TY - JOUR

T1 - Monge and feasibility sequences in general flow problems

AU - Adler, Ilan

AU - Hoffman, Alan J.

AU - Shamir, Ron

N1 - Funding Information:
Correspondence IO: Dr. R. Shamir, Department of Computer Science, School of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel. email: [email protected]. * This work was done while this author was a visitor at RUTCOR, Rutgers University, NJ. Supported by AFOSR grants 89-0512 and 90-0008, and by NSF grant STC 88-09648.

PY - 1993/7/19

Y1 - 1993/7/19

N2 - In a feasible transportation problem, there is always an ordering of the arcs such that greedily sending maximal flow on each arc in turn, according to that order, yields a feasible solution. We characterize those transportation graphs for which there exists a single order which is good for all feasible problems with the same graph. The characterizations are shown to be intimately related to Monge sequences and to totally balanced matrices. We describe efficient algorithms which, for a given graph, construct such order whenever it exists. For a transportation problem with corresponding m×n bipartite graph with e arcs, we show how to generate such an order in O(min(e log e,mn)) steps. Using that order, the feasibility question for any given supply and demand vectors can be determined in O(m+n) time. We also extend the characterization and algorithms to general minimum cost flow problems in which the underlying graph is nonbipartite, and the sources and destinations are not predetermined. We generalize the theory of Monge sequences too to such problems.

AB - In a feasible transportation problem, there is always an ordering of the arcs such that greedily sending maximal flow on each arc in turn, according to that order, yields a feasible solution. We characterize those transportation graphs for which there exists a single order which is good for all feasible problems with the same graph. The characterizations are shown to be intimately related to Monge sequences and to totally balanced matrices. We describe efficient algorithms which, for a given graph, construct such order whenever it exists. For a transportation problem with corresponding m×n bipartite graph with e arcs, we show how to generate such an order in O(min(e log e,mn)) steps. Using that order, the feasibility question for any given supply and demand vectors can be determined in O(m+n) time. We also extend the characterization and algorithms to general minimum cost flow problems in which the underlying graph is nonbipartite, and the sources and destinations are not predetermined. We generalize the theory of Monge sequences too to such problems.

KW - Monge sequences

KW - Network flow

KW - chordal bipartite graphs

KW - greedy algorithms

KW - totally balanced matrices

KW - transshipment problems.

UR - http://www.scopus.com/inward/record.url?scp=38248999236&partnerID=8YFLogxK

U2 - 10.1016/0166-218X(93)90220-I

DO - 10.1016/0166-218X(93)90220-I

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:38248999236

SN - 0166-218X

VL - 44

SP - 21

EP - 38

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

IS - 1-3

ER -