9,792 research outputs found
A cost function for similarity-based hierarchical clustering
The development of algorithms for hierarchical clustering has been hampered
by a shortage of precise objective functions. To help address this situation,
we introduce a simple cost function on hierarchies over a set of points, given
pairwise similarities between those points. We show that this criterion behaves
sensibly in canonical instances and that it admits a top-down construction
procedure with a provably good approximation ratio
Mapping constrained optimization problems to quantum annealing with application to fault diagnosis
Current quantum annealing (QA) hardware suffers from practical limitations
such as finite temperature, sparse connectivity, small qubit numbers, and
control error. We propose new algorithms for mapping boolean constraint
satisfaction problems (CSPs) onto QA hardware mitigating these limitations. In
particular we develop a new embedding algorithm for mapping a CSP onto a
hardware Ising model with a fixed sparse set of interactions, and propose two
new decomposition algorithms for solving problems too large to map directly
into hardware.
The mapping technique is locally-structured, as hardware compatible Ising
models are generated for each problem constraint, and variables appearing in
different constraints are chained together using ferromagnetic couplings. In
contrast, global embedding techniques generate a hardware independent Ising
model for all the constraints, and then use a minor-embedding algorithm to
generate a hardware compatible Ising model. We give an example of a class of
CSPs for which the scaling performance of D-Wave's QA hardware using the local
mapping technique is significantly better than global embedding.
We validate the approach by applying D-Wave's hardware to circuit-based
fault-diagnosis. For circuits that embed directly, we find that the hardware is
typically able to find all solutions from a min-fault diagnosis set of size N
using 1000N samples, using an annealing rate that is 25 times faster than a
leading SAT-based sampling method. Further, we apply decomposition algorithms
to find min-cardinality faults for circuits that are up to 5 times larger than
can be solved directly on current hardware.Comment: 22 pages, 4 figure
Packing Cars into Narrow Roads: PTASs for Limited Supply Highway
In the Highway problem, we are given a path with n edges (the highway), and a set of m drivers, each one characterized by a subpath and a budget. For a given assignment of edge prices (the tolls), the highway owner collects from each driver the total price of the associated path when it does not exceed drivers\u27s budget, and zero otherwise. The goal is to choose the prices to maximize the total profit. A PTAS is known for this (strongly NP-hard) problem [Grandoni,Rothvoss-SODA\u2711, SICOMP\u2716].
In this paper we study the limited supply generalization of Highway, that incorporates capacity constraints. Here the input also includes a capacity u_e >= 0 for each edge e; we need to select, among drivers that can afford the required price, a subset such that the number of drivers that use each edge e is at most u_e (and we get profit only from selected drivers). To the best of our knowledge, the only approximation algorithm known for this problem is a folklore O(log m) approximation based on a reduction to the related Unsplittable Flow on a Path problem (UFP). The main result of this paper is a PTAS for limited supply highway.
As a second contribution, we study a natural generalization of the problem where each driver i demands a different amount d_i of capacity. Using known techniques, it is not hard to derive a QPTAS for this problem. Here we present a PTAS for the case that drivers have uniform budgets. Finding a PTAS for non-uniform-demand limited supply highway is left as a challenging open problem
Strong Effects of Network Architecture in the Entrainment of Coupled Oscillator Systems
Entrainment of randomly coupled oscillator networks by periodic external
forcing applied to a subset of elements is numerically and analytically
investigated. For a large class of interaction functions, we find that the
entrainment window with a tongue shape becomes exponentially narrow for
networks with higher hierarchical organization. However, the entrainment is
significantly facilitated if the networks are directionally biased, i.e.,
closer to the feedforward networks. Furthermore, we show that the networks with
high entrainment ability can be constructed by evolutionary optimization
processes. The neural network structure of the master clock of the circadian
rhythm in mammals is discussed from the viewpoint of our results.Comment: 15 pages, 11 figures, RevTe
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