Costa et al. (Oper. Res. Lett. 31:21–27, 2003) presented a quadratic O(min (Kn,n 2)) greedy algor... more Costa et al. (Oper. Res. Lett. 31:21–27, 2003) presented a quadratic O(min (Kn,n 2)) greedy algorithm to solve the integer multicut and multiflow problems in a rooted tree. (n is the number of nodes of the tree, and K is the number of commodities). Their algorithm is a special case of the greedy type algorithm of Kolen (Location problems on trees and in the rectilinear plane. Ph.D. dissertation, 1982) to solve weighted covering and packing problems defined by general totally balanced (greedy) matrices. In this communication we improve the complexity bound in Costa et al. (Oper. Res. Lett. 31:21–27, 2003) and show that in the case of the integer multicut and multiflow problems in a rooted tree the greedy algorithm of Kolen can be implemented in subquadratic O(K+n+min (K,n)log n) time. The improvement is obtained by identifying additional properties of this model which lead to a subquadratic transformation to greedy form and using more sophisticated data structures.
We consider the optimization problem of locating several new facilities on a tree network, with r... more We consider the optimization problem of locating several new facilities on a tree network, with respect to existing facilities, and to each other. The new facilities are not restricted to be at vertices of the network, but the locations are subject to constraints. Each constraint function, and the objective function, is an arbitrary, nondecreasing function of any finite collection of tree distances between new and existing facilities, and/or between distinct pairs of new facilities, and represents some sort of transport or travel cost. The new facilities are to be located so as to minimize the objective function subject to upper bounds on the constraint functions. We show that such problems are equivalent to mathematical programming problems which, when each function is expressed using only maximization and summation operations on nonnegatively weighted arguments, are linear programming problems of polynomial dimensions. The latter problems can be solved using duality theory with sp...
This paper presents a novel fiber-tracking algorithm, termed combinatorial tracking, which uses s... more This paper presents a novel fiber-tracking algorithm, termed combinatorial tracking, which uses stochastic process modeling and global optimization algorithm for tractography. Combinatorial tracking is a probabilistic tracking algorithm that transforms the brain's white matter into a grid in which each voxel has 26 weighted connections with adjacent voxels. We model the random walk on this graph using a Markov Chain model and suggest two approaches for fiber reconstruction. In the first approach, we find the most probable paths between two voxels with prior connectivity knowledge using a shortest path algorithm. In the second approach, the all-pairs mean first passage time (MFPT) matrix M (or hitting time as referred to in the Spectral Graph theory literature) is calculated analytically. We suggest that M can be interpreted as a global connectivity matrix and use it for fiber reconstruction. We also introduce a simulation framework that can be used to calculate specific elements...
Consider a tree T = (N, E) with “supply” and “demand” regions Σ and Δ, each composed of a finite ... more Consider a tree T = (N, E) with “supply” and “demand” regions Σ and Δ, each composed of a finite number of disjoint, closed and connected subregions of T, some of which may possibly consist of just one point. Given an integer p, we seek a collection of p “centers” x1, …, xp ∈ Σ, which minimize the expression maxy∈δ mini=1,…,pd(y, xi). We present a polynomial algorithm for this problem. Its running time is bounded by O(n log2n) if either Δ or Σ is discrete, and by O(n min|p log2n,n log p|) if both sets contain at least one full edge.
ABSTRACT In many services (e.g., delivery, or customer pickup vehicles) the service unit usually ... more ABSTRACT In many services (e.g., delivery, or customer pickup vehicles) the service unit usually visits a number of demand points on a single multistop tour. Typically, at a specific time of the day, the unit receives the list of waiting calls and immediately starts a tour of the network that includes all waiting customers. The multistop location problem is to find the home location for the service unit. We focus on the minimax criterion for the multistop problem defined on a tree network. Each potential list of customers is associated with the length of its respective tour and with some weight. We seek for the home location of the unit that minimizes the maximum weighted tour length over all feasible customer lists. We consider several weight functions and obtain results that reveal additional properties of the classical absolute center of the tree.
Costa et al. (Oper. Res. Lett. 31:21–27, 2003) presented a quadratic O(min (Kn,n 2)) greedy algor... more Costa et al. (Oper. Res. Lett. 31:21–27, 2003) presented a quadratic O(min (Kn,n 2)) greedy algorithm to solve the integer multicut and multiflow problems in a rooted tree. (n is the number of nodes of the tree, and K is the number of commodities). Their algorithm is a special case of the greedy type algorithm of Kolen (Location problems on trees and in the rectilinear plane. Ph.D. dissertation, 1982) to solve weighted covering and packing problems defined by general totally balanced (greedy) matrices. In this communication we improve the complexity bound in Costa et al. (Oper. Res. Lett. 31:21–27, 2003) and show that in the case of the integer multicut and multiflow problems in a rooted tree the greedy algorithm of Kolen can be implemented in subquadratic O(K+n+min (K,n)log n) time. The improvement is obtained by identifying additional properties of this model which lead to a subquadratic transformation to greedy form and using more sophisticated data structures.
We consider the optimization problem of locating several new facilities on a tree network, with r... more We consider the optimization problem of locating several new facilities on a tree network, with respect to existing facilities, and to each other. The new facilities are not restricted to be at vertices of the network, but the locations are subject to constraints. Each constraint function, and the objective function, is an arbitrary, nondecreasing function of any finite collection of tree distances between new and existing facilities, and/or between distinct pairs of new facilities, and represents some sort of transport or travel cost. The new facilities are to be located so as to minimize the objective function subject to upper bounds on the constraint functions. We show that such problems are equivalent to mathematical programming problems which, when each function is expressed using only maximization and summation operations on nonnegatively weighted arguments, are linear programming problems of polynomial dimensions. The latter problems can be solved using duality theory with sp...
This paper presents a novel fiber-tracking algorithm, termed combinatorial tracking, which uses s... more This paper presents a novel fiber-tracking algorithm, termed combinatorial tracking, which uses stochastic process modeling and global optimization algorithm for tractography. Combinatorial tracking is a probabilistic tracking algorithm that transforms the brain's white matter into a grid in which each voxel has 26 weighted connections with adjacent voxels. We model the random walk on this graph using a Markov Chain model and suggest two approaches for fiber reconstruction. In the first approach, we find the most probable paths between two voxels with prior connectivity knowledge using a shortest path algorithm. In the second approach, the all-pairs mean first passage time (MFPT) matrix M (or hitting time as referred to in the Spectral Graph theory literature) is calculated analytically. We suggest that M can be interpreted as a global connectivity matrix and use it for fiber reconstruction. We also introduce a simulation framework that can be used to calculate specific elements...
Consider a tree T = (N, E) with “supply” and “demand” regions Σ and Δ, each composed of a finite ... more Consider a tree T = (N, E) with “supply” and “demand” regions Σ and Δ, each composed of a finite number of disjoint, closed and connected subregions of T, some of which may possibly consist of just one point. Given an integer p, we seek a collection of p “centers” x1, …, xp ∈ Σ, which minimize the expression maxy∈δ mini=1,…,pd(y, xi). We present a polynomial algorithm for this problem. Its running time is bounded by O(n log2n) if either Δ or Σ is discrete, and by O(n min|p log2n,n log p|) if both sets contain at least one full edge.
ABSTRACT In many services (e.g., delivery, or customer pickup vehicles) the service unit usually ... more ABSTRACT In many services (e.g., delivery, or customer pickup vehicles) the service unit usually visits a number of demand points on a single multistop tour. Typically, at a specific time of the day, the unit receives the list of waiting calls and immediately starts a tour of the network that includes all waiting customers. The multistop location problem is to find the home location for the service unit. We focus on the minimax criterion for the multistop problem defined on a tree network. Each potential list of customers is associated with the length of its respective tour and with some weight. We seek for the home location of the unit that minimizes the maximum weighted tour length over all feasible customer lists. We consider several weight functions and obtain results that reveal additional properties of the classical absolute center of the tree.
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