We give a nearly-linear time reduction that encodes any linear program as a 2-commodity flow problem with only a small blow-up in size. Under mild assumptions similar to those employed by modern fast solvers for linea...
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We give a nearly-linear time reduction that encodes any linear program as a 2-commodity flow problem with only a small blow-up in size. Under mild assumptions similar to those employed by modern fast solvers for linear programs, our reduction causes only a polylogarithmic multiplicative increase in the size of the program, and runs in nearly-linear time. Our reduction applies to high-accuracy approximation algorithms and exact algorithms. Given an approximate solution to the 2-commodity flow problem, we can extract a solution to the linear program in linear time with only a polynomial factor increase in the error. This implies that any algorithm that solves the 2-commodity flow problem can solve linear programs in essentially the same time. Given a directed graph with edge capacities and two source-sink pairs, the goal of the 2-commodity flow problem is to maximize the sum of the flows routed between the two source-sink pairs subject to edge capacities and flow conservation. A 2-commodity flow problem can be formulated as a linear program, which can be solved to high accuracy in almost the current matrix multiplication time (Cohen-Lee-Song JACM'21). Our reduction shows that linear programs can be approximately solved, to high accuracy, using 2-commodity flow as well. Our proof follows the outline of Itai's polynomial-time reduction of a linear program to a 2-commodity flow problem (JACM'78). Itai's reduction shows that exactly solving 2-commodity flow and exactly solving linear programming are polynomial-time equivalent. We improve Itai's reduction to nearly preserve the problem representation size in each step. In addition, we establish an error bound for approximately solving each intermediate problem in the reduction, and show that the accumulated error is polynomially bounded. We remark that our reduction does not run in strongly polynomial time and that it is open whether 2-commodity flow and linear programming are equivalent in strongly polynomial time. Copyri
Food packs are typically given as assistance during relief operations in emergency situations. In distributing these food packs, it is important to consider how to maximize available supplies and make them last until ...
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Food packs are typically given as assistance during relief operations in emergency situations. In distributing these food packs, it is important to consider how to maximize available supplies and make them last until basic urban operations are returned to normal. A food pack planning program that uses integer linear programming is created using Microsoft Excel and Jupyter Notebook with Python programming language. The program is flexible enough to create different food pack plans depending on the conditions of the target beneficiaries. It can be adjusted according to the number of people in the household and the duration for which it would last, among other factors.
In Constraint programming (CP), achieving arc-consistency (AC) of a global constraint with costs consists in removing from the domains of the variables all the values that do not belong to any solution whose cost is b...
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Interval regression modeling for a class of uncertain nonlinear system has not been widely studied to *** proposed modeling method,differently from standard deterministic models,is composed of upper regression model(U...
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Interval regression modeling for a class of uncertain nonlinear system has not been widely studied to *** proposed modeling method,differently from standard deterministic models,is composed of upper regression model(URM) and lower regression model(LRM),and returns an interval output as opposed to a point output(a single value).In this contribution,we address the problem of constructing interval model directly from observed data which is generated from uncertain nonlinear *** method combines sparsity stemming from the idea of linear programming support vector learning approach,and modeling accuracy guaranteed by measuring the minimization of maximum regarding the selection of approximation error between the actual output and the estimated out(namely Min-Max Optimization).First,quadratic programming problem corresponding to support vector regression(S VR) is transferred to linear programming(LP) ***,the min-max optimization is LP problem within the framework of SVR to form new optimization problem,where the sparsity and optimality of the interval modeling can well be reflected and *** optimization problems with constraints in a form of convex inequality and linear equality are solved by ***,the sparsity and optimality of the proposed method are demonstrated by the experimental cases using the two indices,the number of utilised support vectors(SVs) and root mean square error(RMSE).
We study the properties of secret sharing schemes, where a random secret value is transformed into shares distributed among several participants in such a way that only the qualified groups of participants can recover...
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A stable electricity supply and effective allocation of energy are critically important for a country to balance the trade-offs between economic development and sustainable environments. This research presents a novel...
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In this paper, a linear programming problem is investigated in which the feasible region is formed as a special type of fuzzy relational equalities (FRE). In this type of FRE, fuzzy composition is considered as the we...
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In this paper, we propose a Feasible Sequential linear programming (FSLP) algorithm applied to time-optimal control problems (TOCP) obtained through direct multiple shooting discretization. This method is motivated by...
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ISBN:
(数字)9781665467612
ISBN:
(纸本)9781665467629
In this paper, we propose a Feasible Sequential linear programming (FSLP) algorithm applied to time-optimal control problems (TOCP) obtained through direct multiple shooting discretization. This method is motivated by TOCP with nonlinear constraints which arise in motion planning of mechatronic systems. The algorithm applies a trust-region globalization strategy ensuring global convergence. For fully determined problems our algorithm provides locally quadratic convergence. Moreover, the algorithm keeps all iterates feasible enabling early termination at suboptimal, feasible solutions. This additional feasibility is achieved by an efficient iterative strategy using evaluations of constraints, i.e., zero-order information. Convergence of the feasibility iterations can be enforced by reduction of the trust-region radius. These feasibility iterations maintain feasibility for general Nonlinear Programs (NLP). Therefore, the algorithm is applicable to general NLPs. We demonstrate our algorithm’s efficiency and the feasibility update strategy on a TOCP of an overhead crane motion planning simulation case.
The minimax theorem for zero-sum games is easily proved from the strong duality theorem of linear programming. For the converse direction, the standard proof by Dantzig (1951) is known to be incomplete. We explain and...
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