3 research outputs found
On Randomized Algorithms for Matching in the Online Preemptive Model
We investigate the power of randomized algorithms for the maximum cardinality
matching (MCM) and the maximum weight matching (MWM) problems in the online
preemptive model. In this model, the edges of a graph are revealed one by one
and the algorithm is required to always maintain a valid matching. On seeing an
edge, the algorithm has to either accept or reject the edge. If accepted, then
the adjacent edges are discarded. The complexity of the problem is settled for
deterministic algorithms.
Almost nothing is known for randomized algorithms. A lower bound of
is known for MCM with a trivial upper bound of . An upper bound of
is known for MWM. We initiate a systematic study of the same in this paper with
an aim to isolate and understand the difficulty. We begin with a primal-dual
analysis of the deterministic algorithm due to McGregor. All deterministic
lower bounds are on instances which are trees at every step. For this class of
(unweighted) graphs we present a randomized algorithm which is
-competitive. The analysis is a considerable extension of the
(simple) primal-dual analysis for the deterministic case. The key new technique
is that the distribution of primal charge to dual variables depends on the
"neighborhood" and needs to be done after having seen the entire input. The
assignment is asymmetric: in that edges may assign different charges to the two
end-points. Also the proof depends on a non-trivial structural statement on the
performance of the algorithm on the input tree.
The other main result of this paper is an extension of the deterministic
lower bound of Varadaraja to a natural class of randomized algorithms which
decide whether to accept a new edge or not using independent random choices
Submodular Maximization Meets Streaming: Matchings, Matroids, and More
We study the problem of finding a maximum matching in a graph given by an
input stream listing its edges in some arbitrary order, where the quantity to
be maximized is given by a monotone submodular function on subsets of edges.
This problem, which we call maximum submodular-function matching (MSM), is a
natural generalization of maximum weight matching (MWM), which is in turn a
generalization of maximum cardinality matching (MCM). We give two incomparable
algorithms for this problem with space usage falling in the semi-streaming
range---they store only edges, using working memory---that
achieve approximation ratios of in a single pass and in
passes respectively. The operations of these algorithms
mimic those of Zelke's and McGregor's respective algorithms for MWM; the
novelty lies in the analysis for the MSM setting. In fact we identify a general
framework for MWM algorithms that allows this kind of adaptation to the broader
setting of MSM.
In the sequel, we give generalizations of these results where the
maximization is over "independent sets" in a very general sense. This
generalization captures hypermatchings in hypergraphs as well as independence
in the intersection of multiple matroids.Comment: 18 page
Streaming Algorithms for Submodular Function Maximization
We consider the problem of maximizing a nonnegative submodular set function
subject to a -matchoid
constraint in the single-pass streaming setting. Previous work in this context
has considered streaming algorithms for modular functions and monotone
submodular functions. The main result is for submodular functions that are {\em
non-monotone}. We describe deterministic and randomized algorithms that obtain
a -approximation using -space, where is
an upper bound on the cardinality of the desired set. The model assumes value
oracle access to and membership oracles for the matroids defining the
-matchoid constraint.Comment: 29 pages, 7 figures, extended abstract to appear in ICALP 201