Maximization Version Research Articles

Parameterized complexity of coverage in multi-interface IoT networks: Pathwidth

The Internet of Things (IoT) has emerged as one of the growing fields in digital technology over the past decade. A primary goal of IoT is to connect physical objects to the Internet to provide various services. Due to the vast number and diversity of these objects, referred to as devices, IoT must tackle both traditional and novel theoretical and practical network problems. Among these, multi-interface problems are well-known and have been extensively studied.This research focuses on one of the newest multi-interface models that fits well within the IoT context. It is known as the Coverage in the budget-constrained multi-interface problem, where the budget represents the total amount of energy in the network, and coverage refers to the model’s goal of activating all required communications among IoT devices. Since most IoT devices are battery-powered, energy consumption must be considered to extend the network’s lifespan. This means selecting the most energy-efficient interface configuration that allows all desired connections to function. To achieve this, both global energy consumption and the local number of active interfaces are limited. Moreover, this model also incentivize devices to turn on the available interfaces to create a more performant network. Finally, this model also takes into account the performance of the networks assigning a profit to devices that activate interfaces and realize connections.This problem can be represented using an undirected graph where each vertex represents a device, and each edge represents a desired connection. Every device is equipped with a set of available interfaces that can be used to facilitate transmission among the devices. The final goal is to activate a subset of the available interfaces that maximize the total profit, while not violating the constraints.This problem has been recognized as NP-hard, which is why we decided to investigate the decision version from the perspective of fixed-parameter tractability (FPT) theory. FPT is an advanced area of complexity theory that aims to identify the core complexity of a combinatorial problem by incorporating parameters into the time complexity domain.We provide two fixed-parameter tractability results, each describing an FPT algorithm. One algorithm is based on the well-known pathwidth parameter, the number of available interfaces, and the maximum available energy. The other algorithm considers pathwidth, the number of available interfaces, and an upper bound on the optimal profit. Finally, we show that these two algorithms can be applied to the maximization version of the problem.

The Secretary Problem with Predictions

The value maximization version of the secretary problem is the problem of hiring a candidate with the largest value from a randomly ordered sequence of candidates. In this work, we consider a setting where predictions of candidate values are provided in advance. We propose an algorithm that achieves a nearly optimal value if the predictions are accurate and results in a constant-factor competitive ratio otherwise. We also show that the worst-case competitive ratio of an algorithm cannot be higher than some constant [Formula: see text], which is the best possible competitive ratio when we ignore predictions, if the algorithm performs nearly optimally when the predictions are accurate. Additionally, for the multiple-choice secretary problem, we propose an algorithm with a similar theoretical guarantee. We empirically illustrate that if the predictions are accurate, the proposed algorithms perform well; meanwhile, if the predictions are inaccurate, performance is comparable to existing algorithms that do not use predictions.

Computing the quantumguesswork: a quadratic assignment problem

The quantum guesswork quantifies the minimum number of queries needed to guess the state of a quantum ensemble if one is allowed to query only one state at a time. Previous approaches to the computation of the guesswork were based on standard semi-definite programming techniques and therefore lead to approximated results. In contrast, we show that computing the quantum guesswork of qubit ensembles with uniform probability distribution corresponds to solving a quadratic assignment problem and we provide an algorithm that, upon the input of any qubit ensemble over a discrete ring, after finitely many steps outputs the exact closed-form expression of its guesswork. While in general the complexity of our guesswork-computing algorithm is factorial in the number of states, our main result consists of showing a more-than-quadratic speedup for symmetric ensembles, a scenario corresponding to the three-dimensional analog of the maximization version of the turbine-balancing problem. To find such symmetries, we provide an algorithm that, upon the input of any point set over a discrete ring, after finitely many steps outputs its exact symmetries. The complexity of our symmetries-finding algorithm is polynomial in the number of points. As examples, we compute the guesswork of regular and quasi-regular sets of qubit states.

Super-reparametrizations of weighted CSPs: properties and optimization perspective

The notion of reparametrizations of Weighted CSPs (WCSPs) (also known as equivalence-preserving transformations of WCSPs) is well-known and finds its use in many algorithms to approximate or bound the optimal WCSP value. In contrast, the concept of super-reparametrizations (which are changes of the weights that keep or increase the WCSP objective for every assignment) was already proposed but never studied in detail. To fill this gap, we present a number of theoretical properties of super-reparametrizations and compare them to those of reparametrizations. Furthermore, we propose a framework for computing upper bounds on the optimal value of the (maximization version of) WCSP using super-reparametrizations. We show that it is in principle possible to employ arbitrary (under some technical conditions) constraint propagation rules to improve the bound. For arc consistency in particular, the method reduces to the known Virtual AC (VAC) algorithm. We implemented the method for singleton arc consistency (SAC) and compared it to other strong local consistencies in WCSPs on a public benchmark. The results show that the bounds obtained from SAC are superior for many instance groups.

The strong network orientation problem

AbstractThis study presents models and heuristics for solving the strong network orientation problem (SNOP), which can model several tactical optimization problems of setting directions in urban networks. The objective is to set an orientation for each edge in an undirected graph such that the resulting digraph is strongly connected and the total travel distance between any pair of nodes is minimized (or maximized). Investigating tactical optimization problems such as SNOP is motivated by several challenges in urban networks due to the growth of population in urban areas, large number of daily trips, increasing price of maintaining urban networks, and the need to reduce air pollution and passive noise. Thus, a new trend is to utilize the urban networks better. In this context, we first use a multicommodity flow formulation to model the minimization problem. The maximization version is modeled by using the dual formulation of the shortest path problem. Then, scalable heuristic strategies for solving SNOP are investigated. For such purpose, we first propose basic components such as constructive heuristics, perturbations and local searches. They are combined into several metaheuristics based on local searches, multi‐start and evolutionary schemes, i.e. Multistart Local Search, Iterated Local Search (ILS), Relaxed ILS, Evolutionary Local Search (ELS), Relaxed ELS, and Variable Neighborhood Search. Computational experiments have been performed to analyze the proposed methods in terms of efficiency and quality of solutions, using grid instances and a graph from downtown Clermont‐Ferrand in France.

Generalizations, formulations and subgradient based heuristic with dynamic programming procedure for target set selection problems

Target set selection problems model influence processes through networks. Usually, these problems seek a set of individuals to be initially influenced by some content and will help to propagate the content to other individuals of the network. This work considers generalizations for the minimization and maximization versions of these problems. We propose binary linear models and Lagrangian relaxations that, for general cases, offer strictly better bounds than the linear relaxations. To efficiently solve our relaxations we design dynamic programming algorithms and embedded them in subgradient methods to find the best possible bounds. We also use the subgradient methods to construct feasible heuristic solutions and, to assess the effectiveness of our approaches, we run computational experiments on instances generated from real-world datasets. The experiments indicate that our heuristics obtain almost optimal solutions with good bounds in a very short time and, in some cases, they guarantee optimality for the constructed solutions.

The temporal explorer who returns to the base

We study here the problem of exploring a temporal graph when the underlying graph is a star. The aim of the exploration problem in a temporal star is finding a temporal walk which starts and finishes at the center of the star, and visits all leaves. We present a systematic study of the computational complexity of this problem, depending on the number k of time points where each edge can be present in the graph. We distinguish between the decision version StarExp(k), asking whether a complete exploration exists, and the maximization version MaxStarExp(k), asking for an exploration of the greatest possible number of edges. We fully characterize MaxStarExp(k) in terms of complexity. We also partially characterize StarExp(k), showing that it is in P for k<4, but is NP-complete, for every k>5. Finally, we partially characterize classes of “random” temporal stars which are, asymptotically almost surely, yes-instances and no-instances for StarExp(k).

A Stochastic Algorithm Based on Reverse Sampling Technique to Fight Against the Cyberbullying

Cyberbullying has caused serious consequences especially for social network users in recent years. However, the challenge is how to fight against the cyberbullying effectively from the algorithmic perspective. In this article, we study the fighting against the cyberbullying problem, i.e., identify an initial witness set with a budget to spread the positive influence to protect the users in a specific target set such that the number of cybervictim users in the target set being activated by the seed set of cyberbullying is minimized. We first formulate this problem and show its NP-hardness. We further prove that the objective function is submodular with respect to the size of witnesses set when we convert the original problem into the maximal version. Then we propose a stochastic approach to solve this maximal version problem based on the Reverse Sampling Technique with a constant factor guarantee. In addition, we provide theoretical analysis and discuss the relationship between the optimal value and the value returned by the proposed algorithm. To evaluate the proposed approach, we implement extensive experiments on synthetic and real datasets. The experimental results show our approach is superior to the comparison methods.

Coarse geometry and Callias quantisation

Consider a proper, isometric action by a unimodular, locally compact group $G$ on a complete Riemannian manifold $M$. For equivariant elliptic operators that are invertible outside a cocompact subset of $M$, we show that a localised index in the $K$-theory of the maximal group $C^*$-algebra of $G$ is well-defined. The approach is based on the use of maximal versions of equivariant localised Roe algebras, and many of the technical arguments in this paper are used to handle the ways in which they differ from their reduced versions. By using the maximal group $C^*$-algebra instead of its reduced counterpart, we can apply the trace given by integration over $G$ to recover an index defined earlier by the last two authors, and developed further by Braverman, in terms of sections invariant under the group action. As a very special case, this allows one to refine numerical obstructions to positive scalar curvature on a noncompact $\operatorname{Spin}$ manifold $X$ defined via Callias index theory, to obstructions in the $K$-theory of the maximal $C^*$-algebra of the fundamental group $\pi_1(X)$. As a motivating application in another direction, we prove a version of Guillemin and Sternberg's quantisation commutes with reduction principle for equivariant indices of $\operatorname{Spin}^c$ Callias-type operators.

Packing Arc-Disjoint Cycles in Tournaments

A tournament is a directed graph in which there is a single arc between every pair of distinct vertices. Given a tournament T on n vertices, we explore the classical and parameterized complexity of the problems of determining if T has a cycle packing (a set of pairwise arc-disjoint cycles) of size k and a triangle packing (a set of pairwise arc-disjoint triangles) of size k. We refer to these problems as Arc-disjoint Cycles in Tournaments (ACT) and Arc-disjoint Triangles in Tournaments (ATT), respectively. Although the maximization version of ACT can be seen as the dual of the well-studied problem of finding a minimum feedback arc set (a set of arcs whose deletion results in an acyclic graph) in tournaments, surprisingly no algorithmic results seem to exist for ACT. We first show that ACT and ATT are both NP-complete. Then, we show that the problem of determining if a tournament has a cycle packing and a feedback arc set of the same size is NP-complete. Next, we prove that ACT is fixed-parameter tractable via a $$2^{\mathcal {O}(k \log k)} n^{\mathcal {O}(1)}$$ -time algorithm and admits a kernel with $$\mathcal {O}(k)$$ vertices. Then, we show that ATT too has a kernel with $$\mathcal {O}(k)$$ vertices and can be solved in $$2^{\mathcal {O}(k)} n^{\mathcal {O}(1)}$$ time. Afterwards, we describe polynomial-time algorithms for ACT and ATT when the input tournament has a feedback arc set that is a matching. We also prove that ACT and ATT cannot be solved in $$2^{o(\sqrt{n})} n^{\mathcal {O}(1)}$$ time under the exponential-time hypothesis.

An Approximation Algorithm for Fully Planar Edge-Disjoint Paths

We devise a constant-factor approximation algorithm for the maximization version of the edge-disjoint paths problem if the supply graph together with the demand edges form a planar graph. By planar duality this is equivalent to packing cuts in a planar graph such that each cut contains exactly one demand edge. We also show that the natural linear programming relaxations have constant integrality gap, yielding an approximate max-multiflow min-multicut theorem.

AVAILABILITY HEURISTICS AND INSIGHTS FOR CORRUPTION

The most influencing insight which economic analysis of law provides is that human actors respond to incentives. This insight has two implications; first, law can be employed as an apparatus to bring in socially desirable behaviour as well as to discourage socially undesirable behaviour. In other words, law can be used as tool by policy makers to subsidize socially desirable behaviour and tax undesirable behaviour1 . Secondly, law can be used from the perspective of efficiency and distributive concerns i.e. it can be used as a device to encourage or discourage the production of social resources as well as a tool for efficient allocation of such social resources. To exhibit in a coherent manner the repercussions of the incentives on people in a legal system the analysts borrowed from economics the ‘rational choice theory’ which focuses on assumptions relating to how human actors respond to incentives. These assumptions were then applied to the domain of law to understand how law as a tool can be used in incentivising and de-incentivising certain behaviour. Thus, these incentive effects of legal rules were studied and analyzed in designing an efficient legal policy to attain desired social behaviour. The underlying conception of the rational choice theory as pointed out by Richard Posner is that “man is a rational maximizer of his ends”2 the adherents of ‘utility maximization version’ of rational choice theory take the above conception a bit further as according to them:

A polynomial algorithm with asymptotic ratio $2/3$ for the asymmetric maximization version of the $m$-PSP

A polynomial algorithm with asymptotic ratio $2/3$ for the asymmetric maximization version of the $m$-PSP

Reliability and Validity of a 6-Minute Yo-Yo Intermittent Endurance Test Level 2 in Subelite Part Time Male Soccer Players.

Gibson, N, Easton, C, Williams, M, McCunn, R, Gibson, NV. Reliability and validity of a 6-minute Yo-Yo Intermittent Endurance Test Level 2 in subelite part time male soccer players. J Strength Cond Res 36(4): 1011-1018, 2022-The aim of this study was to assess the reliability and relationship to maximal intermittent running performance of the 6-minute Yo-Yo Intermittent Endurance Test Level 2 (YYIET2), among subelite part time soccer players. Twenty male soccer players (15-22 years) completed three 6-minute YYIET2 trials with heart rate (HR), PlayerLoad, and rating of perceived exertion assessed during the protocol and HR and blood lactate assessed during 5 minutes of recovery. Subjects also completed a maximal version of the YYIET2 and the maximal Yo-Yo Intermittent Recovery Test Level 1 (YYIRT1) and 2 (YYIRT2). Heart rate at 4 and 6 minutes, PlayerLoad at 4 minutes, and HR recovery at 2, 3, 4, and 5 minutes during recovery from the 6-minute YYIET2 demonstrated little variance between tests. Correlations between variables measured during and after the 6-minute YYIET2 and distance covered in maximal tests ranged from r = 0.02 to -0.72. The 6-minute YYIET2 provides practitioners with a method of reliably assessing HR responses within subelite part time soccer players, although large correlations with maximal assessments suggest it can be used as a proxy measure for maximal intermittent running performance. Given its ease of administration and low time cost, the 6-minute YYIET2 offers practitioners a useful means of tracking training status and movement efficiency in players longitudinally.

A Polynomial Algorithm with Asymptotic Ratio $$\boldsymbol {2/3}$$ for the Asymmetric Maximization Version of the $$\boldsymbol m $$-PSP

In 2005, Kaplan et al. presented a polynomial-time algorithm with guaranteed approximation ratio $$2/3$$ for the maximization version of the asymmetric TSP. In 2014, Glebov, Skretneva, and Zambalaeva constructed a similar algorithm with approximation ratio $$2/3 $$ and cubic runtime for the maximization version of the asymmetric $$2 $$ -PSP ( $$2 $$ -APSP-max), where it is required to find two edge-disjoint Hamiltonian cycles of maximum total weight in a complete directed weighted graph. The goal of this paper is to construct a similar algorithm for the more general $$m $$ -APSP-max in the asymmetric case and justify an approximation ratio for this algorithm that tends to $$2/3 $$ as $$n$$ grows and the runtime complexity estimate $$O(mn^3)$$ .

Parameterized Complexity of Independent Set in H-Free Graphs

In this paper, we investigate the complexity of Maximum Independent Set (MIS) in the class of H-free graphs, that is, graphs excluding a fixed graph as an induced subgraph. Given that the problem remains NP-hard for most graphs H, we study its fixed-parameter tractability and make progress towards a dichotomy between FPT and W[1]-hard cases. We first show that MIS remains W[1]-hard in graphs forbidding simultaneously $$K_{1, 4}$$ , any finite set of cycles of length at least 4, and any finite set of trees with at least two branching vertices. In particular, this answers an open question of Dabrowski et al. concerning $$C_4$$ -free graphs. Then we extend the polynomial algorithm of Alekseev when H is a disjoint union of edges to an FPT algorithm when H is a disjoint union of cliques. We also provide a framework for solving several other cases, which is a generalization of the concept of iterative expansion accompanied by the extraction of a particular structure using Ramsey’s theorem. Iterative expansion is a maximization version of the so-called iterative compression. We believe that our framework can be of independent interest for solving other similar graph problems. Finally, we present positive and negative results on the existence of polynomial (Turing) kernels for several graphs H.

Approximation algorithms in combinatorial scientific computing

We survey recent work on approximation algorithms for computing degree-constrained subgraphs in graphs and their applications in combinatorial scientific computing. The problems we consider include maximization versions of cardinality matching, edge-weighted matching, vertex-weighted matching and edge-weighted$b$-matching, and minimization versions of weighted edge cover and$b$-edge cover. Exact algorithms for these problems are impractical for massive graphs with several millions of edges. For each problem we discuss theoretical foundations, the design of several linear or near-linear time approximation algorithms, their implementations on serial and parallel computers, and applications. Our focus is on practical algorithms that yield good performance on modern computer architectures with multiple threads and interconnected processors. We also include information about the software available for these problems.

A Polynomial 3/5-Approximate Algorithm for the Asymmetric Maximization Version of the 3-PSP

We present a first polynomial algorithm with guaranteed approximation ratio for the asymmetric maximization version of the asymmetric 3-Peripatetic Salesman Problem (3-APSP). This problem consists in finding the three edge-disjoint Hamiltonian circuits of maximal total weight in a complete weighted digraph. We prove that the algorithm has guaranteed approximation ratio 3/5 and cubic running-time.

Complexity and approximability of extended Spanning Star Forest problems in general and complete graphs

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.

A polynomial 3/5-approximate algorithm for the asymmetric maximization version of 3-PSP

A polynomial 3/5-approximate algorithm for the asymmetric maximization version of 3-PSP