8,287 research outputs found
The Complexity Of The NP-Class
This paper presents a novel and straight formulation, and gives a complete
insight towards the understanding of the complexity of the problems of the so
called NP-Class. In particular, this paper focuses in the Searching of the
Optimal Geometrical Structures and the Travelling Salesman Problems. The main
results are the polynomial reduction procedure and the solution to the Noted
Conjecture of the NP-Class
Routing Games over Time with FIFO policy
We study atomic routing games where every agent travels both along its
decided edges and through time. The agents arriving on an edge are first lined
up in a \emph{first-in-first-out} queue and may wait: an edge is associated
with a capacity, which defines how many agents-per-time-step can pop from the
queue's head and enter the edge, to transit for a fixed delay. We show that the
best-response optimization problem is not approximable, and that deciding the
existence of a Nash equilibrium is complete for the second level of the
polynomial hierarchy. Then, we drop the rationality assumption, introduce a
behavioral concept based on GPS navigation, and study its worst-case efficiency
ratio to coordination.Comment: Submission to WINE-2017 Deadline was August 2nd AoE, 201
An O(1)-Approximation for Minimum Spanning Tree Interdiction
Network interdiction problems are a natural way to study the sensitivity of a
network optimization problem with respect to the removal of a limited set of
edges or vertices. One of the oldest and best-studied interdiction problems is
minimum spanning tree (MST) interdiction. Here, an undirected multigraph with
nonnegative edge weights and positive interdiction costs on its edges is given,
together with a positive budget B. The goal is to find a subset of edges R,
whose total interdiction cost does not exceed B, such that removing R leads to
a graph where the weight of an MST is as large as possible. Frederickson and
Solis-Oba (SODA 1996) presented an O(log m)-approximation for MST interdiction,
where m is the number of edges. Since then, no further progress has been made
regarding approximations, and the question whether MST interdiction admits an
O(1)-approximation remained open.
We answer this question in the affirmative, by presenting a 14-approximation
that overcomes two main hurdles that hindered further progress so far.
Moreover, based on a well-known 2-approximation for the metric traveling
salesman problem (TSP), we show that our O(1)-approximation for MST
interdiction implies an O(1)-approximation for a natural interdiction version
of metric TSP
Advantages of Unfair Quantum Ground-State Sampling
The debate around the potential superiority of quantum annealers over their
classical counterparts has been ongoing since the inception of the field by
Kadowaki and Nishimori close to two decades ago. Recent technological
breakthroughs in the field, which have led to the manufacture of experimental
prototypes of quantum annealing optimizers with sizes approaching the practical
regime, have reignited this discussion. However, the demonstration of quantum
annealing speedups remains to this day an elusive albeit coveted goal. Here, we
examine the power of quantum annealers to provide a different type of quantum
enhancement of practical relevance, namely, their ability to serve as useful
samplers from the ground-state manifolds of combinatorial optimization
problems. We study, both numerically by simulating ideal stoquastic and
non-stoquastic quantum annealing processes, and experimentally, using a
commercially available quantum annealing processor, the ability of quantum
annealers to sample the ground-states of spin glasses differently than
classical thermal samplers. We demonstrate that i) quantum annealers in general
sample the ground-state manifolds of spin glasses very differently than thermal
optimizers, ii) the nature of the quantum fluctuations driving the annealing
process has a decisive effect on the final distribution over ground-states, and
iii) the experimental quantum annealer samples ground-state manifolds
significantly differently than thermal and ideal quantum annealers. We
illustrate how quantum annealers may serve as powerful tools when complementing
standard sampling algorithms.Comment: 13 pages, 11 figure
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