3,566 research outputs found
Interdiction Problems on Planar Graphs
Interdiction problems are leader-follower games in which the leader is
allowed to delete a certain number of edges from the graph in order to
maximally impede the follower, who is trying to solve an optimization problem
on the impeded graph. We introduce approximation algorithms and strong
NP-completeness results for interdiction problems on planar graphs. We give a
multiplicative -approximation for the maximum matching
interdiction problem on weighted planar graphs. The algorithm runs in
pseudo-polynomial time for each fixed . We also show that
weighted maximum matching interdiction, budget-constrained flow improvement,
directed shortest path interdiction, and minimum perfect matching interdiction
are strongly NP-complete on planar graphs. To our knowledge, our
budget-constrained flow improvement result is the first planar NP-completeness
proof that uses a one-vertex crossing gadget.Comment: 25 pages, 9 figures. Extended abstract in APPROX-RANDOM 201
Minor-Embedding in Adiabatic Quantum Computation: I. The Parameter Setting Problem
We show that the NP-hard quadratic unconstrained binary optimization (QUBO)
problem on a graph can be solved using an adiabatic quantum computer that
implements an Ising spin-1/2 Hamiltonian, by reduction through minor-embedding
of in the quantum hardware graph . There are two components to this
reduction: embedding and parameter setting. The embedding problem is to find a
minor-embedding of a graph in , which is a subgraph of
such that can be obtained from by contracting edges. The
parameter setting problem is to determine the corresponding parameters, qubit
biases and coupler strengths, of the embedded Ising Hamiltonian. In this paper,
we focus on the parameter setting problem. As an example, we demonstrate the
embedded Ising Hamiltonian for solving the maximum independent set (MIS)
problem via adiabatic quantum computation (AQC) using an Ising spin-1/2 system.
We close by discussing several related algorithmic problems that need to be
investigated in order to facilitate the design of adiabatic algorithms and AQC
architectures.Comment: 17 pages, 5 figures, submitte
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
We study the family of intersection graphs of low density objects in low
dimensional Euclidean space. This family is quite general, and includes planar
graphs. We prove that such graphs have small separators. Next, we present
efficient -approximation algorithms for these graphs, for
Independent Set, Set Cover, and Dominating Set problems, among others. We also
prove corresponding hardness of approximation for some of these optimization
problems, providing a characterization of their intractability in terms of
density
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