Sung-Pil Hong

An efficient multicast routing algorithm for delay-sensitive applications with dynamic membership

We propose an algorithm for finding a multicast tree in packet-switched networks. The objective is to minimize total cost incurred at the multicast path. The routing model is based on the minimum cost Steiner tree problem. The Steiner problem is extended to incorporate two additional requirements. First, the delay experienced along the path from the source to each destination is bounded. Second, the destinations are allowed to join and leave multicasting anytime during a session. To minimize the disruption to on-going multicasting the algorithm adopts the idea of connecting a new destination to the current multicasting by a minimum cost path satisfying the delay bound. To find such a path is an NP-hard problem and an enumerative method relying on generation of delay bounded paths between node pairs is not likely to find a good routing path in acceptable computation time when network size is large. To cope with such difficulty, the proposed algorithm utilizes an optimization technique called Lagrangian relaxation method. A computational experiment is done on relatively dense and large Waxmans networks. The results seem to be promising. For sparse networks, the algorithm can find near-optimal multicast trees. Also the quality of multicast trees does not seem to deteriorate even when the network size grows. Furthermore, the experimental results shows that the computational efforts for each addition of node to the call are fairly moderate, namely the same as to solve a few shortest path problems.

A fully polynomial bicriteria approximation scheme for the constrained spanning tree problem

We propose a fully polynomial bicriteria approximation scheme for the constrained spanning tree problem. First, an exact pseudo-polynomial algorithm is developed based on a two-variable extension of the well-known matrix-tree theorem. The scaling and approximate binary search techniques are then utilized to yield a fully polynomial approximation scheme.

On the Complexity of the Production-Transportation Problem

The production-transportation problem (PTP) is a generalization of the transportation problem. In PTP, we decide not only the level of shipment from each source to each sink but also the level of supply at each source. A concave production cost function is associated with the assignment of supplies to sources. Thus the objective function of PTP is the sum of the linear transportation costs and the production costs. We show that this problem is generally NP-hard and present some polynomial classes. In particular, we propose a polynomial algorithm for the case in which the transportation cost matrix has the Monge property and the number of sources is fixed. The algorithm generalizes a polynomial algorithm of Tuy, Dan, and Ghannadan [Oper. Res. Lett., 14 (1993), pp. 99–109] for the problem with two sources.

Unification of lower-bound analyses of the lift-and-project rank of combinatorial optimization polyhedra

We present a unifying framework to establish a lower bound on the number of semidefinite-programming-based lift-and-project iterations (rank) for computing the convex hull of the feasible solutions of various combinatorial optimization problems. This framework is based on the maps which are commutative with the lift-and-project operators. Some special commutative maps were originally observed by Lovasz and Schrijver and have been used usually implicitly in the previous lower-bound analyses. In this paper, we formalize the lift-and-project commutative maps and propose a general framework for lower-bound analysis, in which we can recapture many of the previous lower-bound results on the lift-and-project ranks.

A pseudo-polynomial heuristic for path-constrained discrete-time Markovian-target search

We propose a new heuristic for the single-searcher path-constrained discrete-time Markovian-target search. The algorithm minimizes an approximate, instead of exact, nondetection probability computed from the conditional probability that reflects the search history over the time windows of a fixed length, l. Having a pseudo-polynomial complexity, it can solve, in reasonable time, the instances an order of magnitude larger than those solved in the previous studies. By an asymptotic analysis relying on the fast-mixing Markov chain, we show that the relative error of the approximation exponentially diminishes as l increases and the experimental results confirm the analysis. The experiment also reveals a correlation very close to 1 between the approximate and exact nondetection probability of a search path. This means that the heuristic produces near-optimal search paths.

Optimal search-relocation trade-off in Markovian-target searching

In this study, a standard moving-target search model was extended with a multiple-search-speed option, whereby a trade-off is enabled between the increased detection chances owing to the searchers better location and the increased uncertainty of the targets location resulting from the diminished search performance incurred in the relocation. This enhances the detection probability of the output search path and, thereby, the models practicality. However, the scalability of the solution method is essential to its implementation, as the basic model is already NP-hard. We developed an efficient heuristic by combining the idea of approximate nondetection probability minimization and a hybridized shortest-path heuristic that exploits the fast-mixing property of the Markov chain. According to the results of an intensive experiment, the heuristic achieves a near-optimal trade-off within a very reasonable computation time.

Polynomiality of sparsest cuts with fixed number of sources

We show that when the number of sources is constant the sparsest cut problem is solvable in polynomial time.

Handelman rank of zero-diagonal quadratic programs over a hypercube and its applications

It has been observed that the Handelman’s certificate of positivity of a polynomial over a compact polyhedron offers a hierarchical relaxation scheme for polynomial programs. The Handelman hierarchy seems particularly suitable for a class of combinatorial optimizations that are formulated as a zero-diagonal quadratic program over a hypercube. In this paper, we present an error analysis of Handelman hierarchy applied to the special class of polynomial programs and its implications in the computation of the combinatorial optimization problems.

COLUMN GENERATION APPROACH TO LINE PLANNING WITH VARIOUS HALTING PATTERNS — APPLICATION TO THE KOREAN HIGH-SPEED RAILWAY

This study investigated railway line planning optimization models that determine the frequency of trains on each line to satisfy passenger origin–destination demands while minimizing related costs. Most line planning models assume that all trains on the same route run with the same halting pattern. However, to minimize passenger travel time and to provide a train service with faster travel times to as many stations as possible, we must consider various halting patterns; these patterns can be provided in advance or are required to be formulated. Our study addresses two line planning problems that consider halting patterns, describes the computational complexities of each problem, and presents the column generation approach for one model. We also present experimental results obtained for the Korean high-speed railway network.

Rank of Handelman hierarchy for Max-Cut

Abstract We consider a hierarchical relaxation, called Handelman hierarchy, for a class of polynomial optimization problems. We prove that the rank of Handelman hierarchy, if applied to a standard quadratic formulation of Max-Cut, is exactly the same as the number of nodes of the underlying graph. Also we give an error bound for Handelman hierarchy, in terms of its level, applied to the Max-Cut formulation.