algorithmic study of this k -wise Kemeny aggregation problem, we introduce a k -wise counterpart of the majority graph. This graph reveals useful to divide the aggregation problem into several sub-problems, which enables to speed up the exact computation of a consensus ranking. By introducing a k -wise counterpart of the Spearman distance, we also provide a 2-approximation algorithm for the k -wise Kemeny aggregation problem. We conclude with numerical tests. [ABSTRACT FROM AUTHOR]

Future cities promise to be more autonomous than ever, largely owing to our ability of coordinating complex systems in real time: fleets of self-driving cars will offer on-demand transportation services, delivery drones will fly parcels in our skies, power plants will provide renewable energy reliably. In many of these systems, there is no single decision-maker with full information and authority. Instead, the system performance greatly depends on the decisions made by interacting entities with local information and limited communication capabilities. Game theory, intended as the study of multi-agent decision-making, is a fitting paradigm to tackle many of the associated challenges. Moving from this observation, in this paper we review how tools and ideas from game theory can be brought to bear on the coordination of multi-agent systems. At the heart of the proposed approach is the design and influence of agents' preferences so that their local optimization induces a desirable system behavior. Its applicability spans a variety of settings irrespective of whether the decision makers are strategic (e.g., drivers in a road network), or not (e.g., delivery drones). Along the way, we also discuss future research directions and connections with related research areas including algorithmic game theory, incentive and mechanism design, economics, computational complexity, and approximation algorithms. [ABSTRACT FROM AUTHOR]

The length of a tree-decomposition of a graph is the maximum distance between two vertices of a same bag of the decomposition. The treelength of a graph is the minimum length among its tree-decompositions. Treelength of graphs has been studied for its algorithmic applications in classical metric problems such as Traveling Salesman Problem or metric dimension of graphs and also, in compact routing in the context of distributed computing. Deciding whether the treelength of a general graph is at most 2 is NP-complete (graphs of treelength one are precisely the chordal graphs), and it is known that the treelength of a graph cannot be approximated up to a factor less than 3/2 (the best known approximation algorithm for treelength has an approximation ratio of 3). However, nothing is known on the computational complexity of treelength in planar graphs, except that the treelength of any outerplanar graph is equal to the third of the maximum size of its isometric cycles. This work initiates the study of treelength in planar graphs by considering the next natural superclass of outerplanar graphs, namely the one of series-parallel graphs. We first fully describe the treelength of melon graphs (set of pairwise internally disjoint paths linking two vertices), showing that, even in such a restricted graph class, the expression of the treelength is not trivial. Then, we show that treelength can be approximated up to a factor 3/2 in series-parallel graphs. Our main result is a polynomial-time algorithm for deciding whether a series-parallel graph has treelength at most 2. Our latter result relies on a characterization of series-parallel graphs with treelength 2 in terms of infinite families of forbidden isometric subgraphs. [ABSTRACT FROM AUTHOR]

• The approximability and the computational complexity of the Maximum Happy Set problem (MaxHS) are studied. • A (2 Δ + 1) -approximation algorithm for MaxHS on graphs with maximum degree Δ is presented. • The approximation ratio is improved to Δ if the maximum degree Δ is a constant. • The polynomial-time solvability of MaxHS on block/interval graphs is proved. • The NP-hardness of MaxHS on bipartite/cubic graphs is proved. In this paper we study the approximability of the Maximum Happy Set problem (MaxHS) and the computational complexity of MaxHS on graph classes: For an undirected graph G = (V , E) and a subset S ⊆ V of vertices, a vertex v is happy if v and all its neighbors are in S ; otherwise unhappy. Given an undirected graph G = (V , E) and an integer k , the goal of MaxHS is to find a subset S ⊆ V of k vertices such that the number of happy vertices is maximized. MaxHS is known to be NP-hard. In this paper, we design a (2 Δ + 1) -approximation algorithm for MaxHS on graphs with maximum degree Δ. Next, we show that the approximation ratio can be improved to Δ if the maximum degree Δ of the input graph is a constant. Then, we show that MaxHS can be solved in polynomial time if the input graph is restricted to block graphs, or interval graphs. We prove nevertheless that MaxHS on bipartite graphs or on cubic graphs remains NP-hard. [ABSTRACT FROM AUTHOR]

In the Multi-Spreader Crane Scheduling Problem (MSCSP), containers with identical dimensions but variable weights are arranged along a grid. A multi-spreader crane is used to lift all the containers. The crane has m > 1 modes. When it is in the p th mode, the crane can remove p adjacent containers along the same row at the same time as long as the total weight of the containers does not exceed the loading capacity κ p. Such a lift takes h p minutes. It also takes c p , q minutes for the crane to switch from mode p to q when p ≠ q. The goal is to find a crane lift sequence so that the total time it takes to lift all the containers is minimized. This paper investigates the computational complexity of MSCSP. First, we establish a connection between greedy crane lift sequences and supersequences. We then prove that MSCSP is NP-hard when the crane has three or more modes by a reduction from a version of the Shortest Common Supersequence problem. Lastly, we investigate two problems that arise naturally when heuristics are used to solve MSCSP. We show that one can be solved using dynamic programming while the other remains computationally hard. We also provide an approximation algorithm that behaves nicely when the changeover times are not much larger than the lifting times of the crane. [ABSTRACT FROM AUTHOR]

We consider scheduling on identical and unrelated parallel machines with job assignment restrictions. These problems are NP-hard and they do not admit polynomial time approximation algorithms with approximation ratios smaller than 1.5 unless P=NP. However, if we impose limitations on the set of machines that can process a job, the problem sometimes becomes easier in the sense that algorithms with approximation ratios better than 1.5 exist. We introduce a graph framework based on the assignment restrictions and study the computational complexity of the scheduling problem with respect to structural properties of the resulting graphs, in particular, their tree- and cliquewidth. We identify cases that admit polynomial time approximation schemes or FPT algorithms generalizing and extending previous results in this area. [ABSTRACT FROM AUTHOR]

Given a set B of d -dimensional boxes (i.e., axis-aligned hyperrectangles), a minimum coverage kernel is a subset of B of minimum size covering the same region as B. Computing it is NP -hard, but as for many similar NP -hard problems (e.g., Box Cover , and Orthogonal Polygon Covering), the problem becomes solvable in polynomial time under restrictions on B. We show that computing minimum coverage kernels remains NP -hard even when restricting the graph induced by the input to a highly constrained class of graphs. Alternatively, we present two polynomial-time approximation algorithms for this problem: one deterministic with an approximation ratio within O (log ⁡ n) , and one randomized with an improved approximation ratio within O (lg ⁡ OPT) (with high probability). [ABSTRACT FROM AUTHOR]

A mixed shop is a manufacturing infrastructure designed to process a mixture of a set of flow-shop jobs and a set of open-shop jobs. Mixed shops are in general much more complex to schedule than flow-shops and open-shops, and have been studied since the 1980's. We consider the three machine proportionate mixed shop problem denoted as M 3 | p r p t | C max , in which by "proportionate" each job has equal processing times on all three machines. Koulamas and Kyparisis (2015) [6] showed that the problem is solvable in polynomial time in some very special cases; for the non-solvable case, they proposed a 5/3-approximation algorithm. In this paper, we first present an improved 4/3-approximation algorithm and show that this ratio of 4/3 is asymptotically tight; when the largest job is a flow-shop job, we then present a fully polynomial-time approximation scheme (FPTAS). On the negative side, while the F 3 | p r p t | C max problem is polynomial-time solvable, we show an interesting hardness result that adding one open-shop job to the job set makes the problem NP-hard if this open-shop job is larger than any flow-shop job. We are able to design an FPTAS for this special case too. [ABSTRACT FROM AUTHOR]

• Models the scheduling problem as an optimization problem with congruence constraints. • Proposes a new graph representation of the problem. • Provides optimal inapproximability results for the optimization problem. • Exposes the close bounds that relate this problem with the MAX DICUT problem. • Customizes and applies approximation algorithms for MAX DICUT in real case studies. This paper aims to synchronize timetables in a transit network so as to minimize the total passenger transfer waiting time. Assuming a fixed headway for each line, we first formulate the problem as an optimization problem with congruence constraints. We show that the problem is NP-hard, and investigate several special cases of the problem that are solvable in polynomial time. Furthermore, we show that the local search for the general problem is equivalent to the well-studied maximum directed cut problem. As such, we use an approximation algorithm for the maximum directed cut problem to solve the timetable synchronization problem. We assess the quality of the algorithm on a real-world case study, and show that our algorithm significantly outperforms the state-of-the-art in the literature. Lastly, we relax the fixed headway assumption, and propose an efficient recursive quasi-linear time algorithm to minimize the total transfer waiting time in this general setting. [ABSTRACT FROM AUTHOR]

Inspired by scaffold filling, a recent approach for genome reconstruction from incomplete data, we consider a variant of the well-known longest common subsequence problem for the comparison of two sequences. The new problem, called Longest Filled Common Subsequence, aims to compare a complete sequence with an incomplete one, i.e. with some missing elements. Longest Filled Common Subsequence (LFCS), given a complete sequence A , an incomplete sequence B , and a multiset M of symbols missing in B , asks for a sequence B ⁎ obtained by inserting the symbols of M into B so that B ⁎ induces a common subsequence with A of maximum length. We investigate the computational and approximation complexity of the problem and we show that it is NP-hard and APX-hard when A contains at most two occurrences of each symbol, and we give a polynomial time algorithm when the input sequences are over a constant-size alphabet. We give a 3 5 − approximation algorithm for the Longest Filled Common Subsequence problem. Finally, we present a fixed-parameter algorithm for the problem, when it is parameterized by the number of symbols inserted in B that "match" symbols of A. [ABSTRACT FROM AUTHOR]

k -means and k -median clustering are powerful unsupervised machine learning techniques. However, due to complicated dependencies on all the features, it is challenging to interpret the resulting cluster assignments. Moshkovitz, Dasgupta, Rashtchian, and Frost proposed an elegant model of explainable k -means and k -median clustering in ICML 2020. In this model, a decision tree with k leaves provides a straightforward characterization of the data set into clusters. We study two natural algorithmic questions about explainable clustering. (1) For a given clustering, how to find the "best explanation" by using a decision tree with k leaves? (2) For a given set of points, how to find a decision tree with k leaves minimizing the k -means/median objective of the resulting explainable clustering? To address the first question, we introduce a new model of explainable clustering. Our model, inspired by the notion of outliers in robust statistics, is the following. We are seeking a small number of points (outliers) whose removal makes the existing clustering well-explainable. For addressing the second question, we initiate the study of the model of Moshkovitz et al. from the perspective of multivariate complexity. Our rigorous algorithmic analysis sheds some light on the influence of parameters like the input size, dimension of the data, the number of outliers, the number of clusters, and the approximation ratio, on the computational complexity of explainable clustering. [ABSTRACT FROM AUTHOR]

A solution extension problem consists in an instance and a partial feasible solution which is given in advance and the goal is to extend this partial solution to a feasible one. Many well-known problems like Coloring Extension Problems, Traveling Salesman Problem and Routing problems, have been studied in this framework. In this paper, we focus on the edge-weighted spanning star forest problem for both minimization and maximization versions. The goal here is to find a vertex partition of an edge-weighted graph into disjoint non-trivial stars and the value of a solution is given by the sum of the edge-weights of the stars. We propose NP -hardness, parameterized complexity, positive and negative approximation results. • An extension variant of the weighted edge cover problem is studied. • The problem is NP -complete in complete graphs while polynomial solvable in bounded tree-width graphs. • For non-negative weights, the minimization version is in-approximable at all while the maximization version is in RAPX. • If weight function satisfies (extended) c -relaxed triangle inequalities, the minimization version is in RAPX. [ABSTRACT FROM AUTHOR]

Highlights • Algorithm for calibration of car-following model for large-scale microscopic network models. • City-scale case study is carried out, the approach is shown to be computationally efficient and suitable for large-scale network models. • Compared to a traditional approach, it improves the objective function by two orders of magnitude and it improves the fit to link speeds by one order of magnitude. Abstract This paper proposes a simulation-based optimization methodology for the efficient calibration of microscopic traffic flow models (i.e., car-following models) of large-scale stochastic network simulators. The approach is a metamodel simulation-based optimization (SO) method. To improve computational efficiency of the SO algorithm, problem-specific and simulator-specific structural information is embedded into a metamodel. As a closed-form expression is sought, we propose adopting the steady-state solution of the car-following model as an approximation of its simulation-based input-output mapping. This general approach is applied for the calibration of the Gipps car-following model embedded in a microscopic traffic network simulator, on a large network. To this end, a novel formulation for the traffic stream models corresponding to the Gipps car-following law is provided. The proposed approach identifies points with good performance within few simulation runs. Comparing its performances to that of a traditional approach, which does not take advantage of the structural information, the objective function is improved by two orders of magnitude in most experiments. Moreover, this is achieved within tight computational budgets, i.e., few simulation runs. The solutions identified improve the fit to the field measurements by one order of magnitude, on average. The structural information provided to the metamodel is shown to enable the SO algorithm to become robust to both the quality of the initial points and the simulator stochasticity. [ABSTRACT FROM AUTHOR]

Abstract We present a (1 + 3 + ϵ) -approximation algorithm for the k -median problem with uniform penalties, extending the recent result by Li and Svensson for the classical k -median problem without penalties. One important difference of this work from that of Li and Svensson is a new definition of sparse instance to exploit the combinatorial structure of our problem. [ABSTRACT FROM AUTHOR]

Abstract We consider the following problem in which a given number of items has to be chosen from a predefined set. Each item is described by a vector of attributes and for each attribute there is a desired distribution that the selected set should have. We look for a set that fits as much as possible the desired distributions on all attributes. An example of application is the choice of members for a representative committee, where candidates are described by attributes such as gender, age and profession, and where we look for a committee that for each attribute offers a certain representation, i.e., a single committee that contains a certain number of young and old people, certain number of men and women, certain number of people with different professions, etc. Another example of application is the selection of a common set of items to be used by a group of users, where items are labelled by attribute values. With a single attribute the problem collapses to the apportionment problem for party-list proportional representation systems (in such a case the value of the single attribute would be a political affiliation of a candidate). We study the properties of the associated subset selection rules, as well as their computational complexity. [ABSTRACT FROM AUTHOR]

Large graphs are scale-free, ubiquitous having irregular relationships and non-trivial topology. Frequent subgraph mining is a popular method for knowledge extraction from graphs. Most of the existing frequent subgraph mining algorithms are centralized algorithms that cannot handle a single large graph efficiently and incur high communication cost. However, to make the task of subgraph mining less expensive computationally, approximate subgraph mining can be applied which will capture similar structure subgraphs as of exact subgraph mining. In this paper, we propose an approximate subgraph mining algorithm named Ap-FSM implemented on distributed graph environment Pregel. The working of Ap-FSM is divided into three phases. The first phase selects the representative graph from the original graph while preserving the original graph properties. The second phase efficiently performs subgraph extension. Phase 3 introduces a novel two-step optimization for performing subgraph pruning. Analyzing such large graph data will be beneficial from the perspective of expert and intelligent systems, as discovered patterns can be used for knowledge discovery and decision making. To evaluate the performance of Ap-FSM, experiments are performed over three real life datasets having up to billion edges. The results show that the proposed Ap-FSM significantly outperforms the state-of-art frequent subgraph mining algorithms and overcome the challenges of performing frequent subgraph mining on a massive large graph. It is also shown that Ap-FSM achieves high scalability and speedup in distributed graph environment and is highly accurate in finding frequent subgraphs from a single large graph. [ABSTRACT FROM AUTHOR]

With the advent of the age of big data, a typical big data set called limited labeled big data appears. It includes a small amount of labeled data and a large amount of unlabeled data. Some existing neighborhood-based rough set algorithms work well in analyzing the rough data with numerical features. But, they face three challenges: limited labeled property of big data, computational inefficiency and over-fitting in attribute reduction when dealing with limited labeled data. In order to address the three issues, a combination of neighborhood rough set and local rough set called local neighborhood rough set (LNRS) is proposed in this paper. The corresponding concept approximation and attribute reduction algorithms designed with linear time complexity can efficiently and effectively deal with limited labeled big data. The experimental results show that the proposed local neighborhood rough set and corresponding algorithms significantly outperform its original counterpart in classical neighborhood rough set. These results will enrich the local rough set theory and enlarge its application scopes. [ABSTRACT FROM AUTHOR]

In the Canadian traveler problem, we are given an edge weighted graph with two specified vertices s and t and a probability distribution over the edges that tells which edges are present. The goal is to minimize the expected length of a walk from s to t . However, we only get to know whether an edge is active the moment we visit one of its incident vertices. Under the assumption that the edges are active independently, we show NP-hardness on series-parallel graphs and give results on the adaptivity gap. We further show that this problem is NP-hard on disjoint-path graphs and cactus graphs when the distribution is given by a list of scenarios. We also consider a special case called the multi-target graph search problem. In this problem, we are given a probability distribution over subsets of vertices. The distribution specifies which set of vertices has targets. The goal is to minimize the expected length of the walk until finding a target. For the independent decision model, we show that the problem is NP-hard on trees and give a ( 3.59 + ϵ ) -approximation for trees and a ( 14.4 + ϵ ) -approximation for general metrics. For the scenario model, we show NP-hardness on star graphs. [ABSTRACT FROM AUTHOR]

Energy efficiency is lately a major concern for computer engineers, at the levels of both software and hardware. A popular path is the exploitation of heterogeneity and accelerator-based systems, which combine different architectures, each appropriate for specific computational patterns. Approximate computing is another aggressive, yet viable alternative; it minimizes the energy footprint of applications at the expense of output quality.We introduce AcHEe, a set of 12 applications – ranging from real-world applications to kernels – from different domains, exposing a wide range of characteristics, which have been modified to exploit both heterogeneity and approximations. The degree of approximation can be controlled at runtime. The implementation is based on OpenCL and is publicly available.We evaluate these applications on heterogeneous platforms (comprising of CPUs and GPUs) and quantify the isolated and combined effect of heterogeneous and approximate computing. AcHEe can serve as a benchmark suite, to evaluate current and future computational devices from the perspective of heterogeneity and approximations. At the same time, it can serve as a software platform to evaluate and compare different approximation techniques. [ABSTRACT FROM AUTHOR]

Latin Hypercube Designs (LHDs) are crucial in the meta-modeling field. An LHD with maximal minimum distance between any pair of points guarantees a high quality of the model. We introduce a generalization of the Maximin LHD construction, the Maximin LHD completion, in which some points of the design are already fixed. We study the complexity of this problem and prove that it is NP-complete and it does not admit any approximation algorithm in most practical cases. [ABSTRACT FROM AUTHOR]

Comparative genomics is a line of study which aims at determining dis/similarities between genomes. One way of doing this is to infer the large scale mutations (also called genome rearrangements ) that are responsible for transforming one genome into another, under the parsimony principle. Numerous efforts have been made over the past years regarding the traditional approach, in which all genome rearrangements have the same unit cost . However, since long rearrangements are more likely to disturb the organism, some works considered the length-weighted approach, in which the cost of each rearrangement is a function of its length. The problem is thus modeled as follows: given two genomes G 1 and G 2 and a set β of allowed genome rearrangements, each with a given cost, determine their distance , i.e., the total cost of a minimum-cost sequence of rearrangements from β that transforms G 1 into G 2 . In this paper, we study length-weighted reversals and transpositions, mainly considering their prefix variants (in which the leftmost part of the genome must always be included). We present approximation algorithms and bounds on the distances and diameters (i.e., maximum possible distance over any two genomes of same length) for eight problems when genomes are represented as permutations, and another eight problems when genomes are represented as binary strings, in each case considering two different cost functions. [ABSTRACT FROM AUTHOR]

In this paper the following selection problem is discussed. A set of n items is given and we wish to choose a subset of exactly p items of the minimum total cost. This problem is a special case of 0–1 knapsack in which all the item weights are equal to 1. Its deterministic version has an O ( n ) -time algorithm, which consists in choosing p items of the smallest costs. In this paper it is assumed that the item costs are uncertain. Two robust models, namely two-stage and recoverable ones, under discrete and interval uncertainty representations, are discussed. Several positive and negative complexity results for both of them are provided. [ABSTRACT FROM AUTHOR]

Graph burning is a model for the spread of social contagion. The burning number is a graph parameter associated with graph burning that measures the speed of the spread of contagion in a graph; the lower the burning number, the faster the contagion spreads. We prove that the corresponding graph decision problem is NP -complete when restricted to acyclic graphs with maximum degree three, spider graphs and path-forests. We provide polynomial time algorithms for finding the burning number of spider graphs and path-forests if the number of arms and components, respectively, are fixed. Finally, we describe a polynomial time approximation algorithm with approximation factor 3 for general graphs. [ABSTRACT FROM AUTHOR]

Graph Distance Contest (GDC) was organized in the context of ICPR 2016. Its main challenge was to inspect and report performances and effectiveness of exact and approximate graph edit distance methods by comparison with a ground truth. This paper presents the context of this competition, the metrics and datasets used for evaluation, and the results obtained by the eight submitted methods. Results are analyzed and discussed in terms of computation time and accuracy. We also highlight the future challenges in graph edit distance regarding both future methods and evaluation metrics. The contest was supported by the Technical Committee on Graph-Based Representations in Pattern Recognition (TC-15) of the International Association of Pattern Recognition (IAPR). [ABSTRACT FROM AUTHOR]

We present a general technique for approximating bicriteria minimization problems with positive-valued, polynomially computable objective functions. Given 0 < ϵ ≤ 1 and a polynomial-time α -approximation algorithm for the corresponding weighted sum problem, we show how to obtain a bicriteria ( α ⋅ ( 1 + 2 ϵ ) , α ⋅ ( 1 + 2 ϵ ) ) -approximation algorithm for the budget-constrained problem whose running time is polynomial in the encoding length of the input and linear in 1 ϵ . Moreover, we show that our method can be extended to compute an ( α ⋅ ( 1 + 2 ϵ ) , α ⋅ ( 1 + 2 ϵ ) ) -approximate Pareto curve under the same assumptions. Our technique applies to many minimization problems to which most previous algorithms for computing approximate Pareto curves cannot be applied because the corresponding gap problem is NP -hard to solve. For maximization problems, however, we show that approximation results similar to the ones presented here for minimization problems are impossible to obtain in polynomial time unless P = NP . [ABSTRACT FROM AUTHOR]

In this paper we study parallel-machine scheduling with job rejection, where the total processing time of the rejected jobs is required to be no greater than a predefined bound. The objective is to minimize the makespan of the accepted jobs plus the total penalty cost of the rejected jobs. The scheduling problem is NP-hard in the strong sense. We present a 2-approximation algorithm with a time complexity of O ( n log ⁡ n ) by making use of specific data structure. We also develop a polynomial time approximation scheme (PTAS). Due to the job rejection restriction, some techniques for knapsack problems are applied in the development of our PTAS. [ABSTRACT FROM AUTHOR]

A total dominating set of an undirected graph G = ( V , E ) is a set S ⊆ V of vertices such that each vertex of G is adjacent to at least one vertex of S . A secure total dominating set of G is a set D ⊆ V of vertices such that D is a total dominating set of G and each vertex v ∈ V ∖ D is adjacent to at least one vertex u ∈ D with the property that the set ( D ∖ { u } ) ∪ { v } is a total dominating set of G . The secure total domination number γ s t ( G ) of G is the minimum cardinality of a secure total dominating set of G . We establish bounds on the secure total domination number. In particular, we show that every graph G with no isolated vertices satisfies γ s t ( G ) ≤ 2 α ( G ) , where α ( G ) is the independence number of G . Further, we study the problem of finding the secure total domination number. We show that the decision version of the problem is N P -complete for chordal bipartite graphs, planar bipartite graphs with arbitrary large girth and maximum degree 3, split graphs and graphs of separability at most 2. Finally, we show that the optimisation version of the problem can be approximated in polynomial time within a factor of c ln | V | for some constant c > 1 and cannot be approximated in polynomial time within a factor of c ′ ln | V | for some constant c ′ < 1 , unless P = N P . [ABSTRACT FROM AUTHOR]

A graph G is weakly semiregular if there are two numbers a , b , such that the degree of every vertex is a or b . The weakly semiregular number of a graph G , denoted by w r ( G ) , is the minimum number of subsets into which the edge set of G can be partitioned so that the subgraph induced by each subset is a weakly semiregular graph. We present a polynomial time algorithm to determine whether the weakly semiregular number of a given tree is two. On the other hand, we show that determining whether w r ( G ) = 2 for a given bipartite graph G with at most three numbers in its degree set is NP -complete. Among other results, for every tree T , we show that w r ( T ) ≤ 2 log 2 ⁡ Δ ( T ) + O ( 1 ) , where Δ ( T ) denotes the maximum degree of T . A graph G is a [ d , d + s ] -graph if the degree of every vertex of G lies in the interval [ d , d + s ] . A [ d , d + 1 ] -graph is said to be semiregular . The semiregular number of a graph G , denoted by s r ( G ) , is the minimum number of subsets into which the edge set of G can be partitioned so that the subgraph induced by each subset is a semiregular graph. We prove that the semiregular number of a tree T is ⌈ Δ ( T ) 2 ⌉ . On the other hand, we show that determining whether s r ( G ) = 2 for a given bipartite graph G with Δ ( G ) ≤ 6 is NP -complete. In the second part of the work, we consider the representation number. A graph G has a representation modulo r if there exists an injective map ℓ : V ( G ) → Z r such that vertices v and u are adjacent if and only if | ℓ ( u ) − ℓ ( v ) | is relatively prime to r . The representation number , denoted by r e p ( G ) , is the smallest r such that G has a representation modulo r . Narayan and Urick conjectured that the determination of r e p ( G ) for an arbitrary graph G is a difficult problem [38] . In this work, we confirm this conjecture and show that if N P ≠ P , then for any ϵ > 0 , there is no polynomial time ( 1 − ϵ ) n 2 -approximation algorithm for the computation of representation number of regular graphs with n vertices. [ABSTRACT FROM AUTHOR]

After the successful development of the discrete cosine transform (DCT) especially in audio, image and video coding, many state-of-the-arts DCT-based fast algorithms have been developed. However, these algorithms are floating-point and are implemented with high algorithmic complexity and memory intensive. Therefore, they consume large system resources. In particular, those requirements pose a serious limitation on multimedia applications in resource-constrained systems, such as wireless multimedia sensor networks (WMSNs). In this paper we investigate an alternative to DCT transform which known as discrete Tchebichef transform (DTT). Despite its good energy compaction property, low algorithmic complexity and low memory requirements, the DTT is potentially unexploited in the literature. To further reduce its complexity and memory requirements, in this paper we propose a pruned version of DTT (P-DTT). Simulation results show that P-DTT requires a reduced number of arithmetic operations, energy consumption and memory. And it has at the same time competitive compression efficiency compared with Loeffler-DCT and exact DTT. The proposed P-DTT transform is a viable option for image compression and/or communication over WMSNs. [ABSTRACT FROM AUTHOR]

We introduce an adaptive scattered data fitting scheme as extension of local least squares approximations to hierarchical spline spaces. To efficiently deal with non-trivial data configurations, the local solutions are described in terms of (variable degree) polynomial approximations according not only to the number of data points locally available, but also to the smallest singular value of the local collocation matrices. These local approximations are subsequently combined without the need of additional computations with the construction of hierarchical quasi-interpolants described in terms of truncated hierarchical B-splines. A selection of numerical experiments shows the effectivity of our approach for the approximation of real scattered data sets describing different terrain configurations. [ABSTRACT FROM AUTHOR]

Advances in research on quality metrics for bounding volume hierarchies (BVHs) have shown that greedy top-down SAH builders construct BVHs with superior traversal performance despite the fact that the resulting SAH values are higher than those created by more sophisticated builders. Motivated by this observation we examine three construction algorithms that use recursive SAH values of temporarily constructed SAH-built BVHs to guide the construction and perform an extensive evaluation. The resulting BVHs achieve up to 37% better trace performance for primary rays and up to 32% better trace performance for secondary diffuse rays compared to standard plane sweeping without applying spatial splits. Allowing spatial splits still achieves up to 18% resp. 15% better performance. While our approach is not suitable for real-time BVH construction, we show that the proposed algorithm has subquadratic computational complexity in the number of primitives, which renders it usable in practical applications. Additionally, we describe an approach for improving the forecasting abilities of the SAH-EPO ray tracing performance predictor which also increases its relevance for primary rays. [ABSTRACT FROM AUTHOR]

Salient object detection is an important issue in computer vision and image procession in that it can facilitate humans to locate conspicuous visual regions in complex scenes rapidly and improve the performance of object detection and video tracking. In recent years, low-rank matrix approximation has been proved to be favorable in image saliency detection and gained a great deal of attention. An underlying assumption of low-rank recovery is that an image is a combination of background regions being low-rank and salient objects being sparse, which corresponds to tough non-smooth optimization problems. In this paper, by incorporating ℓ 2,1 -norm minimization, we obtain the corresponding smooth optimization problems and propose two effective algorithms with proved convergence. To guarantee the robustness of the proposed methods, the input image is divided into patches and each patch is approximately represented by its mean value. Besides, multi-scale visual features of each patch of the given image are extracted to capture common low-level features such as color, edge, shape and texture. The salient objects of a given image are indicated with sparse coefficients solved by the low-rank matrix approximation problem. Saliency maps are further produced with integration of the high-level prior knowledge. Finally, extensive experiments in four real-world datasets demonstrate that the proposed methods come with competitive performance over the eight compared state-of-the-arts. [ABSTRACT FROM AUTHOR]

Given a graph with labels defined on edges and a source-sink pair ( s , t ) , the Label s - t Cut problem asks for a minimum number of labels such that the removal of edges with these labels disconnects s and t . Similarly, the Global Label Cut problem asks for a minimum number of labels to disconnect G itself. For these two problems, we identify two useful parameters, i.e., l max , the maximum length of any s - t path (only applies to Label s - t Cut ), and f max , the maximum number of appearances of any label in the graph (applies to the two problems). We show that l max = 2 and f max = 2 are two complexity thresholds for Label s - t Cut . Furthermore, we give (i) an O ⁎ ( c k ) time parameterized algorithm for Label s - t Cut with l max bounded from above, where parameter k is the number of labels in a solution, and c is a constant with l max − 1 < c < l max , (ii) a combinatorial l max -approximation algorithm for Label s - t Cut , and (iii) a polynomial time exact algorithm for Global Label Cut with f max bounded from above. [ABSTRACT FROM AUTHOR]

Parameterized Approximation is a topic of considerable interest in the field of Parameterized Complexity. In the past decade, new color coding-related techniques, including the breakthrough representative sets technique, have been proven extremely powerful in the design of fast parameterized algorithms. Furthermore, packing problems, which are often solved via color coding-related techniques, have enjoyed a race towards obtaining the fastest parameterized algorithms that solve them. Therefore, it is natural to ask whether packing problems admit efficient parameterized approximation algorithms. In this paper, we answer this question affirmatively. We present tradeoffs that improve the running times of algorithms for well-known special cases of the 3 -Set k -Packing problem at the cost of their accuracy. Consider a packing problem for which there is no known algorithm with approximation ratio α , and a parameter k . If the value of an optimal solution is at least k , we seek a solution of value at least αk ; otherwise, we seek an arbitrary solution. Clearly, if the best known parameterized algorithm that finds a solution of value t runs in time O ⁎ ( f ( t ) ) for some function f , we are interested in running times better than O ⁎ ( f ( α k ) ) . Our main contribution lies in the adaptation of notions fundamental to the representative sets technique to the design of approximation algorithms: We introduce the definition of “approximate lopsided universal sets”, combine partial executions of representative sets-based algorithms with approximation algorithms, and adapt the iterative expansion framework (in the context of representative sets) to the design of parameterized approximation algorithms. Our ideas are intuitive, and may be relevant to the design of other parameterized approximation algorithms. [ABSTRACT FROM AUTHOR]

Searching in partially ordered structures has been considered in the context of information retrieval and efficient tree-like indices, as well as in hierarchy based knowledge representation. In this paper we focus on tree-like partial orders and consider the problem of identifying an initially unknown vertex in a tree by asking edge queries: an edge query e returns the component of T − e containing the vertex sought for, while incurring some known cost c ( e ) . The Tree Search Problem with Non-Uniform Cost is the following: given a tree T on n vertices, each edge having an associated cost, construct a strategy that minimizes the total cost of the identification in the worst case. Finding the strategy guaranteeing the minimum possible cost is an NP-complete problem already for input trees of degree 3 or diameter 6. The best known approximation guarantee was an O ( log ⁡ n / log ⁡ log ⁡ log ⁡ n ) -approximation algorithm of Cicalese et al. (2012) [4] . We improve upon the above results both from the algorithmic and the computational complexity point of view: We provide a novel algorithm that provides an O ( log ⁡ n log ⁡ log ⁡ n ) -approximation of the cost of the optimal strategy. In addition, we show that finding an optimal strategy is NP-hard even when the input tree is a spider of diameter 6, i.e., at most one vertex has degree larger than 2. [ABSTRACT FROM AUTHOR]

In 2011, Khattri and Abbasbandy developed an optimal two-step Jarratt-like method for approximating simple roots of a nonlinear equation. We develop their method for solving nonlinear systems of equations. The main feature of the extended methods is that it uses only one LU factorization which preserves and reduces computational complexities. Following this aim, the suggested method is generalized in such a way that we increase the order of convergence but we do not need new LU factorization. Convergence and complexity analysis are provided rigorously. Using some small and large systems, applicability along with some comparisons are illustrated. [ABSTRACT FROM AUTHOR]

Time series subsequence matching has importance in a variety of areas in healthcare informatics. These include case-based diagnosis and treatment as well as discovery of trends among patients. However, few medical systems employ subsequence matching due to high computational and memory complexities. This paper proposes a randomized Monte Carlo sampling method to broaden search criteria with minimal increases in computational and memory complexities over R -NN indexing. Information gain improves while producing result sets that approximate the theoretical result space, query results increase by several orders of magnitude, and recall is improved with no significant degradation to precision over R -NN matching. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of the representation of simplicial complexes by trees. We introduce and analyze local and global tree representations. We prove that the global tree representation is more efficient in terms of time complexity for searching a given simplex and we show that the local tree representation is more efficient in terms of size of the structure. The simplicial complexes are modeled by hypergraphs. We then prove that the associated combinatorial optimization problems are very difficult to solve and to approximate even if the set of maximal simplices induces a planar graph of maximum degree at most three or a bounded degree hypergraph. However, we prove polynomial time algorithms that compute constant factor approximations and optimal solutions for some classes of instances. [ABSTRACT FROM AUTHOR]

The prohibitive computational complexity caused by redundant dictionary which is used in signal sparse decomposition has always been perplexing researchers of signal processing all the time. However, large over-complete dictionaries are essential to approximate the signal. In this paper, a decomposition algorithm based on modulation correlation partition (PBMC) is introduced, which improves the process of searching matched atoms in the redundant dictionary of functions. Through analyzing the structure of signals, we unite the frequency factor and phase factor to obtain the 2D modulation factor. The over-complete dictionary is partitioned into several sub-dictionaries according to the modulation correlation of atoms. Each sub-dictionary is represented by a single selected atom which is used in the greedy algorithm. At the end of this paper, experimental results show that the computational complexity of signal sparse decomposition can be adequately reduced by partitioning over-complete dictionary without impacting the decomposition result. [ABSTRACT FROM AUTHOR]

In large wireless multi-hop networks, routing is a main issue as they include many nodes that span over relatively a large area. In such a scenario, finding smallest set of dominant nodes for forwarding packets would be a good approach for better communication. Connected dominating set (CDS) computation is one of the method to find important nodes in the network. As CDS computation is an NP problem, several approximation algorithms are available but these algorithms have high message complexity. This paper discusses the design and implementation of a distributed algorithm to compute connected dominating sets in a wireless network with the help of network spectral properties. Based on local neighborhood, each node in the network finds its ego centric network. To identify dominant nodes, it uses bridge centrality value of ego centric network. A distributed algorithm is proposed to find nodes to connect dominant nodes which approximates CDS. The algorithm has been applied on networks with different network sizes and varying edge probability distributions. The algorithm outputs 40% important nodes in the network to form back haul communication links with an approximation ratio ≤ 0.04 * ∂ + 1, where ∂ is the maximum node degree. The results confirm that the algorithm contributes to a better performance with reduced message complexity. [ABSTRACT FROM AUTHOR]