3,704 research outputs found
Reduction techniques for the prize collecting Steiner tree problem and the maximumâweight connected subgraph problem
The concept of reduction has frequently distinguished itself as a pivotal ingredient of exact solving approaches for the Steiner tree problem in graphs. In this article we broaden the focus and consider reduction techniques for three Steiner problem variants that have been extensively discussed in the literature and entail various practical applications: The prizeâcollecting Steiner tree problem, the rooted prizeâcollecting Steiner tree problem and the maximumâweight connected subgraph problem. By introducing and subsequently deploying numerous new reduction methods, we are able to drastically decrease the size of a large number of benchmark instances, already solving more than 90% of them to optimality. Furthermore, we demonstrate the impact of these techniques on exact solving, using the example of the stateâofâtheâart Steiner problem solver SCIPâJack
Combinatorial persistency criteria for multicut and max-cut
In combinatorial optimization, partial variable assignments are called
persistent if they agree with some optimal solution. We propose persistency
criteria for the multicut and max-cut problem as well as fast combinatorial
routines to verify them. The criteria that we derive are based on mappings that
improve feasible multicuts, respectively cuts. Our elementary criteria can be
checked enumeratively. The more advanced ones rely on fast algorithms for upper
and lower bounds for the respective cut problems and max-flow techniques for
auxiliary min-cut problems. Our methods can be used as a preprocessing
technique for reducing problem sizes or for computing partial optimality
guarantees for solutions output by heuristic solvers. We show the efficacy of
our methods on instances of both problems from computer vision, biomedical
image analysis and statistical physics
Optimality of Treating Interference as Noise: A Combinatorial Perspective
For single-antenna Gaussian interference channels, we re-formulate the
problem of determining the Generalized Degrees of Freedom (GDoF) region
achievable by treating interference as Gaussian noise (TIN) derived in [3] from
a combinatorial perspective. We show that the TIN power control problem can be
cast into an assignment problem, such that the globally optimal power
allocation variables can be obtained by well-known polynomial time algorithms.
Furthermore, the expression of the TIN-Achievable GDoF region (TINA region) can
be substantially simplified with the aid of maximum weighted matchings. We also
provide conditions under which the TINA region is a convex polytope that relax
those in [3]. For these new conditions, together with a channel connectivity
(i.e., interference topology) condition, we show TIN optimality for a new class
of interference networks that is not included, nor includes, the class found in
[3].
Building on the above insights, we consider the problem of joint link
scheduling and power control in wireless networks, which has been widely
studied as a basic physical layer mechanism for device-to-device (D2D)
communications. Inspired by the relaxed TIN channel strength condition as well
as the assignment-based power allocation, we propose a low-complexity
GDoF-based distributed link scheduling and power control mechanism (ITLinQ+)
that improves upon the ITLinQ scheme proposed in [4] and further improves over
the heuristic approach known as FlashLinQ. It is demonstrated by simulation
that ITLinQ+ provides significant average network throughput gains over both
ITLinQ and FlashLinQ, and yet still maintains the same level of implementation
complexity. More notably, the energy efficiency of the newly proposed ITLinQ+
is substantially larger than that of ITLinQ and FlashLinQ, which is desirable
for D2D networks formed by battery-powered devices.Comment: A short version has been presented at IEEE International Symposium on
Information Theory (ISIT 2015), Hong Kon
On the performance of a cavity method based algorithm for the Prize-Collecting Steiner Tree Problem on graphs
We study the behavior of an algorithm derived from the cavity method for the
Prize-Collecting Steiner Tree (PCST) problem on graphs. The algorithm is based
on the zero temperature limit of the cavity equations and as such is formally
simple (a fixed point equation resolved by iteration) and distributed
(parallelizable). We provide a detailed comparison with state-of-the-art
algorithms on a wide range of existing benchmarks networks and random graphs.
Specifically, we consider an enhanced derivative of the Goemans-Williamson
heuristics and the DHEA solver, a Branch and Cut Linear/Integer Programming
based approach. The comparison shows that the cavity algorithm outperforms the
two algorithms in most large instances both in running time and quality of the
solution. Finally we prove a few optimality properties of the solutions
provided by our algorithm, including optimality under the two post-processing
procedures defined in the Goemans-Williamson derivative and global optimality
in some limit cases
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
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