  This paper proposes a new class of online policies for scheduling in
input-buffered crossbar switches. Our policies are throughput optimal for a
large class of arrival processes which satisfy strong-law of large numbers.
Given an initial configuration and no further arrivals, our policies drain all
packets in the system in the minimal amount of time (providing an online
alternative to the batch approach based on Birkhoff-VonNeumann decompositions).
We show that it is possible for policies in our class to be throughput optimal
even if they are not constrained to be maximal in every time slot.
  Most algorithms for switch scheduling take an edge based approach; in
contrast, we focus on scheduling (a large enough set of) the most congested
ports. This alternate approach allows for lower-complexity algorithms, and also
requires a non-standard technique to prove throughput-optimality. One algorithm
in our class, Maximum Vertex-weighted Matching (MVM) has worst-case complexity
similar to Max-size Matching, and in simulations shows slightly better delay
performance than Max-(edge)weighted-Matching (MWM).
