128 research outputs found
Few Cuts Meet Many Point Sets
We study the problem of how to breakup many point sets in into
smaller parts using a few splitting (shared) hyperplanes. This problem is
related to the classical Ham-Sandwich Theorem. We provide a logarithmic
approximation to the optimal solution using the greedy algorithm for submodular
optimization
Scalable Bicriteria Algorithms for Non-Monotone Submodular Cover
In this paper, we consider the optimization problem \scpl (\scp), which is to
find a minimum cost subset of a ground set such that the value of a
submodular function is above a threshold . In contrast to most
existing work on \scp, it is not assumed that is monotone. Two bicriteria
approximation algorithms are presented for \scp that, for input parameter , give ratio to the optimal cost and ensures
the function is at least . A lower bound shows that
under the value query model shows that no polynomial-time algorithm can ensure
that is larger than . Further, the algorithms presented are
scalable to large data sets, processing the ground set in a stream. Similar
algorithms developed for \scp also work for the related optimization problem of
\smpl (\smp). Finally, the algorithms are demonstrated to be effective in
experiments involving graph cut and data summarization functions
Robust Fault Tolerant uncapacitated facility location
In the uncapacitated facility location problem, given a graph, a set of
demands and opening costs, it is required to find a set of facilities R, so as
to minimize the sum of the cost of opening the facilities in R and the cost of
assigning all node demands to open facilities. This paper concerns the robust
fault-tolerant version of the uncapacitated facility location problem (RFTFL).
In this problem, one or more facilities might fail, and each demand should be
supplied by the closest open facility that did not fail. It is required to find
a set of facilities R, so as to minimize the sum of the cost of opening the
facilities in R and the cost of assigning all node demands to open facilities
that did not fail, after the failure of up to \alpha facilities. We present a
polynomial time algorithm that yields a 6.5-approximation for this problem with
at most one failure and a 1.5 + 7.5\alpha-approximation for the problem with at
most \alpha > 1 failures. We also show that the RFTFL problem is NP-hard even
on trees, and even in the case of a single failure
Minimal Actuator Placement with Optimal Control Constraints
We introduce the problem of minimal actuator placement in a linear control
system so that a bound on the minimum control effort for a given state transfer
is satisfied while controllability is ensured. We first show that this is an
NP-hard problem following the recent work of Olshevsky. Next, we prove that
this problem has a supermodular structure. Afterwards, we provide an efficient
algorithm that approximates up to a multiplicative factor of O(logn), where n
is the size of the multi-agent network, any optimal actuator set that meets the
specified energy criterion. Moreover, we show that this is the best
approximation factor one can achieve in polynomial-time for the worst case.
Finally, we test this algorithm over large Erdos-Renyi random networks to
further demonstrate its efficiency.Comment: This version includes all the omitted proofs from the one to appear
in the American Control Conference (ACC) 2015 proceeding
Approximating Source Location and Star Survivable Network Problems
In Source Location (SL) problems the goal is to select a mini-mum cost source
set such that the connectivity (or flow) from
to any node is at least the demand of . In many SL problems
if , namely, the demand of nodes selected to is
completely satisfied. In a node-connectivity variant suggested recently by
Fukunaga, every node gets a "bonus" if it is selected to
. Fukunaga showed that for undirected graphs one can achieve ratio for his variant, where is the maximum demand. We
improve this by achieving ratio \min\{p^*\lnk,k\}\cdot O(\ln (k/q^*)) for a
more general version with node capacities, where is
the maximum bonus and is the minimum capacity. In
particular, for the most natural case considered by Fukunaga, we
improve the ratio from to . We also get ratio
for the edge-connectivity version, for which no ratio that depends on only
was known before. To derive these results, we consider a particular case of the
Survivable Network (SN) problem when all edges of positive cost form a star. We
give ratio for this variant, improving over the best
ratio known for the general case of Chuzhoy and Khanna
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