2,012 research outputs found
Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling
A distance labeling scheme is an assignment of bit-labels to the vertices of
an undirected, unweighted graph such that the distance between any pair of
vertices can be decoded solely from their labels. An important class of
distance labeling schemes is that of hub labelings, where a node
stores its distance to the so-called hubs , chosen so that for
any there is belonging to some shortest
path. Notice that for most existing graph classes, the best distance labelling
constructions existing use at some point a hub labeling scheme at least as a
key building block. Our interest lies in hub labelings of sparse graphs, i.e.,
those with , for which we show a lowerbound of
for the average size of the hubsets.
Additionally, we show a hub-labeling construction for sparse graphs of average
size for some , where is the
so-called Ruzsa-Szemer{\'e}di function, linked to structure of induced
matchings in dense graphs. This implies that further improving the lower bound
on hub labeling size to would require a
breakthrough in the study of lower bounds on , which have resisted
substantial improvement in the last 70 years. For general distance labeling of
sparse graphs, we show a lowerbound of , where is the communication complexity of the
Sum-Index problem over . Our results suggest that the best achievable
hub-label size and distance-label size in sparse graphs may be
for some
A More Reliable Greedy Heuristic for Maximum Matchings in Sparse Random Graphs
We propose a new greedy algorithm for the maximum cardinality matching
problem. We give experimental evidence that this algorithm is likely to find a
maximum matching in random graphs with constant expected degree c>0,
independent of the value of c. This is contrary to the behavior of commonly
used greedy matching heuristics which are known to have some range of c where
they probably fail to compute a maximum matching
The Query-commit Problem
In the query-commit problem we are given a graph where edges have distinct
probabilities of existing. It is possible to query the edges of the graph, and
if the queried edge exists then its endpoints are irrevocably matched. The goal
is to find a querying strategy which maximizes the expected size of the
matching obtained. This stochastic matching setup is motivated by applications
in kidney exchanges and online dating.
In this paper we address the query-commit problem from both theoretical and
experimental perspectives. First, we show that a simple class of edges can be
queried without compromising the optimality of the strategy. This property is
then used to obtain in polynomial time an optimal querying strategy when the
input graph is sparse. Next we turn our attentions to the kidney exchange
application, focusing on instances modeled over real data from existing
exchange programs. We prove that, as the number of nodes grows, almost every
instance admits a strategy which matches almost all nodes. This result supports
the intuition that more exchanges are possible on a larger pool of
patient/donors and gives theoretical justification for unifying the existing
exchange programs. Finally, we evaluate experimentally different querying
strategies over kidney exchange instances. We show that even very simple
heuristics perform fairly well, being within 1.5% of an optimal clairvoyant
strategy, that knows in advance the edges in the graph. In such a
time-sensitive application, this result motivates the use of committing
strategies
- …