The paper studies a robust mixed integer program with a single unrestricted continuous variable. ... more The paper studies a robust mixed integer program with a single unrestricted continuous variable. The purpose of the paper is the polyhedral study of the robust solution set using submodularity. A submodular function is a set function with a diminishing returns property, and little work has been studied on the utilization of submodularity in the study of optimization problems considering data uncertainty. In this paper, we propose valid inequalities using submodularity. Valid inequalities for the robust mixed integer program are defined. A polynomial separation algorithm is proposed, and we show that the convex hull of the problem can be completely described using the proposed inequalities. In computational tests, we showed the proposed cuts are effective when they are applied to general robust discrete optimization problems with one or multiple constraints.
We consider chance-constrained binary knapsack problems, where the weights of items are independe... more We consider chance-constrained binary knapsack problems, where the weights of items are independent random variables with the means and standard deviations known. The chance constraint can be reformulated as a second-order cone constraint under some assumptions for the probability distribution of the weights. The problem becomes a second-order cone-constrained binary knapsack problem, which is equivalent to a robust binary knapsack problem with an ellipsoidal uncertainty set. We demonstrate that optimal solutions to robust binary knapsack problems with inner and outer polyhedral approximations of the ellipsoidal uncertainty set can provide both upper and lower bounds on the optimal value of the second-order cone-constrained binary knapsack problem, and they can be obtained by solving ordinary binary knapsack problems repeatedly. Moreover, we prove that the solution providing the upper bound converges to the optimal solution to the secondorder cone-constrained binary knapsack problem...
Industrial Engineering and Management Systems, 2011
One of the critical issues in wireless sensor network is the design of a proper routing protocol.... more One of the critical issues in wireless sensor network is the design of a proper routing protocol. One possible approach is utilizing a virtual infrastructure, which is a subset of sensors to connect all the sensors in the network. Among the many virtual infrastructures, the connected dominating set is widely used. Since a small connected dominating set can help to decrease the protocol overhead and energy consumption, it is preferable to find a small sized connected dominating set. Although many algorithms have been suggested to construct a minimum connected dominating set, there have been few exact approaches. In this paper, we suggest an improved optimal algorithm for the minimum connected dominating set problem, and extensive computational results showed that our algorithm outperformed the previous exact algorithms. Also, we suggest a new heuristic algorithm to find the connected dominating set and computational results show that our algorithm is capable of finding good quality solutions quite fast.
We consider a chance-constrained binary knapsack problem where weights of items are independent a... more We consider a chance-constrained binary knapsack problem where weights of items are independent and normally distributed. Probabilistic cover inequalities can be defined for the problem. The lifting problem for probabilistic cover inequalities is NP-hard. We propose a polynomial time approximate lifting method for probabilistic cover inequalities based on the robust optimization approach. We present computational experiments on multidimensional chance-constrained knapsack problems. The results show that our lifting method reduces the computation time substantially.
We consider two types of path selection problems for arc capacitated network. Given an arc capaci... more We consider two types of path selection problems for arc capacitated network. Given an arc capacitated network and a set of commodities, one problem is to find a subset of commodities to be routed and an assignment of them to the paths so that profit is maximized. The other problem is to route all given commodities in the network so that cost is minimized. Bifurcation of flow is not allowed in both cases. We formulate the problems as integer programming models and solve them. Column generation technique to solve the linear programming relaxation is proposed with two types of columns. To obtain an optimum integer solution for the problems, we propose a branching strategy in the branch-and-price scheme. Computational exper-iments show that the algorithm gives optimal solutions within reasonably small computing time.
We consider a vehicle routing problem with uncertain travel times in which a penalty is incurred ... more We consider a vehicle routing problem with uncertain travel times in which a penalty is incurred for each vehicle that exceeds a given time limit. A traditional stochastic programming approach would require precise knowledge of the underlying probability distributions of random data. In a novel approach presented here, we assume that only rough information on future travel times is available, leading to the multiple range forecasts of travel times and the probabilities of each range being realized. In this setting, we replace the point estimates of travel times on a scenario by range estimates. For each scenario, we then find the robust routes that protect the solution against the worst case within the given ranges, and finally we find the routes with the minimum expected cost. We propose a branch-and-cut algorithm to solve the problem and report computational results on both randomly generated and the well-known Solomon's instances. The results demonstrate that our approach is ...
This paper considers the problem of designing a capacitated network with a tree configuration (CT... more This paper considers the problem of designing a capacitated network with a tree configuration (CTP). For a given set of nodes with their capacities, k types of link facilities with various characteristics, and installation cost for connecting each pair of nodes using each type of link facility, the problem is to find a tree network which satisfies the given traffic requirements between all pairs of nodes and minimizes total installation cost. We formulate (CTP) as an integer programming problem using path variables. To solve the linear programming relaxation which has exponentially many variables, we develop a polynomial-time column generation procedure. Moreover, to tighten the formulation, an efficient preprocessing procedure is devised and some classes of valid inequalities are found. Using the results, we develop a branch-and-cut algorithm with column generation where an efficient branching rule is adopted. Computational results show that the algorithm can solve practically-sized problems to optimality within a reasonable time.
We formulate the edge coloring problem on a simple graph as the integer program of covering edges... more We formulate the edge coloring problem on a simple graph as the integer program of covering edges by matchings. For the NP-hard case of 3-regular graphs we show that it is sufficient to solve the linear programming relaxation with the additional constraints that each odd circuit be covered by at least three matchings. We give an efficient separation algorithm for recognizing violated odd circuit constraints and a linear programming based constrained weighted matching algorithm for pricing. Computational experiments with the overall linear programming system are presented. edge coloring, integer programming
The ring loading problem with integer demand splitting is that of routing κ traffic requirements ... more The ring loading problem with integer demand splitting is that of routing κ traffic requirements on an undirected ring network. We present a compact polyhedral description of the set of feasible solutions to the problem, whose number of variables and constraints is O(κ).
We consider a network design problem in which flow bifurcations are allowed. The demand data are ... more We consider a network design problem in which flow bifurcations are allowed. The demand data are assumed to be uncertain, and the uncertainties of demands are expressed by an uncertainty set. The goal is to install facilities on the edges at minimum cost. The solution should be able to deliver any of the demand requirements defined in the uncertainty set. We propose an exact solution algorithm based on a decomposition approach in which the problem is decomposed into two distinct problems: (1) designing edge capacities; and (2) checking the feasibility of the designed edge capacities with respect to the uncertain demand requirements. The algorithm is a special case of the Benders decomposition method. We show that the robust version of the Benders subproblem can be formulated as a linear program whose size is polynomially bounded. We also propose a simultaneous cut generation scheme to accelerate convergence of the Benders decomposition algorithm. Computational results on real-life telecommunication problems are reported, and these demonstrate that robust solutions with very small penalties in the objective values can be obtained.
In this article, we investigate the vehicle routing problem with deadlines, whose goal is to sati... more In this article, we investigate the vehicle routing problem with deadlines, whose goal is to satisfy the requirements of a given number of customers with minimum travel distances while respecting both of the deadlines of the customers and vehicle capacity. It is assumed that the travel time between any two customers and the demands of the customer are uncertain. Two types of uncertainty sets with adjustable parameters are considered for the possible realizations of travel time and demand. The robustness of a solution against the uncertain data can be achieved by making the solution feasible for any travel time and demand defined in the uncertainty sets. We propose a Dantzig-Wolfe decomposition approach, which enables the uncertainty of the data to be encapsulated in the column generation subproblem. A dynamic programming algorithm is proposed to solve the subproblem with data uncertainty. The results of computational experiments involving two well-known test problems show that the robustness of the solution can be greatly improved.
The bandwidth packing problem (BWP) concerns the selection of calls from a given set and the assi... more The bandwidth packing problem (BWP) concerns the selection of calls from a given set and the assignment of one path to each selected call. The ultimate aim of the BWP is to maximize profit while the routings of the selected calls observe the capacity constraints of the links. Here, we additionally consider queueing delays in the network, which may cause a deterioration in the quality of service to users if they exceed the acceptable limits. The integer programming formulation for the BWP with the queueing delay restriction contains a nonlinear constraint that is intrinsic to the model. We apply the Dantzig-Wolfe decomposition to this nonlinear constraint, and since the Dantzig-Wolfe decomposition has exponentially many variables, we propose the branch-and-price procedure to find optimal solutions. We also propose a generalized Dantzig-Wolfe reformulation based on the aggregation of variables, which makes our branch-and-price algorithm more competitive. Computational results on cases...
IEEE Journal on Selected Areas in Communications, 2000
We consider the routing and wavelength assignment (RWA) problem on WDM ring networks without wave... more We consider the routing and wavelength assignment (RWA) problem on WDM ring networks without wavelength conversion. When the physical network and required connections are given, RWA is the problem to select a suitable path and wavelength among the many possible choices for each connection such that no two paths using the same wavelength pass through the same link. We give an integer programming formulation of the problem and propose an algorithm to solve it. Although the formulation has exponentially many variables, we solve the linear programming relaxation of it by using the column generation technique. We solve the column generation problem efficiently by decomposing the problem into several subproblems. After solving the linear programming relaxation, we apply the branch-and-price procedure to get an optimal solution. We test the proposed algorithm on some randomly generated data. Test results show that the algorithm gives optimal solutions to almost all instances under the given node limit of the branch-and-bound tree.
We consider the multicast routing and wavelength assignment (MC-RWA) problem on WDM bidirectional... more We consider the multicast routing and wavelength assignment (MC-RWA) problem on WDM bidirectional ring networks without wavelength conversion. We give an integer programming formulation of the problem and propose an algorithm to solve it optimally. The algorithm is based on column generation and branch-and-price. We test the proposed algorithm on randomly generated data and the test results show that the algorithm gives optimal solutions to all of the test problems.
We consider the problem of designing a local network in a two‐level telecommunication network. Gi... more We consider the problem of designing a local network in a two‐level telecommunication network. Given one or two hub nodes, central offices (COs) and conduits, the problem is to find a set of unidirectional self‐healing rings (USHRs) which covers all COs and satisfies all demands at minimum cost. The solution approach used is the decomposition and column generation. Master problem
We consider an extended formulation approach to the edge-weighted maximal clique problem. The pro... more We consider an extended formulation approach to the edge-weighted maximal clique problem. The problem is formulated by using additional variables for the set of nodes with the natural variables for the set of edges. We show that the proposed formulation is superior to the natural formulation both theoretically and practically. By using the projection technique, we can also derive new classes of facet-defining inequalities for the lower-dimensional polytope of the natural variables. Computational results are reported.
We consider the problem of designing an ATM VP-based leased line backbone network. Given point-to... more We consider the problem of designing an ATM VP-based leased line backbone network. Given point-to-point communication demands having predefined sizes in a network, the problem is to find configurations of demand routes and link facilities installed on each edge satisfying all demands at minimum cost under some constraints. One of the most important constraints is that a single demand cannot be split over multiple link facilities. This is a sort of bin packing constraint. We propose an integer programming formulation of the problem and an algorithm to solve it. An efficient column generation technique to solve the linear programming relaxation is proposed, and a valid inequality is used to strengthen the integer programming formulation. The algorithm incorporates the column generation technique and the cutting plane approach into a branch-and-bound scheme. We test the proposed algorithm on some real problems. The results show that the algorithm can be used to solve the problems within reasonably small computing times.
This paper considers the hop-constrained multicast route packing problem with a bandwidth reserva... more This paper considers the hop-constrained multicast route packing problem with a bandwidth reservation to build QoS-guaranteed multicast routing trees with a minimum installation cost. Given a set of multicast sessions, each of which has a hop limit constraint and a bandwidth requirement, the problem is to determine the set of multicast routing trees in an arc-capacitated network with the objective of minimizing the cost. For the problem, we propose a branch-and-cut-and-price algorithm, which can be viewed as a branch-and-bound method incorporating both the strong cutting plane algorithm and the column generation method. We implemented and tested the proposed algorithm on randomly generated problem instances with sizes up to 30 nodes, 570 arcs, and 10 multicast sessions. The test results show that the algorithm can obtain the optimal solution to practically sized problem instances within a reasonable time limit in most cases.
This paper considers the discrete two-hub location problem. We need to choose two hubs from a set... more This paper considers the discrete two-hub location problem. We need to choose two hubs from a set of nodes. The remaining nodes are to be connected to one of the two hubs which act as switching points for intemodal flows. A configuration which minimizes the total flow cost needs to be found. We show that the problem can be solved in polynomial dme when the hub locations are fixed. Since there are at most ½n(n-1) ways to choose the hub locations, the two-hub location problem can be solved in polynomial time. We transform the quadratic 0-1 integer program of the single allocation problem in the fixed two-hub system into a linear program and show that all extreme points of the polytope defined by the LP are integral. Also, the problem can be transformed into a minimum cut problem which can be solved efficiently by any polynomial time algorithm.
Classical column generation often shows desperately slow convergence. Recently, many acceleration... more Classical column generation often shows desperately slow convergence. Recently, many acceleration techniques are proposed. We propose Chebyshev center based column generation. In this method, the Chebyshev center is used for centering dual solutions within dual polyhedron. The Chebyshev center can be obtained by solving a linear program, so that our method can be applied with small modification of the classical column generation scheme. Numerical experiments show the effectiveness of our method.
The paper studies a robust mixed integer program with a single unrestricted continuous variable. ... more The paper studies a robust mixed integer program with a single unrestricted continuous variable. The purpose of the paper is the polyhedral study of the robust solution set using submodularity. A submodular function is a set function with a diminishing returns property, and little work has been studied on the utilization of submodularity in the study of optimization problems considering data uncertainty. In this paper, we propose valid inequalities using submodularity. Valid inequalities for the robust mixed integer program are defined. A polynomial separation algorithm is proposed, and we show that the convex hull of the problem can be completely described using the proposed inequalities. In computational tests, we showed the proposed cuts are effective when they are applied to general robust discrete optimization problems with one or multiple constraints.
We consider chance-constrained binary knapsack problems, where the weights of items are independe... more We consider chance-constrained binary knapsack problems, where the weights of items are independent random variables with the means and standard deviations known. The chance constraint can be reformulated as a second-order cone constraint under some assumptions for the probability distribution of the weights. The problem becomes a second-order cone-constrained binary knapsack problem, which is equivalent to a robust binary knapsack problem with an ellipsoidal uncertainty set. We demonstrate that optimal solutions to robust binary knapsack problems with inner and outer polyhedral approximations of the ellipsoidal uncertainty set can provide both upper and lower bounds on the optimal value of the second-order cone-constrained binary knapsack problem, and they can be obtained by solving ordinary binary knapsack problems repeatedly. Moreover, we prove that the solution providing the upper bound converges to the optimal solution to the secondorder cone-constrained binary knapsack problem...
Industrial Engineering and Management Systems, 2011
One of the critical issues in wireless sensor network is the design of a proper routing protocol.... more One of the critical issues in wireless sensor network is the design of a proper routing protocol. One possible approach is utilizing a virtual infrastructure, which is a subset of sensors to connect all the sensors in the network. Among the many virtual infrastructures, the connected dominating set is widely used. Since a small connected dominating set can help to decrease the protocol overhead and energy consumption, it is preferable to find a small sized connected dominating set. Although many algorithms have been suggested to construct a minimum connected dominating set, there have been few exact approaches. In this paper, we suggest an improved optimal algorithm for the minimum connected dominating set problem, and extensive computational results showed that our algorithm outperformed the previous exact algorithms. Also, we suggest a new heuristic algorithm to find the connected dominating set and computational results show that our algorithm is capable of finding good quality solutions quite fast.
We consider a chance-constrained binary knapsack problem where weights of items are independent a... more We consider a chance-constrained binary knapsack problem where weights of items are independent and normally distributed. Probabilistic cover inequalities can be defined for the problem. The lifting problem for probabilistic cover inequalities is NP-hard. We propose a polynomial time approximate lifting method for probabilistic cover inequalities based on the robust optimization approach. We present computational experiments on multidimensional chance-constrained knapsack problems. The results show that our lifting method reduces the computation time substantially.
We consider two types of path selection problems for arc capacitated network. Given an arc capaci... more We consider two types of path selection problems for arc capacitated network. Given an arc capacitated network and a set of commodities, one problem is to find a subset of commodities to be routed and an assignment of them to the paths so that profit is maximized. The other problem is to route all given commodities in the network so that cost is minimized. Bifurcation of flow is not allowed in both cases. We formulate the problems as integer programming models and solve them. Column generation technique to solve the linear programming relaxation is proposed with two types of columns. To obtain an optimum integer solution for the problems, we propose a branching strategy in the branch-and-price scheme. Computational exper-iments show that the algorithm gives optimal solutions within reasonably small computing time.
We consider a vehicle routing problem with uncertain travel times in which a penalty is incurred ... more We consider a vehicle routing problem with uncertain travel times in which a penalty is incurred for each vehicle that exceeds a given time limit. A traditional stochastic programming approach would require precise knowledge of the underlying probability distributions of random data. In a novel approach presented here, we assume that only rough information on future travel times is available, leading to the multiple range forecasts of travel times and the probabilities of each range being realized. In this setting, we replace the point estimates of travel times on a scenario by range estimates. For each scenario, we then find the robust routes that protect the solution against the worst case within the given ranges, and finally we find the routes with the minimum expected cost. We propose a branch-and-cut algorithm to solve the problem and report computational results on both randomly generated and the well-known Solomon's instances. The results demonstrate that our approach is ...
This paper considers the problem of designing a capacitated network with a tree configuration (CT... more This paper considers the problem of designing a capacitated network with a tree configuration (CTP). For a given set of nodes with their capacities, k types of link facilities with various characteristics, and installation cost for connecting each pair of nodes using each type of link facility, the problem is to find a tree network which satisfies the given traffic requirements between all pairs of nodes and minimizes total installation cost. We formulate (CTP) as an integer programming problem using path variables. To solve the linear programming relaxation which has exponentially many variables, we develop a polynomial-time column generation procedure. Moreover, to tighten the formulation, an efficient preprocessing procedure is devised and some classes of valid inequalities are found. Using the results, we develop a branch-and-cut algorithm with column generation where an efficient branching rule is adopted. Computational results show that the algorithm can solve practically-sized problems to optimality within a reasonable time.
We formulate the edge coloring problem on a simple graph as the integer program of covering edges... more We formulate the edge coloring problem on a simple graph as the integer program of covering edges by matchings. For the NP-hard case of 3-regular graphs we show that it is sufficient to solve the linear programming relaxation with the additional constraints that each odd circuit be covered by at least three matchings. We give an efficient separation algorithm for recognizing violated odd circuit constraints and a linear programming based constrained weighted matching algorithm for pricing. Computational experiments with the overall linear programming system are presented. edge coloring, integer programming
The ring loading problem with integer demand splitting is that of routing κ traffic requirements ... more The ring loading problem with integer demand splitting is that of routing κ traffic requirements on an undirected ring network. We present a compact polyhedral description of the set of feasible solutions to the problem, whose number of variables and constraints is O(κ).
We consider a network design problem in which flow bifurcations are allowed. The demand data are ... more We consider a network design problem in which flow bifurcations are allowed. The demand data are assumed to be uncertain, and the uncertainties of demands are expressed by an uncertainty set. The goal is to install facilities on the edges at minimum cost. The solution should be able to deliver any of the demand requirements defined in the uncertainty set. We propose an exact solution algorithm based on a decomposition approach in which the problem is decomposed into two distinct problems: (1) designing edge capacities; and (2) checking the feasibility of the designed edge capacities with respect to the uncertain demand requirements. The algorithm is a special case of the Benders decomposition method. We show that the robust version of the Benders subproblem can be formulated as a linear program whose size is polynomially bounded. We also propose a simultaneous cut generation scheme to accelerate convergence of the Benders decomposition algorithm. Computational results on real-life telecommunication problems are reported, and these demonstrate that robust solutions with very small penalties in the objective values can be obtained.
In this article, we investigate the vehicle routing problem with deadlines, whose goal is to sati... more In this article, we investigate the vehicle routing problem with deadlines, whose goal is to satisfy the requirements of a given number of customers with minimum travel distances while respecting both of the deadlines of the customers and vehicle capacity. It is assumed that the travel time between any two customers and the demands of the customer are uncertain. Two types of uncertainty sets with adjustable parameters are considered for the possible realizations of travel time and demand. The robustness of a solution against the uncertain data can be achieved by making the solution feasible for any travel time and demand defined in the uncertainty sets. We propose a Dantzig-Wolfe decomposition approach, which enables the uncertainty of the data to be encapsulated in the column generation subproblem. A dynamic programming algorithm is proposed to solve the subproblem with data uncertainty. The results of computational experiments involving two well-known test problems show that the robustness of the solution can be greatly improved.
The bandwidth packing problem (BWP) concerns the selection of calls from a given set and the assi... more The bandwidth packing problem (BWP) concerns the selection of calls from a given set and the assignment of one path to each selected call. The ultimate aim of the BWP is to maximize profit while the routings of the selected calls observe the capacity constraints of the links. Here, we additionally consider queueing delays in the network, which may cause a deterioration in the quality of service to users if they exceed the acceptable limits. The integer programming formulation for the BWP with the queueing delay restriction contains a nonlinear constraint that is intrinsic to the model. We apply the Dantzig-Wolfe decomposition to this nonlinear constraint, and since the Dantzig-Wolfe decomposition has exponentially many variables, we propose the branch-and-price procedure to find optimal solutions. We also propose a generalized Dantzig-Wolfe reformulation based on the aggregation of variables, which makes our branch-and-price algorithm more competitive. Computational results on cases...
IEEE Journal on Selected Areas in Communications, 2000
We consider the routing and wavelength assignment (RWA) problem on WDM ring networks without wave... more We consider the routing and wavelength assignment (RWA) problem on WDM ring networks without wavelength conversion. When the physical network and required connections are given, RWA is the problem to select a suitable path and wavelength among the many possible choices for each connection such that no two paths using the same wavelength pass through the same link. We give an integer programming formulation of the problem and propose an algorithm to solve it. Although the formulation has exponentially many variables, we solve the linear programming relaxation of it by using the column generation technique. We solve the column generation problem efficiently by decomposing the problem into several subproblems. After solving the linear programming relaxation, we apply the branch-and-price procedure to get an optimal solution. We test the proposed algorithm on some randomly generated data. Test results show that the algorithm gives optimal solutions to almost all instances under the given node limit of the branch-and-bound tree.
We consider the multicast routing and wavelength assignment (MC-RWA) problem on WDM bidirectional... more We consider the multicast routing and wavelength assignment (MC-RWA) problem on WDM bidirectional ring networks without wavelength conversion. We give an integer programming formulation of the problem and propose an algorithm to solve it optimally. The algorithm is based on column generation and branch-and-price. We test the proposed algorithm on randomly generated data and the test results show that the algorithm gives optimal solutions to all of the test problems.
We consider the problem of designing a local network in a two‐level telecommunication network. Gi... more We consider the problem of designing a local network in a two‐level telecommunication network. Given one or two hub nodes, central offices (COs) and conduits, the problem is to find a set of unidirectional self‐healing rings (USHRs) which covers all COs and satisfies all demands at minimum cost. The solution approach used is the decomposition and column generation. Master problem
We consider an extended formulation approach to the edge-weighted maximal clique problem. The pro... more We consider an extended formulation approach to the edge-weighted maximal clique problem. The problem is formulated by using additional variables for the set of nodes with the natural variables for the set of edges. We show that the proposed formulation is superior to the natural formulation both theoretically and practically. By using the projection technique, we can also derive new classes of facet-defining inequalities for the lower-dimensional polytope of the natural variables. Computational results are reported.
We consider the problem of designing an ATM VP-based leased line backbone network. Given point-to... more We consider the problem of designing an ATM VP-based leased line backbone network. Given point-to-point communication demands having predefined sizes in a network, the problem is to find configurations of demand routes and link facilities installed on each edge satisfying all demands at minimum cost under some constraints. One of the most important constraints is that a single demand cannot be split over multiple link facilities. This is a sort of bin packing constraint. We propose an integer programming formulation of the problem and an algorithm to solve it. An efficient column generation technique to solve the linear programming relaxation is proposed, and a valid inequality is used to strengthen the integer programming formulation. The algorithm incorporates the column generation technique and the cutting plane approach into a branch-and-bound scheme. We test the proposed algorithm on some real problems. The results show that the algorithm can be used to solve the problems within reasonably small computing times.
This paper considers the hop-constrained multicast route packing problem with a bandwidth reserva... more This paper considers the hop-constrained multicast route packing problem with a bandwidth reservation to build QoS-guaranteed multicast routing trees with a minimum installation cost. Given a set of multicast sessions, each of which has a hop limit constraint and a bandwidth requirement, the problem is to determine the set of multicast routing trees in an arc-capacitated network with the objective of minimizing the cost. For the problem, we propose a branch-and-cut-and-price algorithm, which can be viewed as a branch-and-bound method incorporating both the strong cutting plane algorithm and the column generation method. We implemented and tested the proposed algorithm on randomly generated problem instances with sizes up to 30 nodes, 570 arcs, and 10 multicast sessions. The test results show that the algorithm can obtain the optimal solution to practically sized problem instances within a reasonable time limit in most cases.
This paper considers the discrete two-hub location problem. We need to choose two hubs from a set... more This paper considers the discrete two-hub location problem. We need to choose two hubs from a set of nodes. The remaining nodes are to be connected to one of the two hubs which act as switching points for intemodal flows. A configuration which minimizes the total flow cost needs to be found. We show that the problem can be solved in polynomial dme when the hub locations are fixed. Since there are at most ½n(n-1) ways to choose the hub locations, the two-hub location problem can be solved in polynomial time. We transform the quadratic 0-1 integer program of the single allocation problem in the fixed two-hub system into a linear program and show that all extreme points of the polytope defined by the LP are integral. Also, the problem can be transformed into a minimum cut problem which can be solved efficiently by any polynomial time algorithm.
Classical column generation often shows desperately slow convergence. Recently, many acceleration... more Classical column generation often shows desperately slow convergence. Recently, many acceleration techniques are proposed. We propose Chebyshev center based column generation. In this method, the Chebyshev center is used for centering dual solutions within dual polyhedron. The Chebyshev center can be obtained by solving a linear program, so that our method can be applied with small modification of the classical column generation scheme. Numerical experiments show the effectiveness of our method.
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Papers by Sungsoo Park