Sorting by Genome Rearrangements is a classic problem in Computational Biology. Several models have been considered so far, each of them defines how a genome is modeled (for example, permutations when assuming no dupl...
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Sorting by Genome Rearrangements is a classic problem in Computational Biology. Several models have been considered so far, each of them defines how a genome is modeled (for example, permutations when assuming no duplicated genes, strings if duplicated genes are allowed, and/or use of signs on each element when gene orientation is known), and which rearrangements are allowed. Recently, a new problem, called Sorting by Multi-Cut Rearrangements, was proposed. It uses the k-cut rearrangement which cuts a permutation (or a string) at k >= 2 places and rearranges the generated blocks to obtain a new permutation (or string) of same size. This new rearrangement may model chromoanagenesis, a phenomenon consisting of massive simultaneous rearrangements. Similarly as the Double-Cut-and-Join, this new rearrangement also generalizes several genome rearrangements such as reversals, transpositions, revrevs, transreversals, and block-interchanges. In this paper, we extend a previous work based on unsigned permutations and strings to signed permutations. We show the complexity of this problem for different values of k, and that the approximation algorithm proposed for unsigned permutations with any value of k can be adapted to signed permutations. We also show a 1.5-approximation algorithm for the specific case k = 4, as well as a generic approximation algorithm applicable for any k >= 5, that always reaches constant ratio. The latter makes use of the cycle graph, a well-known structure in genome rearrangements. We implemented and tested the proposed algorithms on simulated data.
We present approximation algorithms for the orthogonal z-oriented three-dimensional packing problem (TPPz) and analyze their asymptotic performance bound. This problem consists in packing a list of rectangular boxes L...
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We present approximation algorithms for the orthogonal z-oriented three-dimensional packing problem (TPPz) and analyze their asymptotic performance bound. This problem consists in packing a list of rectangular boxes L = (b(1),b(2),...,b(n)) into a rectangular box B = (l, w, infinity), orthogonally and oriented in the z-axis, in such a way that the height of the packing is minimized. We say that a packing is oriented in the z-axis when the boxes in L are allowed to be rotated (by ninety degrees) around the z-axis. This problem has some nice applications but has been less investigated than the well-known variant of it-denoted by TPP (three-dimensional orthogonal packing problem)-in which rotations of the boxes are not allowed. The problem TPP can be reduced to TPPz. Given an algorithm for TPPz, we can obtain an algorithm for TPP with the same asymptotic bound. We present an algorithm for TPPz, called R, and three other algorithms, called LS, BS, and SS, for special cases of this problem in which the instances are more restricted. The algorithm LS is for the case in which all boxes in L have square bottoms;BS is for the case in which the box B has a square bottom, and SS is for the case in which the box B and all boxes in L have square bottoms. For an algorithm A, we denote by r(A) the asymptotic performance bound of A. We show that 2.5 less than or equal to r(R) < 2.67, 2.5 less than or equal to r(LS) less than or equal to 2.528, 2.5 less than or equal to r(BS) less than or equal to 2.543, and 2.333 less than or equal to r(SS) less than or equal to 2.361. The algorithms presented here have the same complexity O(n log n) as the other known algorithms for these problems, but they have better asymptotic performance bounds.
In this paper, we address the trip-constrained vehicle routing cover problem (theTcVRC problem). Specifically, given a metric complete graphG=(V,E;w)with a set D(subset of V)of depots, a setJ(=V\D)of customer location...
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In this paper, we address the trip-constrained vehicle routing cover problem (theTcVRC problem). Specifically, given a metric complete graphG=(V,E;w)with a set D(subset of V)of depots, a setJ(=V\D)of customer locations, each customerhaving unsplittable demand 1, andkvehicles with capacityQ, it is asked to find a setC={C-i|=1,2,...,k}ofktours forkvehicles to service all customers, each tourfor a vehicle starts and ends at one depot inDand permits to be replenished at someother depots inDbefore continuously servicing at mostQcustomers, i.e., the numberof customers continuously serviced in per trip of each tour is at mostQ(except thetwo end-vertices of that trip), where each trip is a path or cycle, starting at a depot andending at other depot (maybe the same depot) inD, such that there are no other depotsin the interior of that path or cycle, the objective is to minimize the maximum weightof suchktours inC, i.e., minCmax{w(C-i)|i=1,2,...,k}, wherew(Ci)is thetotal weight of edges in that tourCi. Consideringkvehicles whether to have commondepot or suppliers, we consider three variations of the TcVRC problem, i.e., (1) the trip-constrained vehicle routing cover problem with multiple suppliers (the TcVRC-MSproblem) is asked to find a setC={Ci|i=1,2,...,k}ofktours mentioned-above,the objective is to minimize the maximum weight of suchktours inC;(2) the trip-constrained vehicle routing cover problem with common depot and multiple suppliers(the TcVRC-CDMS problem) is asked to find a setC={Ci|i=1,2,...,k}ofk tours mentioned-above, where each tour starts and ends at same depotvinD, eachvehicle having its suppliers at some depots inD(possibly includingv), the objectiveis to minimize the maximum weight of suchktours inC;(3) the trip-constrainedk-traveling salesman problem with non-suppliers (the TckTS-NS problem, simply theTckTSP-NS) is asked to find a setC={C-i=1,2,...,k}of k tours mentioned-above, where each tour starts and ends at same depotvinD, each vehicle havingnon-suppli
In this paper, we prove that the Max Morse Matching Problem is approximable, thus resolving an open problem posed by Joswig and Pfetsch [1]. For D-dimensional simplicial complexes, we obtain a (D+1)/(D-2+D+1)-factor a...
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In this paper, we prove that the Max Morse Matching Problem is approximable, thus resolving an open problem posed by Joswig and Pfetsch [1]. For D-dimensional simplicial complexes, we obtain a (D+1)/(D-2+D+1)-factor approximation ratio using a simple edge reorientation algorithm that removes cycles. For D >= 5, we describe a 2/D-factor approximation algorithm for simplicial manifolds by processing the simplices in increasing order of dimension. This algorithm leads to 1/2-factor approximation for 3-manifolds and 4/9-factor approximation for 4-manifolds. This algorithm may also be applied to non-manifolds resulting in a 1/(D+1)-factor approximation ratio. One application of these algorithms is towards efficient homology computation of simplicial complexes. Experiments using a prototype implementation on several datasets indicate that the algorithm computes near optimal results. (C) 2016 Elsevier B.V. All rights reserved.
We study the approximability of minimum total weighted tardiness with a modified objective which includes an additive constant. This ensures the existence of a positive lower bound for the minimum value. Moreover the ...
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We study the approximability of minimum total weighted tardiness with a modified objective which includes an additive constant. This ensures the existence of a positive lower bound for the minimum value. Moreover the new objective has a natural interpretation in just-in-time production systems. (C) 2007 Elsevier B.V. All rights reserved.
The time-dependent orienteering problem is dual to the time-dependent traveling salesman problem. It consists of visiting a maximum number of sites within a given deadline. The traveling time between two sites is in g...
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The time-dependent orienteering problem is dual to the time-dependent traveling salesman problem. It consists of visiting a maximum number of sites within a given deadline. The traveling time between two sites is in general dependent on the starting time. For any epsilon > 0, we provide a (2 + epsilon)-approximation algorithm for the time-dependent orienteering problem which runs in polynomial time if the ratio between the maximum and minimum traveling time between any two sites is constant. No prior upper approximation bounds were known for this time-dependent problem. (C) 2001 Elsevier Science B.V. All rights reserved.
In the minimum-cost k-(S, T) connected digraph (abbreviated as k-(S, T) connectivity) problem we are given a positive integer k, a directed graph G = (V, E) with nonnegative costs on the edges, and two subsets S, T of...
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In the minimum-cost k-(S, T) connected digraph (abbreviated as k-(S, T) connectivity) problem we are given a positive integer k, a directed graph G = (V, E) with nonnegative costs on the edges, and two subsets S, T of V;the goal is to find a subset of edges (E) over cap of minimum cost such that the subgraph (V, (E) over cap) has k edge-disjoint directed paths from each vertex in S to each vertex in T. Most of our results focus on a specialized version of the problem that we call the standard version, where every edge of positive cost has its tail in S and its head in T. This version of the problem captures NP-hard problems such as the minimum-cost k-vertex connected spanning subgraph problem. We give an approximation algorithm with a guarantee of O((log k)(log n)) for the standard version of the k-(S, T) connectivity problem, where n denotes the number of vertices. For k = 1, we give a simple 2-approximation algorithm that generalizes a well-known 2-approximation algorithm for the minimum-cost strongly connected spanning subgraph problem. For k = 2, we give a 3-approximation algorithm;this matches the best approximation guarantee known for the special case of the minimum-cost 2-vertex connected spanning subgraph problem. Besides the standard version, we study another version that is intermediate between the standard version and the problem in its full generality. In the relaxed version of the (S, T) connectivity problem, each edge of positive cost has its head in T but there is no restriction on the tail. We study the relaxed version with the connectivity parameter k fixed at one and observe that this version is at least as hard to approximate as the directed Steiner tree problem. We match this by giving an algorithm that achieves an approximation guarantee of alpha(n)+1 for the relaxed (S, T) connectivity problem, where alpha(n) denotes the best approximation guarantee available for the directed Steiner tree problem. The key to the analysis is a structural result
Training a one-node neural network with the ReLU activation function via optimization, which we refer to as the ON-ReLU problem, is a fundamental problem in machine learning. In this paper, we begin by proving the NP-...
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Training a one-node neural network with the ReLU activation function via optimization, which we refer to as the ON-ReLU problem, is a fundamental problem in machine learning. In this paper, we begin by proving the NP-hardness of the ON-ReLU problem. We then present an approximation algorithm to solve the ON-ReLU problem, whose running time is O(n(k)) where n is the number of samples, and k is a predefined integral constant as an algorithm parameter. We analyze the performance of this algorithm under two regimes and show that: (1) given any arbitrary set of training samples, the algorithm guarantees an (n/k)-approximation for the ON-ReLU problem - to the best of our knowledge, this is the first time that an algorithm guarantees an approximation ratio for arbitrary data scenario;thus, in the ideal case (i.e., when the training error is zero) the approximation algorithm achieves the globally optimal solution for the ON-ReLU problem;and (2) given training sample with Gaussian noise, the same approximation algorithm achieves a much better asymptotic approximation ratiowhich is independent of the number of samples n. Extensive numerical studies show that our approximation algorithm can perform better than the gradient descent algorithm. Our numerical results also show that the solution of the approximation algorithm can provide a good initialization for gradient descent, which can significantly improve the performance.
We consider the following two deterministic inventory optimization problems with non-stationary demands. Submodular joint replenishment problem. This involves multiple item types and a single retailer who faces demand...
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We consider the following two deterministic inventory optimization problems with non-stationary demands. Submodular joint replenishment problem. This involves multiple item types and a single retailer who faces demands over a finite planning horizon of T periods. In each time period, any subset of item-types can be ordered incurring a joint ordering cost which is submodular. Moreover, items can be held in inventory while incurring a holding cost. The objective is to find a sequence of orders that satisfies all demands and minimizes the total ordering and holding costs. Inventory routing problem. This involves a single depot that stocks items, and multiple retailer locations facing demands over a finite planning horizon of T periods. In each time period, any subset of locations can be visited using a vehicle originating from the depot. There is also cost incurred for holding items at any retailer. The objective here is to satisfy all demands while minimizing the sum of routing and holding costs. We present a unified approach that yields -factor approximation algorithms for both problems when the holding costs are polynomial functions. A special case is the classic linear holding cost model, wherein this is the first sub-logarithmic approximation ratio for either problem.
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...
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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.
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