One-Pass Algorithms for Some Generalized Network Problems

Abstract

The generalized network problem and the closely related restricted dyadic problem are two special model types that occur frequently in applications of linear programming. Although they are next in order after pure network or distribution problems with respect to ease of computation, the jump in degree of difficulty is such that, in the most general problem, there exist no algorithms for them comparable in speed or efficiency to those for pure network or distribution problems. There are, however, numerous examples in which some additional special structure leads one to anticipate the existence of algorithms that compare favorably with the efficiency of those for the corresponding pure cases. Also, these more special structures may be encountered as part of larger or more complicated models. In this paper we designate by topological properties two special structures that permit evolution of efficient algorithms. These follow by extensions of methods of Charnes and Cooper and of Dijkstra for the corresponding pure network problems. We obtain easily implemented algorithms that provide an optimum in one “pass” through the network. The proofs provided for these extended theorems differ in character from those provided (or not provided) in the more special “pure” problem algorithms published.

Chance-Constrained Generalized Networks

“…Thus we have solved an extension of the wastewater problem which permits process factors to be random. This result secures for us the possibility of solving the wastewater model, deterministic or chance-constrained, by one pass through the network using the techniques developed in [6], a fact which was not previously known even for the deterministic case. …”

“…In accordance with procedures such as those discussed in [6] and I? ]» the constraints of (3) give rise to the deterministic equivalent constraints (7) is of the exact same form as (2) and can be similarly interpreted.…”

“…Under these assumptions the deterministic problem (6) can be solved using the one pass algorithm discussed in [6].…”

“…Specifically, the deterministic equivalent for our problem turns out to be a deterministic generalized network, and thus can be solved using existing algorithms for such problems (see [5j, [6j, [?]). In particular, we show that for a certain class of problems the deterministic equivalent can be solved using the one pass algorithms presented in [6]. Extensions of these one pass algorithms to include problems which involve decision rules more general than the zero order rule are under investigation.…”

Combinatorial Approximation Algorithms for Generalized Flow Problems

“…Since Khachiyan's polynomial-time algorithm for solving linear programs [32], the generalized shortest path problem has had a polynomialtime algorithm. Prior to his work, Charnes and Raike [7] showed how to solve the GSP using a variant of Dijkstra's algorithm [12], but problem instances are restricted to have nonnegative costs and flow multipliers at most one. The goal is to obtain one unit of flow at a sink, not a source, vertex.…”