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In this paper, we focus on Hitting-Set , a fundamental problem in combinatorial optimization, through the lens of sublinear time algorithms. Given access to the hypergraph through a subset query oracle in the query model, we give sublinear time algorithms for Hitting-Set with almost tight parameterized query complexity. In parameterized query complexity , we estimate the number of queries to the oracle based on the parameter k , the size of the Hitting-Set. The subset query oracle we use in this paper is called Generalized d -partite Independent Set query oracle (GPIS) and it was introduced by Bishnu et al. (ISAAC'18). GPIS is a generalization to hypergraphs of the Bipartite Independent Set query oracle (BIS) introduced by Beame et al. (ITCS'18 and TALG'20) for estimating the number of edges in graphs. Since its introduction GPIS query oracle has been used for estimating the number of hyperedges independently by Dell et al. (SODA'20 and SICOMP'22) and Bhattacharya et al. (STACS'22), and for estimating the number of triangles in a graph by Bhattacharya et al. (ISAAC'19 and TOCS'21). Formally, GPIS is defined as follows: GPIS oracle for a d-uniform hypergraph H takes as input d pairwise disjoint non-empty subsets A 1 , ... , A d of vertices in H and answers whether there is a hyperedge in H that intersects each set A i , where i ∈ { 1 , 2 , ... , d }. For d = 2 , the GPIS oracle is nothing but BIS oracle. We show that d - Hitting-Set , the hitting set problem for d -uniform hypergraphs, can be solved using O ˜ d (k d log ⁡ n) GPIS queries. Additionally, we also showed that d - Decision-Hitting-Set , the decision version of d - Hitting-Set can be solved with O ˜ d (min ⁡ { k d log ⁡ n , k 2 d 2 }) GPIS queries. We complement these parameterized upper bounds with an almost matching parameterized lower bound that states that any algorithm that solves d - Decision-Hitting-Set requires Ω (( k + d d )) GPIS queries. [ABSTRACT FROM AUTHOR]

• The adaptive bulk search is a QUBO solver alternative to quantum annealing. • The cyclic-min algorithm is a SA-like algorithm without random number generation. • The bit permutated cyclic-min algorithm provides near-optimal solutions for QUBO. • Our scalable implementation attains linear improvement of the search rate. A wide range of combinatorial optimization problems can be reduced to the Ising model, and equivalently the quadratic unconstrained binary optimization (QUBO) problem. Thus, in recent years, researchers have proposed to solve QUBO on FPGAs, GPUs, and special-purpose processors. The adaptive bulk search (ABS) is a previously-proposed framework for solving QUBO in parallel on multiple GPUs. In the ABS, a CPU host performs a GA-based global search while GPUs asynchronously perform many local searches in parallel. An original ABS adopts a simple local search algorithm called a cyclic-min algorithm, which does not use pseudo random numbers. However, the lack of randomness may cause a potential drawback of restricted bit-flipping operations in a local search. To avoid this drawback, this paper proposes a cyclic-min algorithm with randomly-generated multiple bit permutations, which enables a more effective local search with random number generation in CPUs (not in GPUs). Furthermore, this paper introduces a scalable implementation of the ABS with MPI and OpenMP. Our experimental results on TSUBAME3.0 show that the solution quality improves and the throughput linearly increases as the number of GPUs increases; with 256 GPUs, it evaluates 20.1 × 10 12 solutions per second. [ABSTRACT FROM AUTHOR]

• Formulate a non-cooperative game with both UEs and MECs as players. • Prove the existence of and convergence to a Nash equilibrium. • Develop a set of algorithms to find the Nash equilibrium. • Demonstrate numerical examples of non-cooperative games with and without MECs' participation. In a newly emerged fog computing environment, various user equipments (UE) enhance their computing power and extend their battery lifetime by computation offloading to mobile edge cloud (MEC) servers. Such an environment is distributed and competitive in nature. In this paper, we take a game theoretical approach to computation offloading optimization in a fog computing environment. Such an approach captures and characterizes the nature of a competitive environment. The main contributions of the paper can be summarized as follows. First, we formulate a non-cooperative game with both UEs and MECs as players. Each UE attempts to minimize the execution time of its tasks with an energy constraint. Each MEC attempts to minimize the product of its power consumption for computation and execution time for allocated tasks. Second, we develop a heuristic algorithm for a UE to determine its "heuristically" best response to the current situation, an algorithm for an MEC to determine its best response to the current situation, and an iterative algorithm to find the Nash equilibrium. Third, we prove that our iterative algorithm converges to a Nash equilibrium. We demonstrate numerical examples of our non-cooperative games with and without MECs' participation. We observe that our iterative algorithm always quickly converges to a Nash equilibrium. The uniqueness of our non-cooperative games is that the strategy set of a player can be discrete and the payoff function of a player can be obtained by a heuristic algorithm for combinatorial optimization. To the best of the author's knowledge, there has been no such investigation of non-cooperative games based on combinatorial optimization for computation offloading optimization in a fog computing environment. [ABSTRACT FROM AUTHOR]

In this paper, we study task scheduling with or without energy constraint in mobile edge computing with multiple cloud-assisted edge servers as combinatorial optimization problems within the framework of classical scheduling theory. The first problem is to schedule a list of independent tasks on a mobile device and several heterogeneous edge servers and cloud servers, such that the makespan is minimized. The second problem is to schedule a list of independent tasks and to determine the computation and communication speeds of a mobile device, such that the makespan is minimized and the energy consumption of the mobile device does not exceed certain energy budget. The paper makes the following tangible contributions. We design heuristic task scheduling algorithms for both problems by considering the heterogeneity of computation and communication speeds. We derive a lower bound for the optimal schedule and prove an asymptotic performance bound for our heuristic algorithms. We experimentally evaluate the performance of our heuristic algorithms and show that their performance is very close to that of an optimal algorithm. Our analysis employs three key techniques, namely, the method of communication unification (i.e., all tasks have the same communication to computation ratio), the concept of effective speed of an edge server or a cloud server (i.e., the perceived speed of a server by ignoring the details and differences of communication speed and computation speed, wireless communication time and wired communication time, a regular edge server and a cloud-assisted edge server, execution time and waiting time), and the construction of virtual tasks (i.e., imaginary tasks which do not exist). Such unique techniques make it possible to derive a lower bound for the optimal solution, to derive an upper bound for the heuristic solution, to prove an asymptotic performance bound, and to find the best edge server order. To the best of the author's knowledge, this is the first paper in the literature which optimizes the makespan of task scheduling with or without energy constraint in mobile edge computing with multiple cloud-assisted edge servers. • Study task scheduling in mobile edge computing with multiple cloud-assisted edge servers. • Design heuristic algorithms by considering the heterogeneity of computation and communication speeds. • Derive a lower bound for the optimal schedule and prove an asymptotic performance bound. • Experimentally evaluate the performance of the heuristic algorithms. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of computing valuations of the Dieudonné determinants of matrices over discrete valuation skew fields (DVSFs). Under a reasonable computational model, we propose two algorithms for a class of DVSFs, called split. Our algorithms are extensions of the combinatorial relaxation of Murota (1995) and the matrix expansion by Moriyama and Murota (2013) , both of which are based on combinatorial optimization. While our algorithms require an upper bound on the output, we give an estimation of the bound for skew polynomial matrices and show that the estimation is valid only for skew polynomial matrices. We consider two applications of this problem. The first one is the noncommutative weighted Edmonds' problem (nc-WEP), which is to compute the degree of the Dieudonné determinants of matrices having noncommutative symbols. We show that the presented algorithms reduce the nc-WEP to the unweighted problem in polynomial time. In particular, we show that the nc-WEP over the rational field is solvable in time polynomial in the input bit-length. We also present an application to analyses of degrees of freedom of linear time-varying systems by establishing formulas on the solution spaces of linear differential/difference equations. [ABSTRACT FROM AUTHOR]

In this paper, we describe the first parallel Branch-and-Bound (B&B) algorithm with a history-based domination technique. Although history-based domination substantially speeds up a B&B search, it makes parallelization much more challenging. Our algorithm is the first parallel exact algorithm for the Sequential Ordering Problem using a pure B&B approach. To effectively explore the solution space, we have developed three novel parallelization techniques: thread restart, parallel history domination, and history-table memory management. The proposed algorithm was experimentally evaluated using the SOPLIB and TSPLIB benchmarks on multi-core processors. Using 32 threads with a time limit of one hour, the algorithm gives geometric-mean speedups of 72x and 20x on the medium-difficulty SOPLIB and TSPLIB instances, respectively. On the hard instances, it solves 12 instances that the sequential algorithm does not solve, with geometric-mean speedups of 16x on SOPLIB and 32x on TSPLIB. Super-linear speedups up to 366x are seen on 16 instances. • We propose the first parallel B&B algorithm that involves a history-based domination technique. • The proposed algorithm includes three novel parallelization techniques: thread restart, parallel history-based domination, and history-table memory management. • The proposed algorithm is the first parallel B&B algorithm for the Sequential Ordering Problem, which is an NP-hard sequencing problem with precedence constraints. • We present a thorough experimental evaluation of the proposed parallel algorithm on SOPLIB and TSPLIB. The results show that super-linear speedup is not an anomaly; it can be achieved on many instances. We report a speedup ratio as high as 285 on a 32-core processor. [ABSTRACT FROM AUTHOR]

Food Supply Chains (FSCs) have become increasingly complex with the average distance between producers and consumers rising considerably in the past two decades. Consequently, FSCs are a major source of carbon emissions and reducing transportation costs a major challenge for businesses. To address this, we present a mathematical model to promote the three core dimensions of sustainability (economic, environmental, and social), based on the Mixed-Integer Linear Programming (MILP) method. The model addresses the environmental dimension by intending to decrease the carbon emissions of different transport modes involved in the logistics network. Several supply chain network characteristics are incorporated and evaluated, with a consideration of social sustainability (job generation from operating various facilities). The mathematical model's robustness is demonstrated by testing and deploying it to a variety of problem instances. A real-life case study (Norwegian salmon supply chain) helps to comprehend the model's applicability. To understand the importance of optimizing food supply networks holistically, the paper investigates the impact of multiple supply chain permutations on total cost, demand fluctuations and carbon emissions. To address fluctuations in retail demand, we undertook sensitivity analysis for variations in demand, enabling the proposed model to revamp Norway's salmon supply chain network. Subsequently, the results are thoroughly examined to identify managerial implications. • Food Supply Chains often lack economic, social, and environmental sustainability. • MILP model employed to optimize food supply chain networks' sustainability. • Modelling considers the effect of electrical vehicles on costs and carbon emissions. • Sensitivity analysis demonstrates the model's resilience. • Case study (salmon in Norway) helps comprehend the model's applicability. • Logistical operations can be reconfigured to improve sustainability outcomes. [ABSTRACT FROM AUTHOR]