Target-based computer-assisted orchestration can be thought of as the process of searching for combina-tions of orchestral sounds in a database of sound samples to match a given sound (called target ) while respecting...
详细信息
Target-based computer-assisted orchestration can be thought of as the process of searching for combina-tions of orchestral sounds in a database of sound samples to match a given sound (called target ) while respecting specific symbolic constraints (such as the musical instruments that we can use). It is modeled as a combinatorial optimization problem, where the feasible solutions are subsets of orchestral sounds, and the function to minimize measures a distance between the chosen subset of orchestral sounds and the target. In this article, we focus on this optimization problem and analyse it through the lens of com-putational complexity and approximation algorithms. We first study the so-called static case where there is no temporality in the target sound (we want to reproduce a 'single' constant sound). We show that the problem is already NP-hard, but is solvable by a pseudo-polynomial-time algorithm. We also provide an approximation algorithm with additive error. We then consider the more general case with tempo-rality, when the target sound can be seen as a sequence of sounds over a time horizon. We show how the previous results in the static case generalize in this temporal case. We conclude our study with some experiments on real target sounds.(c) 2022 Elsevier B.V. All rights reserved.
We describe a half-approximation algorithm, b-SUITOR, for computing a b-MATCHING of maximum weight in a graph with weights on the edges. b-MATCHING is a generalization of the well-known MATCHING problem in graphs, whe...
详细信息
We describe a half-approximation algorithm, b-SUITOR, for computing a b-MATCHING of maximum weight in a graph with weights on the edges. b-MATCHING is a generalization of the well-known MATCHING problem in graphs, where the objective is to choose a subset of M edges in the graph such that at most a specified number b(v) of edges in M are incident on each vertex v. Subject to this restriction we maximize the sum of the weights of the edges in M. We prove that the b-SUITOR algorithm computes the same b-MATCHING as the one obtained by the GREEDY algorithm for the problem. We implement the algorithm on serial and shared-memory parallel processors and compare its performance against a collection of approximation algorithms that have been proposed earlier. Our results show that the b-SUITOR algorithm outperforms the GREEDY and locally dominant edge algorithms by one to two orders of magnitude on a serial processor. The b-SUITOR algorithm has a high degree of concurrency, and it scales well up to 240 threads on a shared-memory multiprocessor. The b-SUITOR algorithm outperforms the locally dominant edge algorithm by a factor of 14 on 16 cores of an Intel Xeon multiprocessor.
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.
Recent studies have shown that the chromosomal recombination only takes places at some narrow hotspots. Within the chromosomal region between these hotspots (called haplotype block), little or even no recombination oc...
详细信息
In Constrained Fault-Tolerant Resource Allocation (FTRA) problem, we are given a set of sites containing facilities as resources and a set of clients accessing these resources. Each site i can open at most R-i facilit...
详细信息
In Constrained Fault-Tolerant Resource Allocation (FTRA) problem, we are given a set of sites containing facilities as resources and a set of clients accessing these resources. Each site i can open at most R-i facilities with opening cost f(i). Each client j requires an allocation of r(j) open facilities and connecting j to any facility at site i incurs a connection cost c(ij). The goal is to minimize the total cost of this resource allocation scenario. FTRA generalizes the Unconstrained Fault-Tolerant Resource Allocation (FTRA(infinity)) [1] and the classical Fault-Tolerant Facility Location (FTFL) [2] problems: for every site i, FTRA(infinity) does not have the constraint R-i, whereas FTFL sets R-i = 1. These problems are said to be uniform if all r(j)'s are the same, and general otherwise. For the general metric FTRA, we first give an LP-rounding algorithm achieving an approximation ratio of 4. Then we show the problem reduces to FTFL, implying the ratio of 1.7245 from [3]. For the uniform FTRA, we provide a 1.52-approximation primal-dual algorithm in O (n(4)) time, where n is the total number of sites and clients. (C) 2015 Elsevier B.V. All rights reserved.
This paper deals with approximation algorithms for the prize collecting generalized Steiner forest problem, defined as follows. The input is an undirected graph G = (V, E), a collection T = {T-1, ... ,T-k}, each a sub...
详细信息
This paper deals with approximation algorithms for the prize collecting generalized Steiner forest problem, defined as follows. The input is an undirected graph G = (V, E), a collection T = {T-1, ... ,T-k}, each a subset of V of size at least 2, a weight function w: E -> R+, and a penalty function p: T -> R+. The goal is to find a forest F that minimizes the cost of the edges of F plus the penalties paid for subsets T-i whose vertices are not all connected by F. Our main result is a (3 - 4/n)-approximation for the prize collecting generalized Steiner forest problem, where n >= 2 is the number of vertices in the graph. This obviously implies the same approximation for the special case called the prize collecting Steiner forest problem (all subsets T-i are of size 2). The approximation algorithm is obtained by applying the local ratio method, and is much simpler than the best known combinatorial algorithm for this problem. Our approach gives a (2 - 1/n-1)-approximation for the prize collecting Steiner tree problem (all subsets T-i are of size 2 and there n _T is some root vertex r that belongs to all of them). This latter algorithm is in fact the local ratio version of the primal-dual algorithm of Goemans and Williamson [M.X. Goemans, D.P. Williamson, A general approximation technique for constrained forest problems, SIAM Journal on Computing 24 (2) (April 1995) 296-317]. Another special case of our main algorithm is Bar-Yehuda's local ratio (2 - 2/n)-approximation for the generalized Steiner forest problem (all the penalties are infinity) [R. Bar-Yehuda, One for the price n of two: a unified approach for approximating covering problems, Algorithmica 27 (2) (June 2000) 131-144]. Thus, an important contribution of this paper is in providing a natural generalization of the framework presented by Goemans and Williamson, and later by Bar-Yehuda. (C) 2008 Elsevier B.V. All rights reserved.
A star graph is a tree of diameter at most two. A star forest is a graph that consists of node-disjoint star graphs. In the spanning star forest problem, given an unweighted graph G, the objective is to find a star fo...
详细信息
A star graph is a tree of diameter at most two. A star forest is a graph that consists of node-disjoint star graphs. In the spanning star forest problem, given an unweighted graph G, the objective is to find a star forest that contains all vertices of G and has the maximum number of edges. This problem is the complement of the dominating set problem in the following sense: On a graph with n vertices, the size of the maximum spanning star forest is equal to n minus the size of the minimum dominating set. We present a 0.71-approximation algorithm for this problem, improving upon the approximation factor of 0.6 of Nguyen et al. (SIAM J. Comput. 38:946-962, 2008). We also present a 0.64-approximation algorithm for the problem on node-weighted graphs. Finally, we present improved hardness of approximation results for the weighted (both edge-weighted and node-weighted) versions of the problem. Our algorithms use a non-linear rounding scheme, which might be of independent interest.
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.
Uncertainty about data appears in many real-world applications and an important issue is how to manage, analyze and solve optimization problems over such data. An important tool for data analysis is clustering. When t...
详细信息
Uncertainty about data appears in many real-world applications and an important issue is how to manage, analyze and solve optimization problems over such data. An important tool for data analysis is clustering. When the data set is uncertain, we can model them as a set of probabilistic points each formalized as a probability distribution function which describes the possible locations of the points. In this paper, we study k-center problem for probabilistic points in a general metric space. First, we present a fast greedy approximation algorithm that builds k centers using a farthest-first traversal in k iterations. This algorithm improves the previous approximation factor of the unrestricted assigned k-center problem from 10 (see [1]) to 6. Next, we restrict the centers to be selected from all the probabilistic locations of the given points and we show that an optimal solution for this restricted setting is a 2-approximation factor solution for an optimal solution of the assigned k-center problem with expected distance assignment. Using this idea, we improve the approximation factor of the unrestricted assigned k-center problem to 4 by increasing the running time. The algorithm also runs in polynomial time when k is a constant. Additionally, we implement our algorithms on three real data sets. The experimental results show that in practice the approximation factors of our algorithms are better than in theory for these data sets. Also we compare the results of our algorithm with the previous works and discuss about the achieved results. At the end, we present our theoretical results for probabilistic k-median clustering.
Given a graph G with a set of terminals, two weight functions c and d defined on the edge set of G, and a bound D, a popular NP-hard problem in designing networks is to find the minimum cost Steiner tree (under functi...
详细信息
Given a graph G with a set of terminals, two weight functions c and d defined on the edge set of G, and a bound D, a popular NP-hard problem in designing networks is to find the minimum cost Steiner tree (under function c) in G, to connect all terminals in such a way that its diameter (under function d) is bounded by D. Marathe et al. (J. Algoritm. 28(1), 142-171, 1998) proposed an (O(ln n), O(ln n)) approximation algorithm for this bicriteria problem, where n is the number of terminals. The first factor reflects the approximation ratio on the diameter bound D, and the second factor indicates the cost-approximation ratio. Later, Kapoor and Sarwat (Theory Comput. Syst. 41(4), 779-794, 2007) introduced a parameterized approximation algorithm with performance guarantee of (O(p . ln n/ln p), O(ln n/ln p)) for any input value p > 1, by which one can improve the approximation factor for cost at the price of worsening the approximation factor of diameter. In this paper, we consider the reverse scenario in which minimizing the diameter of the solution is more important. We first propose a parameterized approximation algorithm with performance guarantee of (O(ln n/ln p), O(p. H-p . ln n/ln p)), where H-p = 1+ 1/ 2+...+ 1/ p is the pth harmonic number. Parameter p is part of the input and this algorithm runs in polynomial time for constant values of p. We also present another algorithm with approximation ratio of (O(ln n/n p), O(mu . p . ln n/ln p)) which relies on the approximation factor (mu) of the NP-hard problem min-degree constrained minimum spanning tree.
暂无评论