1,745 research outputs found
Minimum Makespan Multi-vehicle Dial-a-Ride
Dial a ride problems consist of a metric space (denoting travel time between
vertices) and a set of m objects represented as source-destination pairs, where
each object requires to be moved from its source to destination vertex. We
consider the multi-vehicle Dial a ride problem, with each vehicle having
capacity k and its own depot-vertex, where the objective is to minimize the
maximum completion time (makespan) of the vehicles. We study the "preemptive"
version of the problem, where an object may be left at intermediate vertices
and transported by more than one vehicle, while being moved from source to
destination. Our main results are an O(log^3 n)-approximation algorithm for
preemptive multi-vehicle Dial a ride, and an improved O(log t)-approximation
for its special case when there is no capacity constraint. We also show that
the approximation ratios improve by a log-factor when the underlying metric is
induced by a fixed-minor-free graph.Comment: 22 pages, 1 figure. Preliminary version appeared in ESA 200
Network Design with Coverage Costs
We study network design with a cost structure motivated by redundancy in data
traffic. We are given a graph, g groups of terminals, and a universe of data
packets. Each group of terminals desires a subset of the packets from its
respective source. The cost of routing traffic on any edge in the network is
proportional to the total size of the distinct packets that the edge carries.
Our goal is to find a minimum cost routing. We focus on two settings. In the
first, the collection of packet sets desired by source-sink pairs is laminar.
For this setting, we present a primal-dual based 2-approximation, improving
upon a logarithmic approximation due to Barman and Chawla (2012). In the second
setting, packet sets can have non-trivial intersection. We focus on the case
where each packet is desired by either a single terminal group or by all of the
groups, and the graph is unweighted. For this setting we present an O(log
g)-approximation.
Our approximation for the second setting is based on a novel spanner-type
construction in unweighted graphs that, given a collection of g vertex subsets,
finds a subgraph of cost only a constant factor more than the minimum spanning
tree of the graph, such that every subset in the collection has a Steiner tree
in the subgraph of cost at most O(log g) that of its minimum Steiner tree in
the original graph. We call such a subgraph a group spanner.Comment: Updated version with additional result
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