We present an efficient algorithm for isometrically embedding weighted trees into low-dimensional l & INFIN;-normed spaces. The proposed algorithm takes an n-vertex weighted tree as input and maps each of its vert...
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We present an efficient algorithm for isometrically embedding weighted trees into low-dimensional l & INFIN;-normed spaces. The proposed algorithm takes an n-vertex weighted tree as input and maps each of its vertices onto an integer-valued vector in Rd, where d = theta(log n). The overall running time of the proposed algorithm is theta(n log2 n). We also present many applications of this algorithm in the context of competitive programming, including novel tasks for which we do not know any alternative efficient solution. Essentially, the low dimensionality of the resulting vectors allows the efficient answering of various distance-related queries. By means of experiment, we have also compared different implementations of the proposed algorithm with regards to both running time and dimensionality of the resulting embedding for different classes of trees.(c) 2022 Elsevier B.V. All rights reserved.
We consider the WEAK ROMAN DOMINATION problem. Given an undirected graph G = (V, E), the aim is to find a weak Roman domination function (wrd-function for short) of minimum cost, i.e. a function f : V -> {0, 1, 2} ...
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We consider the WEAK ROMAN DOMINATION problem. Given an undirected graph G = (V, E), the aim is to find a weak Roman domination function (wrd-function for short) of minimum cost, i.e. a function f : V -> {0, 1, 2} such that every vertex v is an element of V is defended (i.e. there exists a neighbor u of v, possibly u = v, such that f (u) >= 1) and for every vertex v is an element of V with f (v) = 0 there exists a neighbor u of v such that f (u) >= 1 and the function f(u -> v) defined by f(u -> v)(v) = 1, f(u -> v)(u) = f (u) - 1 and f(u -> v)(x) = f(x) otherwise does not contain any undefended vertex. The cost of a wrd-function f is defined by cost(f) = Sigma(v is an element of V)f(v). The trivial enumeration algorithm runs in time O*(3(n)) and polynomial space and is the best one known for the problem so far. We are breaking the trivial enumeration barrier by providing two faster algorithms: we first prove that the problem can be solved in O*(2(n)) time needing exponential space, and then describe an O*(2.2279(n)) algorithm using polynomial space. Our results rely on structural properties of a wrd-function, as well as on the best polynomial space algorithm for the RED-BLUE DOMINATING SET problem. Moreover we show that the problem can be solved in linear-time on interval graphs. (C) 2017 Elsevier B.V. All rights reserved.
Overtime is a common phenomenon in surgery departments, causing stress to physicians, dissatisfaction to patients, and financial loss to hospitals. We help risk-averse managers of operating rooms (ORs) to mitigate ove...
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Overtime is a common phenomenon in surgery departments, causing stress to physicians, dissatisfaction to patients, and financial loss to hospitals. We help risk-averse managers of operating rooms (ORs) to mitigate overtime in tactical surgery scheduling, which determines the assignment of elective patients to available ORs in upcoming time periods. We model the uncertain surgical durations via partial, full, or empirical distributions. To mitigate overtime, our model maximizes the risk aversion level of the OR manager (and thus the risk-hedging ability of the solution) while ensuring that the certainty equivalent of surgery duration in each OR at each time period does not exceed the stipulated working hours. The corresponding decision criterion, termed the maximized risk aversion level, is demonstrated in theory and in numerical experiments to be able to mitigate both the overtime probability and the expected overtime duration. To solve the problem, we develop an exact hill-climbing algorithm and demonstrate its convergence and correctness. Numerical experiments based on real-life surgery data show that our method outperforms the existing methods in several indicators that of concern to OR managers. In particular, this method is computationally amiable and hence is applicable to larger-scale instances. (C) 2019 Elsevier Ltd. All rights reserved.
In this paper, a new branch-and-price-and-cut algorithm is proposed to solve the one-dimensional bin-packing problem (1D-BPP). The 1D-BPP is one of the most fundamental problems in combinatorial optimization and has b...
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In this paper, a new branch-and-price-and-cut algorithm is proposed to solve the one-dimensional bin-packing problem (1D-BPP). The 1D-BPP is one of the most fundamental problems in combinatorial optimization and has been extensively studied for decades. Recently, a set of new 500 test instances were proposed for the 1D-BPP, and the best exact algorithm proposed in the literature can optimally solve 167 of these new instances, with a time limit of 1 hour imposed on each execution of the algorithm. The exact algorithm proposed in this paper is based on the classical set-partitioning model for the 1DBPPs and the subset row inequalities. We describe an ad hoc label-setting algorithm to solve the pricing problem, dominance, and fathoming rules to speed up its computation and a new primal heuristic. The exact algorithm can easily handle some practical constraints, such as the incompatibility between the items, and therefore, we also apply it to solve the one-dimensional bin-packing problem with conflicts (1D-BPPC). The proposed method is tested on a large family of 1D-BPP and 1D-BPPC classes of instances. For the 1DBPP, the proposed method can optimally solve 237 instances of the new set of difficult instances;the largest instance involves 1,003 items and bins of capacity 80,000. For the 1D-BPPC, the experiments show that the method is highly competitive with state-of-the-art methods and that it successfully closed several open 1D-BPPC instances.
We study the following optimization problem over a dynamical system that consists of several linear subsystems: Given a finite set of n x n matrices and an n-dimensional vector, find a sequence of K matrices, each cho...
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We study the following optimization problem over a dynamical system that consists of several linear subsystems: Given a finite set of n x n matrices and an n-dimensional vector, find a sequence of K matrices, each chosen from the given set of matrices, to maximize a convex function over the product of the K matrices and the given vector. This simple problem has many applications in operations research and control, yet a moderate-sized instance is challenging to solve to optimality for state-of-the-art optimization software. We propose a simple exact algorithm for this problem. Our algorithm runs in polynomial time when the given set of matrices has the oligo-vertex property, a concept we introduce in this paper for a finite set of matrices. We derive several sufficient conditions for a set of matrices to have the oligo-vertex property. Numerical results demonstrate the clear advantage of our algorithm in solving large-sized instances of the problem over one state-of-the-art global optimization solver. We also propose several open questions on the oligo-vertex property and discuss its potential connection with the finiteness property of a set of matrices, which may be of independent interest.
The problem is, given a set of n vectors in a d-dimensional normed space, find a subset with the largest length of the sum vector. We prove that, in the case of the lp norm, the problem is APX-complete for any p is an...
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The problem is, given a set of n vectors in a d-dimensional normed space, find a subset with the largest length of the sum vector. We prove that, in the case of the lp norm, the problem is APX-complete for any p is an element of [1, 2] and is not in APX if p is an element of (2, infinity). In the case of an arbitrary norm, we propose an algorithm which finds an optimal solution in time 0 (n(d-1)(d + logn)), improving previously known algorithms. In particular, the two-dimensional problem can be solved in nearly linear time. We also present an improved algorithm for the cardinality-constrained version of the problem with running time 0 (dn(d+1)). In the two-dimensional case, this version is shown to be solvable in nearly quadratic time. (C) 2018 Elsevier B.V. All rights reserved.
We revisit the exact algorithm to compute the treewidth of a graph of Tamaki and present it in a way that facilitates improvements. The so-called I-blocks and O-blocks enumerated by the algorithm are interpreted as su...
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Discrete optimization is a branch of mathematical optimization where some of the decision variables are restricted to real values in a discrete set. The use of discrete decision variables greatly expands the scope and...
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Discrete optimization is a branch of mathematical optimization where some of the decision variables are restricted to real values in a discrete set. The use of discrete decision variables greatly expands the scope and capacity of mathematical optimization models. In the era of big data, efficiency and scalability are increasingly important in evaluating the performance of an algorithm. However, discrete optimization problems usually are challenging to solve. In this thesis, we develop new fast exact algorithms for discrete optimization problems arising in the field of resource allocation and switched linear systems. The first problem is the discrete resource allocation problem with nested bound constraints. It is a fundamental problem with a wide variety of applications in search theory, economics, inventory systems, etc. Given B units of resource and n activities, each of which associated with a convex allocation cost fi(·), we aim to find an allocation of resources to the n activities, denoted by x ∈ Zn, to minimize the total allocation cost ∑ni=0di subject to the total amount of resource constraint as well as lower and upper bound constraints on total resource allocated to subsets of activities. We develop a Θ(n2logB/n)-time algorithm for it. It is an infeasibility-guided divide-and-conquer algorithm and the worst-case complexity is usually not achieved. Numerical experiments demonstrate that our algorithm significantly outperforms a state-of-the-art optimization solver and the performance of our algorithm is competitive compared to the algorithm with the best worst-case complexity for this problem in the literature. The second problem is the minimum convex cost network flow problem on the dynamic lot size network. In the dynamic lot size network, there are one source node and n sink nodes with demand di, i = 1,...,n. Let B = ∑ni=0di be the total demand. We aim to find a flow x to minimize the total arc cost and satisfy all the flow balance and capacity constrain
Seru Production is widely used in the Japanese electronics industry owing to its benefits. The total tardiness can be significantly reduced by Seru Production. We focus on investigating the fundamental principle of th...
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Seru Production is widely used in the Japanese electronics industry owing to its benefits. The total tardiness can be significantly reduced by Seru Production. We focus on investigating the fundamental principle of the total tardiness reduction brought by Seru Production. We formulate the seru system operation with minimising the total tardiness and analyse the solution space. We clarify that the model is non-linear. To exactly obtain the optimal solution of the non-linear model, we decompose the non-linear model into seru formation and seru scheduling which is formulated as a linear model. Thus, the small-scale seru system operation with minimising the total tardiness is solved exactly. For the large-scale problems, we propose a cooperative coevolution algorithm, where two evolution algorithms deal with the seru formation and seru scheduling. In the coevolution process, the two algorithms perform cooperation to seek the better solutions of seru system operation with minimising the total tardiness. Extensive experiments are tested to investigate how Seru Production reduces the total tardiness.
The remarkable growth of biological data is a motivation to accelerate the discovery of solutions in many domains of computational bioinformatics. In different phases of the computational pipelines, pattern matching i...
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The remarkable growth of biological data is a motivation to accelerate the discovery of solutions in many domains of computational bioinformatics. In different phases of the computational pipelines, pattern matching is a very practical operation. For example, pattern matching enables users to find the locations of particular DNA subsequences in a database or DNA sequence. Furthermore, in these expanding biological databases, some patterns are updated over time. To perform faster searches, high-speed pattern matching algorithms are needed. The present paper introduces three pattern matching algorithms that are specially formulated to speed up searches on large DNA sequences. The proposed algorithms raise performance by utilizing word processing (in place of the character processing presented in previous works) and also by searching the least frequent word of the pattern in the sequence. In terms of time cost, the experimental results demonstrate the superiority of the presented algorithms over the other simulated algorithms.
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