Demand Vertex Research Articles

Bike rebalancing: How to find a balanced matching in the k center problem?

In the bike-sharing system, inspired by the problem proposed by O’Mahony and Shmoys (AAAI 2015), we present a new model called the balanced k center problem with matching constraints. Given a network containing the same number of two types of points, supply points and demand points, we aim to allocate k (supply) centers for supply points, and k (demand) centers for demand points. The k supply centers should be perfectly matched with the k demand centers, within an allowed distance. The goal is to minimize the maximum distance from each supply (or demand) point to its closest supply (or demand) centers.The model is motivated by the real-world application in rebalancing bikes, where the bikes are collected in supply centers, and delivered to the matched demand centers, and then distributed from there to serve their demand vertices. Therefore, to successfully complete this rebalancing procedure, two matched groups must have the same number of points. Otherwise, the bikes collected at the supply center may not be enough to serve the demand stations assigned to the demand center.To fully capture the problem, we bring an identical restriction L, which restricts the exact number of vertices assigned to a single center. As a result, all matched centers must serve the same number of bikes. Hence, they are balanced. Our main result is a 9 approximation algorithm. The algorithm is built on the technique called transfer preserving network that successfully combines the approach for the capacitated k center problem with the maximum flow network.

Multi-supply multi-capacitated p-median location optimization via a hybrid bi-level intelligent algorithm

The Capacitated P-Median Problem (CPMP) is a classic discrete facility location problem in which p capacitated facilities (medians) are selected from an economic point of view to serve a set of demand vertexes so that the total demand assigned to each of the facilities (medians) does not exceed its capacity. Since the capacity used by each facility is fixed and does not depend on the vertex demand, the traditional capacitated p-median problem’s total capacity is underutilized. To make full use of the total capacity, a more generalized model, namely, the Multi-supply Multi-Capacitated Location Problem (MMCLP), is considered in this paper, which assumes not only that each facility is allowed to open with different capacities while meeting the constraint of total capacity but also that each demand vertex can accept supplies from different facilities. To solve the MMCLP effectively, we propose a hybrid bi-level optimization framework based on decomposition. First, the original MMCLP is decomposed into numerous mixed-integer programming (MIP) sub-problems, i.e., inner optimization is performed, and for outer optimization, a Genetic Algorithm based on Co-Evolution of Population and Neighborhood (GACEPN) is proposed to find better sub-problems near the good sub-problems among all the candidate sub-problems. The inner optimization problem is solved by a commercial MIP solver. The results obtained are returned to the outer layer so that the neighborhood of the outer layer optimization can be adjusted appropriately to generate better sub-problems, thereby approaching the optimal solution of the original problem. The experiments show that for a large-scale MMCLP, compared to the Gurobi solver, the proposed bi-level algorithm (a hybrid of the GACEPN and Gurobi) can find a better solution in less time.

A strongly polynomial time algorithm for the maximum supply rate problem on trees

Suppose that we are given a graph whose each vertex is either a supply vertex or a demand vertex and is assigned a nonnegative integer supply or demand value. We consider partitioning G into connected components by removing edges from G so that each connected component has exactly one supply vertex and there exists a flow in each connected component satisfying the supply/demand constraints. The problem that determines the existence of such a partition is called the partition problem. Ito et al. (2005) showed that the partition problem is NP-complete in general and it can be solved in linear time if the given graph is a tree. When the graph does not have such a partition, we scale the demand values uniformly by scale factor r so that the obtained graph has a desired partition. The maximum supply rate problem is the problem that finds the maximum value of such r. Whereas the maximum supply rate problem is NP-hard in general in the same way as the partition problem, Morishita and Nishizeki (2015) gave a weakly polynomial-time algorithm for the problem on trees.In this paper, we give a first strongly polynomial-time algorithm for the maximum supply rate problem on trees. Our algorithm is based on the dynamic programming technique, in which we compute “surplus” and “deficit” of the supply in subproblems from leaves to the root. We use piecewise linear functions of r to represent them, and one of our important contributions is to bound the size of the representation of each function.

A biased random-key genetic algorithm for wireless backhaul network design

This paper describes a biased random-key genetic algorithm for a real-world wireless backhaul network design problem. This is a novel problem, closely related to variants of the Steiner tree problem and the facility location problem. Given a parameter h, we want to build a forest where each tree has at most h hops from the demand nodes, where traffic originates, to the root nodes where each tree is rooted. Candidate Steiner nodes do not have any demand but represent locations where we can install cellsites to cover the traffic and equipment to backhaul the traffic to the cellular core network. Each Steiner node can cover demand nodes within a given distance, subject to a capacity constraint. The aggregate set of constraints may make it impossible to cover or backhaul all demands. A revenue function computes the revenue associated with the total amount of traffic covered and backhauled to the root nodes. The objective of the problem is to build a forest that maximizes the difference between the total revenue and the cost associated with the installed equipment. Although we will have a forest when we consider only the backhaul links and root nodes, the addition of demand vertices can induce undirected cycles, resulting in a directed acyclic graph. We consider instances of this problem with several additional constraints that are motivated by the requirements of real-world telecommunication networks.

Parametric power supply networks

Suppose that each vertex of a graph $$G$$ G is either a supply vertex or a demand vertex and is assigned a supply or a demand. All demands and supplies are nonnegative constant numbers in a steady network, while they are functions of a variable $$\lambda $$ ? in a parametric network. Each demand vertex can receive "power" from exactly one supply vertex through edges in $$G$$ G . One thus wishes to partition $$G$$ G to connected components by deleting edges from $$G$$ G so that each component has exactly one supply vertex whose supply is at least the sum of demands in the component. The "partition problem" asks whether $$G$$ G has such a partition. If $$G$$ G has no such partition, one wishes to find the maximum number $$r^*$$ r ? , $$0\le r^* <1$$ 0 ≤ r ? < 1 , such that $$G$$ G has such a partition when every demand is reduced to $$r^*$$ r ? times the original demand. The "maximum supply rate problem" asks to compute $$r^*$$ r ? . In this paper, we deal with a network in which $$G$$ G is a tree, and first give a polynomial-time algorithm for the maximum supply rate problem for a steady tree network, and then give an algorithm for the partition problem on a parametric tree network, which takes pseudo-polynomial time if all the supplies and demands are piecewise linear functions of $$\lambda $$ ? .

Connecting Supply and Demand Vertices with Fault Tolerance

Connecting Supply and Demand Vertices with Fault Tolerance

Partitioning Trees with Supply, Demand and Edge-Capacity

Let T be a given tree. Each vertex of T is either a supply vertex or a demand vertex, and is assigned a positive number, called the supply or demand. Each demand vertex v must be supplied an amount of “power,” equal to the demand of v, from exactly one supply vertex through edges in T . Each edge is assigned a positive number called the capacity. One wishes to partition T into subtrees by deleting edges from T so that each subtree contains exactly one supply vertex whose supply is no less than the sum of all demands in the subtree and the power flow through each edge is no more than capacity of the edge. The “partition problem” is a decision problem to ask whether T has such a partition. The “maximum partition problem” is an optimization version of the partition problem. In this paper, we give three algorithms for the problems. First is a linear-time algorithm for the partition problem. Second is a pseudo-polynomial-time algorithm for the maximum partition problem. Third is a fully polynomial-time approximation scheme (FPTAS) for the maximum partition problem.

Minimum Cost Partitions of Trees with Supply and Demand

Let T be a given tree. Each vertex of T is either a supply vertex or a demand vertex, and is assigned a positive integer, called the supply or the demand. Every demand vertex v of T must be supplied an amount of "power," equal to the demand of v, from exactly one supply vertex through edges in T. Each edge e of T has a direction, and is assigned a positive integer which represents the cost required to delete e from T or reverse the direction of e. Then one wishes to obtain subtrees of T by deleting edges or reversing the directions of edges so that (a) each subtree contains exactly one supply vertex whose supply is no less than the sum of all demands in the subtree and (b) every edge is directed away from the supply vertex in each subtree. One wishes to minimize the total cost to obtain such subtrees from T. In the paper, we first show that this minimization problem is NP-hard, and then give a pseudo-polynomial-time algorithm to solve the problem. We finally give a fully polynomial-time approximation scheme (FPTAS) for the problem.

An efficient heuristic method for capacitated P-Median problem

Capacitated p-median problem (CPMP) is an important variation of facility location problem in which p capacitated medians are economically selected to serve a set of demand vertices so that the total assigned demand to each of the candidate medians must not exceed its capacity. The classical CPMP uses a network in a Euclidean plane such that distance between any two points in the network is the straight-line distance between them. So this paper surveys and analyses the effect of using different metrics on this problem. This paper presents a genetic algorithm to solve the CPMP. Two different assignment techniques namely, classical assignment method and assignment through urgencies are used to assign the demand points to the p selected medians. The behavior and efficiency of the assignment scenarios are examined and compared on CPMP. According to the results, the classical scenario shows superiority in time consuming, whereas the assignment through urgencies scenario is absolutely superior in quality of the obtained solutions over the classical one. In order to check for quality and validity of the suggestive method, we compare the final solution produced over the 10 test problems of Osman and Christofides[16]. Comparison of the results indicates good quality and solutions.

Partitioning graphs of supply and demand

Assume that each vertex of a graph G is either a supply vertex or a demand vertex and is assigned a positive integer, called a supply or a demand. Each demand vertex can receive “power” from at most one supply vertex through edges in G . One thus wishes to partition G into connected components by deleting edges from G so that each component C has exactly one supply vertex whose supply is no less than the sum of demands of all demand vertices in C . If G does not have such a partition, one wishes to partition G into connected components so that each component C either has no supply vertex or has exactly one supply vertex whose supply is no less than the sum of demands in C , and wishes to maximize the sum of demands in all components with supply vertices. We deal with such a maximization problem, which is NP-hard even for trees and strongly NP-hard for general graphs. In this paper, we show that the problem can be solved in pseudo-polynomial time for series–parallel graphs and partial k -trees–that is, graphs with bounded tree-width.

Approximability of partitioning graphs with supply and demand

Suppose that each vertex of a graph G is either a supply vertex or a demand vertex and is assigned a positive real number, called the supply or the demand. Each demand vertex can receive “power” from at most one supply vertex through edges in G. One thus wishes to partition G into connected components by deleting edges from G so that each component C either has no supply vertex or has exactly one supply vertex whose supply is at least the sum of demands in C, and wishes to maximize the fulfillment, that is, the sum of demands in all components with supply vertices. This maximization problem is known to be NP-hard even for trees having exactly one supply vertex and strongly NP-hard for general graphs. In this paper, we focus on the approximability of the problem. We first show that the problem is MAXSNP-hard and hence there is no polynomial-time approximation scheme (PTAS) for general graphs unless P = NP . We then present a fully polynomial-time approximation scheme (FPTAS) for series-parallel graphs having exactly one supply vertex.

Experimental Evaluation of Maximum-Supply Partitioning Algorithms for Demand-Supply Graphs

Let G = (D ∪ S,E) be an undirected graph with a vertex set D ∪ S and an (undirected) edge set E, where the vertex set is partitioned into two subsets, a demand vertex set D and a supply vertex set S. We assume that D ≠ o and S ≠ o in this paper. Each demand or supply vertex v has a positive real number d(v) or s(v), called the demand or supply of v, respectively. For any subset V' ⊆ D ∪ S, the demand of V' is defined by d(V') = ΣveV'∩ Dd(v) if V'∩ D≠o or d(V') = 0 if V'∩ D =o. Let s(S) = ΣveV'S(v). Any partition π ={V1, ..., Vr} (|S| ≤ r ≤ |D ∪ S|) of D ∪ S is called a feasible partition of G if and only if π satisfies the following (1) and (2) for any k = 1, ..., r: (1) |Vk∩ S|≤q 1, and (2) if Vk ∩ S = {uk} then the induced subgraph G[Vk] of G is connected and d(Vk)≤qs(uk). The demand d(π) of π is defined by d(π)=ΣVk ∩ S ≠ od(Vk). The Maximum-Supply Partitioning Problem (MSPP) is to find a feasible partition π of G such that d(π) is maximum among all feasible partitions of G. We implemented not only existing algorithms for obtainity heuristic or optimum solutions to MSPP but also those that are corrected or improved from existing ones. In this paper we show comparison of their capability based on computational experiments.

PARTITIONING TREES OF SUPPLY AND DEMAND

Assume that a tree T has a number ns of "supply vertices" and all the other vertices are "demand vertices." Each supply vertex is assigned a positive number called a supply, while each demand vertex is assigned a positive number called a demand. One wishes to partition T into exactly ns subtrees by deleting edges from T so that each subtree contains exactly one supply vertex whose supply is no less than the sum of demands of all demand vertices in the subtree. The "partition problem" is a decision problem to ask whether T has such a partition. The "maximum partition problem" is an optimization version of the partition problem. In this paper, we give three algorithms for the problems. The first is a linear-time algorithm for the partition problem. The second is a pseudo-polynomial-time algorithm for the maximum partition problem. The third is a fully polynomial-time approximation scheme (FPTAS) for the maximum partition problem.

Exact algorithms for the minimum latency problem

Let G = (V, E,w) be an undirected graph with positive weight w(e) on each edge e ∈ E. Given a starting vertex s ∈ V and a subset U ⊂ V as the demand vertex set, the minimum latency problem (MLP) asks for a tour P starting at s and visiting each demand vertex at least once such that the total latency of all demand vertices is minimized, in which the latency of a vertex is the length of the path from s to the first visit of the vertex. The MLP is an important problem in computer science and operations research, and is also known as the delivery man problem or the traveling repairman problem. Similar to the well-known traveling salesperson problem (TSP), in the MLP we are asked to find an “optimal” way for routing a server passing through the demand vertices. The difference is the objective functions. The latency of a vertex can be thought of as the delay of the service. In the MLP we care about the total delay (service quality), while the total length (service cost) is concerned in the TSP. The MLP on a metric space is NP-hard and also MAX-SNP-hard [4]. Polynomial time algorithms are only known for very special graphs, such as paths [1, 6], edge-unweighted trees [9], trees of diameter 3 [4], trees of constant number of leaves [8], or graphs with similar structure [12]. Even for caterpillars (paths with edges sticking out), no polynomial time algorithm has been reported. In a recent work, it is shown that the MLP on edge-weighted trees is NP-hard [11]. Due to the NP-hardness, many works ∗corresponding author (bangye@mail.stu.edu.tw)

Local Search Heuristics for Capacitated p -Median Problems

The search for p-median vertices on a network (graph) is a classical location problem. The p facilities (medians) must be located so as to minimize the sum of the distances from each demand vertex to its nearest facility. The Capacitated p-Median Problem (CPMP) considers capacities for the service to be given by each median. The total service demanded by vertices identified by p-median clusters cannot exceed their service capacity. Primal-dual based heuristics are very competitive and provide simultaneously upper and lower bounds to optimal solutions. The Lagrangean/surrogate relaxation has been used recently to accelerate subgradient like methods. The dual lower bound have the same quality of the usual Lagrangean relaxation dual but is obtained using modest computational times. This paper explores improvements on upper bounds applying local search heuristics to solutions made feasible by the Lagrangean/surrogate optimization process. These heuristics are based on location-allocation procedures that swap medians and vertices inside the clusters, reallocate vertices, and iterate until no improvements occur. Computational results consider instances from the literature and real data obtained using a geographical information system.

Abordagens complementares para problemas de p-medianas

A localização de p-medianas é um problema clássico de otimização combinatória. O objetivo é localizar em uma rede p nós (denominados medianas), de forma a minimizar a soma das distâncias de cada nó de demanda até sua mediana mais próxima. Neste trabalho aborda-se a relaxação lagrangeana/surrogate como técnica para resolver tais problemas. Discute-se a utilização desta relaxação em combinação com métodos de otimização por subgradientes e com métodos de geração de colunas. O trabalho apresenta testes computacionais que demonstram a eficiência dos algoritmos propostos, considerando problemas obtidos da literatura e problemas reais obtidos a partir de Sistemas de Informações Geográficas.

Diversification strategies in local search for a nonbifurcated network loading problem

This paper examines a variant of the network loading problem, a network design problem found in the telecommunications industry. In this problem, facilities of fixed capacity must be installed on the edges of an undirected network to carry the flow from a central vertex to a set of demand vertices. The objective is to minimize the total installation costs. In this work, the nonbifurcated version of the problem is considered, where the demand at any given vertex must be satisfied through a single path. The proposed heuristics alternate between a construction phase and a local search phase. Each new construction phase, except the first one, is part of a diversification strategy aimed at providing a new starting point for the following local search phase. Different diversification strategies are tested and compared on large-scale instances with up to 500 vertices.

Tabu Search for a Network Loading Problem with Multiple Facilities

This paper examines a network design problem that arises in the telecommunications industry. In this problem, communication between a gateway vertex and a number of demand vertices is achieved through a network of fiber optic cables. Since each cable has an associated capacity (bandwidth), enough capacity must be installed on the links of the network to satisfy the demand, using possibly different types of cables. Starting with a network with no capacity or some capacity already installed, a tabu search heuristic is designed to find a solution that minimizes the cost of installing any additional capacity on the network. This tabu search applies a k-shortest path algorithm to find alternative paths from the gateway to the demand vertices. Numerical results are presented on different types of networks with up to 200 vertices and 100 demand vertices.

Parallel Algorithms for Zero-One Supply-Demand Problems

A technique that yields fast parallel algorithms for several zero-one supply-demand problems is presented. $NC$ algorithms are given for the following related problems: (1) Given a sequence of supplies $a_1 , \cdots ,a_n $ and demands $b_1 , \cdots ,b_m $, construct a zero-one flow pattern satisfying these constraints, where every supply vertex can send at most one unit of flow to each demand vertex. (2) Given a sequence of positive and negative integers summing to zero, representing supplies and demands, respectively, construct a zero-one flow pattern so that the net flow out of (into) each vertex is its supply (demand), where every vertex can send at most one unit of flow to every other vertex. (3) Construct a digraph without self-loops with specified in- and out-degrees. The results are extended to the case where the input represents upper bounds on supplies and lower bounds on demands.

ポリマトロイド対の普遍基の特徴付け

This note gives a characterization of the universal pair of bases of a pair of polymatroids as the nearest pair of bases with respect to a class of pseudo-distances including the Kullback-Leibler divergence. N. Megiddo considered the lexico-optimal flow problem in a multiterminal network N = (V, A, c; S^+, S^-) (V: vertex set, A: arc set, c: capacity, S^+: supply (source) vertices, S^-: demand (sink) vertices), which is to find a maximal flow such that the supply flow (s^+ (v) | v ∈ S^+)[resp., the demand flow (s^- (v) | v ∈ S^-)] is as proportional as possible to a given weight vector. This problem is treated by S. Fujishige as a special case of the lexico-optimal base problem for a single polymatroid. This paper considers the problem of finding a maximal flow such that the supply flow s^+ and the demand flow s^- are as "near" as possible (where a one-to-one correspondence between S^+ and S^- is assumed to be given), and generalizes it to the problem of finding a "nearest" pair of bases of a pair of polymatroids It is shown that the "nearest" pair coincides with the universal pair if either of the following criteria is adopted. (1) The f-divergence (a generalization of the Kullback-Leibler divergence) between the bases should be minimized; (2) The vector consisting of the ratios of the corresponding components of the bases should be lexicographically maximized.