Metatheorems for dynamic weighted matching

Abstract

We consider the maximum weight matching (MWM) problem in dynamic graphs. We provide two reductions. The first reduces the dynamic MWM problem on m-edge, n-node graphs with weights bounded by N to the problem with weights bounded by (n/ϵ)2;), so that if the MWM problem can be α-Approximated with update time t(m, n,N), then it can also be (1+ϵ)α-Approximated with update time O(t(m, n, (n/ϵ)2;)) log2 n + log n log logN)). The second reduction reduces the dynamic MWM problem to the dynamic maximum cardinality matching (MCM) problem in which the graph is unweighted. This reduction shows that if there is an --Approximation algorithm for MCM with update time t(m, n) in m-edge n-node graphs, then there is also a (2 + ϵ)α-approximation algorithm for MWM with update time O(t(m, n) ϵ-2 log2N). We also obtain better bounds in our reductions if the ratio between the largest and the smallest edge weight is small. Combined with recent work on MCM, these two reductions substantially improve upon the state-of-The-Art of dynamic MWM algorithms.