Implicit Flow Maximization by Iterative Squaring

Abstract

Application areas like logic design and network analysis produce large graphs G=(V,E) on which traditional algorithms, which work on adjacency list representations, are not practicable anymore. These large graphs often contain regular structures that enable compact implicit representations by decision diagrams like OBDDs [1, 2, 3]. To solve problems on such implicitly given graphs, specialized algorithms are needed. These are considered as heuristics with typically higher worst-case runtimes than traditional methods. In this paper, an implicit algorithm for flow maximization in 0–1 networks is presented, which works on OBDD-representations of node and edge sets. Because it belongs to the class of layered-network methods, it has to construct blocking-flows. In contrast to previous implicit methods, it avoids breadth-first searches and layer-wise proceeding, and uses iterative squaring instead. In this way, the algorithm needs to execute only O(log²|V|) operations on the OBDDs to obtain a layered-network or at least one augmenting path, respectively. Moreover, each OBDD-operation is efficient if the node and edge sets are represented by compact OBDDs during the flow computation. In order to investigate the algorithm’s behavior on large and structured networks, it has been analyzed on grid networks, on which a maximum flow is computed in polylogarithmic time O(log³|V|) and space O(log²|V|). In contrast, previous methods need time and space Ω(|V|^1/2log|V|) on grids, and are beaten also in experiments for |V| ≥ 2²⁶.

An extended version of this paper can be obtained via http://ls2-www.cs.uni-dortmund.de/~sawitzki/IFMbIS.pdf.