20,282 research outputs found
Phase Transitions of the Typical Algorithmic Complexity of the Random Satisfiability Problem Studied with Linear Programming
Here we study the NP-complete -SAT problem. Although the worst-case
complexity of NP-complete problems is conjectured to be exponential, there
exist parametrized random ensembles of problems where solutions can typically
be found in polynomial time for suitable ranges of the parameter. In fact,
random -SAT, with as control parameter, can be solved quickly
for small enough values of . It shows a phase transition between a
satisfiable phase and an unsatisfiable phase. For branch and bound algorithms,
which operate in the space of feasible Boolean configurations, the empirically
hardest problems are located only close to this phase transition. Here we study
-SAT () and the related optimization problem MAX-SAT by a linear
programming approach, which is widely used for practical problems and allows
for polynomial run time. In contrast to branch and bound it operates outside
the space of feasible configurations. On the other hand, finding a solution
within polynomial time is not guaranteed. We investigated several variants like
including artificial objective functions, so called cutting-plane approaches,
and a mapping to the NP-complete vertex-cover problem. We observed several
easy-hard transitions, from where the problems are typically solvable (in
polynomial time) using the given algorithms, respectively, to where they are
not solvable in polynomial time. For the related vertex-cover problem on random
graphs these easy-hard transitions can be identified with structural properties
of the graphs, like percolation transitions. For the present random -SAT
problem we have investigated numerous structural properties also exhibiting
clear transitions, but they appear not be correlated to the here observed
easy-hard transitions. This renders the behaviour of random -SAT more
complex than, e.g., the vertex-cover problem.Comment: 11 pages, 5 figure
Random subcubes as a toy model for constraint satisfaction problems
We present an exactly solvable random-subcube model inspired by the structure
of hard constraint satisfaction and optimization problems. Our model reproduces
the structure of the solution space of the random k-satisfiability and
k-coloring problems, and undergoes the same phase transitions as these
problems. The comparison becomes quantitative in the large-k limit. Distance
properties, as well the x-satisfiability threshold, are studied. The model is
also generalized to define a continuous energy landscape useful for studying
several aspects of glassy dynamics.Comment: 21 pages, 4 figure
Phase transitions in project scheduling.
The analysis of the complexity of combinatorial optimization problems has led to the distinction between problems which are solvable in a polynomially bounded amount of time (classified in P) and problems which are not (classified in NP). This implies that the problems in NP are hard to solve whereas the problems in P are not. However, this analysis is based on worst-case scenarios. The fact that a decision problem is shown to be NP-complete or the fact that an optimization problem is shown to be NP-hard implies that, in the worst case, solving it is very hard. Recent computational results obtained with a well known NP-hard problem, namely the resource-constrained project scheduling problem, indicate that many instances are actually easy to solve. These results are in line with those recently obtained by researchers in the area of artificial intelligence, which show that many NP-complete problemsexhibit so-called phase transitions, resulting in a sudden and dramatic change of computational complexity based on one or more order parameters that are characteristic of the system as a whole. In this paper we provide evidence for the existence of phase transitions in various resource-constrained project scheduling problems. We discuss the use of network complexity measures and resource parameters as potential order parameters. We show that while the network complexity measures seem to reveal continuous easy-hard or hard-easy phase-transitions, the resource parameters exhibit an easy-hard-easy transition behaviour.Networks; Problems; Scheduling; Algorithms;
From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz
The next few years will be exciting as prototype universal quantum processors
emerge, enabling implementation of a wider variety of algorithms. Of particular
interest are quantum heuristics, which require experimentation on quantum
hardware for their evaluation, and which have the potential to significantly
expand the breadth of quantum computing applications. A leading candidate is
Farhi et al.'s Quantum Approximate Optimization Algorithm, which alternates
between applying a cost-function-based Hamiltonian and a mixing Hamiltonian.
Here, we extend this framework to allow alternation between more general
families of operators. The essence of this extension, the Quantum Alternating
Operator Ansatz, is the consideration of general parametrized families of
unitaries rather than only those corresponding to the time-evolution under a
fixed local Hamiltonian for a time specified by the parameter. This ansatz
supports the representation of a larger, and potentially more useful, set of
states than the original formulation, with potential long-term impact on a
broad array of application areas. For cases that call for mixing only within a
desired subspace, refocusing on unitaries rather than Hamiltonians enables more
efficiently implementable mixers than was possible in the original framework.
Such mixers are particularly useful for optimization problems with hard
constraints that must always be satisfied, defining a feasible subspace, and
soft constraints whose violation we wish to minimize. More efficient
implementation enables earlier experimental exploration of an alternating
operator approach to a wide variety of approximate optimization, exact
optimization, and sampling problems. Here, we introduce the Quantum Alternating
Operator Ansatz, lay out design criteria for mixing operators, detail mappings
for eight problems, and provide brief descriptions of mappings for diverse
problems.Comment: 51 pages, 2 figures. Revised to match journal pape
The Scaling Window of the 2-SAT Transition
We consider the random 2-satisfiability problem, in which each instance is a
formula that is the conjunction of m clauses of the form (x or y), chosen
uniformly at random from among all 2-clauses on n Boolean variables and their
negations. As m and n tend to infinity in the ratio m/n --> alpha, the problem
is known to have a phase transition at alpha_c = 1, below which the probability
that the formula is satisfiable tends to one and above which it tends to zero.
We determine the finite-size scaling about this transition, namely the scaling
of the maximal window W(n,delta) = (alpha_-(n,delta),alpha_+(n,delta)) such
that the probability of satisfiability is greater than 1-delta for alpha <
alpha_- and is less than delta for alpha > alpha_+. We show that
W(n,delta)=(1-Theta(n^{-1/3}),1+Theta(n^{-1/3})), where the constants implicit
in Theta depend on delta. We also determine the rates at which the probability
of satisfiability approaches one and zero at the boundaries of the window.
Namely, for m=(1+epsilon)n, where epsilon may depend on n as long as |epsilon|
is sufficiently small and |epsilon|*n^(1/3) is sufficiently large, we show that
the probability of satisfiability decays like exp(-Theta(n*epsilon^3)) above
the window, and goes to one like 1-Theta(1/(n*|epsilon|^3)) below the window.
We prove these results by defining an order parameter for the transition and
establishing its scaling behavior in n both inside and outside the window.
Using this order parameter, we prove that the 2-SAT phase transition is
continuous with an order parameter critical exponent of 1. We also determine
the values of two other critical exponents, showing that the exponents of 2-SAT
are identical to those of the random graph.Comment: 57 pages. This version updates some reference
Three Phase Relative Permeability Models for WAG Simulation
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Near Optimal Exploration-Exploitation in Non-Communicating Markov Decision Processes
While designing the state space of an MDP, it is common to include states
that are transient or not reachable by any policy (e.g., in mountain car, the
product space of speed and position contains configurations that are not
physically reachable). This leads to defining weakly-communicating or
multi-chain MDPs. In this paper, we introduce \tucrl, the first algorithm able
to perform efficient exploration-exploitation in any finite Markov Decision
Process (MDP) without requiring any form of prior knowledge. In particular, for
any MDP with communicating states, actions and
possible communicating next states,
we derive a regret bound, where is the diameter
(i.e., the longest shortest path) of the communicating part of the MDP. This is
in contrast with optimistic algorithms (e.g., UCRL, Optimistic PSRL) that
suffer linear regret in weakly-communicating MDPs, as well as posterior
sampling or regularised algorithms (e.g., REGAL), which require prior knowledge
on the bias span of the optimal policy to bias the exploration to achieve
sub-linear regret. We also prove that in weakly-communicating MDPs, no
algorithm can ever achieve a logarithmic growth of the regret without first
suffering a linear regret for a number of steps that is exponential in the
parameters of the MDP. Finally, we report numerical simulations supporting our
theoretical findings and showing how TUCRL overcomes the limitations of the
state-of-the-art
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