Moldable tasks allow schedulers to determine the number of processors assigned to each task, thus enabling efficient use of large-scale parallel processing systems. We consider the problem of scheduling independent mo...
详细信息
Moldable tasks allow schedulers to determine the number of processors assigned to each task, thus enabling efficient use of large-scale parallel processing systems. We consider the problem of scheduling independent moldable tasks on processors and propose a new perspective of the existing speedup models: as the number pof processors assigned to a task increases, the speedup is linear if pis small and becomes sublinear after pexceeds a threshold. Based on this, we propose an efficient approximation algorithm to minimize the makespan. As a by-product, we also propose an approximation algorithm to maximize the sum of values of tasks completed by a deadline;this scheduling objective is considered for moldable tasks for the first time while similar works have been done for other types of parallel tasks. (C) 2023 Elsevier B.V. All rights reserved.
In this study, we present a general framework of outer approximation algorithms to solve convex vector optimization problems, in which the Pascoletti-Serafini (PS) scalarization is solved iteratively. This scalarizati...
详细信息
In this study, we present a general framework of outer approximation algorithms to solve convex vector optimization problems, in which the Pascoletti-Serafini (PS) scalarization is solved iteratively. This scalarization finds the minimum 'distance' from a reference point, which is usually taken as a vertex of the current outer approximation, to the upper image through a given direction. We propose efficient methods to select the parameters (the reference point and direction vector) of the PS scalarization and analyse the effects of these on the overall performance of the algorithm. Different from the existing vertex selection rules from the literature, the proposed methods do not require solving additional single-objective optimization problems. Using some test problems, we conduct an extensive computational study where three different measures are set as the stopping criteria: the approximation error, the runtime, and the cardinality of the solution set. We observe that the proposed variants have satisfactory results, especially in terms of runtime compared to the existing variants from the literature.
In Two-dimensional Bin Packing (2BP), we are given n rectangles as input and our goal is to find an axis-aligned nonoverlapping packing of these rectangles into the minimum number of unit square bins. 2BP admits no AP...
详细信息
In Two-dimensional Bin Packing (2BP), we are given n rectangles as input and our goal is to find an axis-aligned nonoverlapping packing of these rectangles into the minimum number of unit square bins. 2BP admits no APTAS and the current best approximation ratio is 1.406 by Bansal and Khan (ACM-SIAM symposium on discrete algorithms (SODA), pp 13-25, 2014. https://***/ 10.1137/1.9781611973402.2). A well-studied variant of 2BP is Guillotine Two-dimensional Bin Packing (G2BP), where rectangles must be packed in such a way that every rectangle in the packing can be obtained by applying a sequence of end-to-end axis-parallel cuts, also called guillotine cuts. Bansal et al. (Symposium on foundations of computer science (FOCS). IEEE, pp 657-666, 2005. https://***/ 10.1109/SFCS.2005.10) gave an APTAS for G2BP. Let lambda be the smallest constant such that for every set I of items, the number of bins in the optimal solution to G2BP for I is upper bounded by lambda opt(I) + c, where opt( I) is the number of bins in the optimal solution to 2BP for I and c is a constant. It is known that 4/3 <= lambda <= 1.692. Bansal and Khan (2014) conjectured that lambda = 4/3. The conjecture, if true, will imply a (4/3 + epsilon)-approximation algorithm for 2BP. Given a small constant delta > 0, a rectangle is called large if both its height and width are at least d, else it is called skewed. We make progress towards the conjecture by showing that lambda = 4/3 when all input rectangles are skewed. We also give an APTAS for 2BP for skewed items, though general 2BP does not admit an APTAS.
We study the approximability of two related problems on graphs with n nodes and m edges: n-Pairs Shortest Paths (n-PSP), where the goal is to find a shortest path between O(n) prespecified pairs, and All Node Shortest...
详细信息
ISBN:
(纸本)9781665455190
We study the approximability of two related problems on graphs with n nodes and m edges: n-Pairs Shortest Paths (n-PSP), where the goal is to find a shortest path between O(n) prespecified pairs, and All Node Shortest Cycles (ANSC), where the goal is to find the shortest cycle passing through each node. Approximate n-PSP has been previously studied, mostly in the context of distance oracles. We ask the question of whether approximate n-PSP can be solved faster than by using distance oracles or All Pair Shortest Paths (APSP). ANSC has also been studied previously, but only in terms of exact algorithms, rather than approximation. We provide a thorough study of the approximability of n-PSP and ANSC, providing a wide array of algorithms and conditional lower bounds that trade off between running time and approximation ratio. A highlight of our conditional lower bounds results is that for any integer k >= 1, under the combinatorial 4k-clique hypothesis, there is no combinatorial algorithm for unweighted undirected n-PSP with approximation ratio better than 1 + 1/k that runs in O(m(2-2/(k+1)) n(1/(k+1)-epsilon)) time. This nearly matches an upper bound implied by the result of Agarwal (2014). Our algorithms use a surprisingly wide range of techniques, including techniques from the girth problem, distance oracles, approximate APSP, spanners, fault-tolerant spanners, and link-cut trees. A highlight of our algorithmic results is that one can solve both n-PSP and ANSC in (O) over tilde (m + n(3/2+epsilon)) time1 with approximation factor 2 + epsilon (and additive error that is function of epsilon), for any constant epsilon > 0. For n-PSP, our conditional lower bounds imply that this approximation ratio is nearly optimal for any subquadratic-time combinatorial algorithm. We further extend these algorithms for n-PSP and ANSC to obtain a time/accuracy trade-off that includes near-linear time algorithms. Additionally, for ANSC, for all integers k >= 1, we extend the very recent a
In this article we study a generalized team orienteering problem (GTOP), which is to find service paths for multiple homogeneous vehicles in a network such that the profit sum of serving the nodes in the paths is maxi...
详细信息
In this article we study a generalized team orienteering problem (GTOP), which is to find service paths for multiple homogeneous vehicles in a network such that the profit sum of serving the nodes in the paths is maximized, subject to the cost budget of each vehicle. This problem has many potential applications in IoTs and smart cities, such as dispatching energy-constrained mobile chargers to charge as many energy-critical sensors as possible to prolong the network lifetime. In this article, we first formulate the GTOP problem, where each node can be served by different vehicles, and the profit of serving the node is a submodular function of the number of vehicles serving it. We then propose a novel (1-(1/epsilon)1/2+epsilon)-approximation algorithm for the problem, where epsilon is a given constant with 0 < epsilon <= 1 and e is the base of the natural logarithm. In particular, the approximation ratio is about 0.33 when epsilon = 0.5. In addition, we devise an improved approximation algorithm for a special case of the problem where the profit is the same by serving a node once and multiple times. We finally evaluate the proposed algorithms with simulation experiments, and the results of which are very promising. Especially, the profit sums delivered by the proposed algorithms are up to 14% higher than those by existing algorithms, and about 93.6% of the optimal solutions.
Sorting permutations with various operations has applications in macro rearrangement of genes in a genome and the design of computer interconnection networks. Block-interchange is a powerful operation that swaps two s...
详细信息
Sorting permutations with various operations has applications in macro rearrangement of genes in a genome and the design of computer interconnection networks. Block-interchange is a powerful operation that swaps two substrings that are called as blocks in literature, in a given permutation. When the blocks are restricted to be adjacent then one obtains a well studied operation: transposition. We call either a prefix or a suffix as an extreme. Restricting one of the swapped blocks to be an extreme in block-interchange operation yields a prefix or a suffix block-interchange respectively, the two types of extreme block-interchanges. For prefix block-interchange operation over permutations we design: (i) an optimum algorithm to sort reverse permutation, R-n, in n/2 moves, (ii) a simple 2-approximation algorithm, and (iii) for permutations with O(1) cycles, a 4/3 approximation algorithm. Due to symmetry, these results apply to suffix block-interchange operation also. (C) 2021 Elsevier B.V. All rights reserved.
We consider the robust (min-max regret) version of identical parallel machine scheduling problem, in which jobs may be outsourced to balance total cost against production efficiency. The total cost is measured in term...
详细信息
We consider the robust (min-max regret) version of identical parallel machine scheduling problem, in which jobs may be outsourced to balance total cost against production efficiency. The total cost is measured in terms of the total completion time of jobs processed in-house and the cost of outsourcing the rest. Processing times of in-house jobs are uncertain and they are described as two types of scenarios: discrete and interval. The objective is to obtain a robust (min-max regret) decision that minimises the absolute deviation of total cost from the optimal solution under the worst-case scenario. We first prove the worst-case scenario for any feasible solution. For the interval scenario, we further prove that the maximum regret value can be obtained in polynomial time for any feasible schedule. We also prove that for any discrete scenario, the minimum total cost can be obtained in polynomial time. Since the problem with the interval scenario is strongly NP-hard, we then transform the problem into an equivalent robust single machine scheduling problem. Finally, we develop 2-approximation algorithms for the problem with discrete and interval scenarios, respectively. These results are helpful for bridging the scheduling theory and practice in identical parallel machining environments with outsourcing and uncertain processing times.
Given a weighted graph G=(V,E)with weight w:E→Z+,a k-cycle transversal is an edge subset A of E such that G−A has no *** minimum weight of kcycle transversal is the weighted transversal number on k-cycle,denoted byτ...
详细信息
Given a weighted graph G=(V,E)with weight w:E→Z+,a k-cycle transversal is an edge subset A of E such that G−A has no *** minimum weight of kcycle transversal is the weighted transversal number on k-cycle,denoted byτk(Gw).In this paper,we design a(k−1/2)-approximation algorithm for the weighted transversal number on k-cycle when k is *** a weighted graph G=(V,E)with weight w:E→Z+,a k-clique transversal is an edge subset A of E such that G−A has no *** minimum weight of k-clique transversal is the weighted transversal number on k-clique,denoted byτapproximation algorithm for the weighted transversal number on k(Gw).In this paper,we design a(k2−k−1)/***,we discuss the relationship between k-clique covering and k-clique packing in complete graph Kn.
In this thesis, we will have discussions on two main topics, max-min allocation and scheduling jobs with precedent constraints on machines with communication delays. New approximation algorithms are given in Chapter 2...
详细信息
In this thesis, we will have discussions on two main topics, max-min allocation and scheduling jobs with precedent constraints on machines with communication delays. New approximation algorithms are given in Chapter 2, 4 and 5, where linear programming plays a fairly important role on algorithm designs, while Chapter 3 contains partial results on the general max-min allocation. The Santa Claus problem is also known as the restricted max-min fair allocation. In this problem, Santa Claus has a set of gifts, and he wants to distribute them among a set of children so that the least happy child is made as happy as possible. Here, the value that a child i has for a present j is of the form pij ∈ {0, pj}. Based on a modification of Haxell’s hypergraph matching argument, a polynomial time algorithm by Annamalai et al. gives a 12.33-approximation. In joint work with Sami Davies and Thomas Rothvoss, a matroid version of the Santa Claus problem is introduced. The algorithm is based on Haxell’s augmenting tree, but with the introduction of the matroid structure. Our result can then be used as a blackbox to obtain a (4 + ε)-approximation for Santa Claus, comparing against a natural, compact LP. A recent work of Cheng and Mao [CM19] also gets the factor (4 + ε). On the second half, we first consider the classic problem of scheduling jobs with precedence constraints on identical machines to minimize makespan, in the presence of communication delays. In this setting, denoted by P | prec, c | Cmax, if two dependent jobs are scheduled on different machines, then at least c units of time must pass between their executions. Despite its relevance to many applications, the best known approximation ratio was O(c), whereas Graham’s greedy list scheduling algorithm already gives a (c + 1)-approximation in that setting. An outstanding open problem in the top-10 list by Schuurman and Woeginger and its recent update by Bansal asks whether there exists a constant-factor approximation algorithm.
We present a (1-e)-approximation algorithms for maximum cardinality matchings in disk intersection graphs - all with near linear running time. We also present an estimation algorithm that returns (1 ± e)-approxim...
详细信息
暂无评论