237,536 research outputs found
Solving Hard Computational Problems Efficiently: Asymptotic Parametric Complexity 3-Coloring Algorithm
Many practical problems in almost all scientific and technological
disciplines have been classified as computationally hard (NP-hard or even
NP-complete). In life sciences, combinatorial optimization problems frequently
arise in molecular biology, e.g., genome sequencing; global alignment of
multiple genomes; identifying siblings or discovery of dysregulated pathways.In
almost all of these problems, there is the need for proving a hypothesis about
certain property of an object that can be present only when it adopts some
particular admissible structure (an NP-certificate) or be absent (no admissible
structure), however, none of the standard approaches can discard the hypothesis
when no solution can be found, since none can provide a proof that there is no
admissible structure. This article presents an algorithm that introduces a
novel type of solution method to "efficiently" solve the graph 3-coloring
problem; an NP-complete problem. The proposed method provides certificates
(proofs) in both cases: present or absent, so it is possible to accept or
reject the hypothesis on the basis of a rigorous proof. It provides exact
solutions and is polynomial-time (i.e., efficient) however parametric. The only
requirement is sufficient computational power, which is controlled by the
parameter . Nevertheless, here it is proved that the
probability of requiring a value of to obtain a solution for a
random graph decreases exponentially: , making
tractable almost all problem instances. Thorough experimental analyses were
performed. The algorithm was tested on random graphs, planar graphs and
4-regular planar graphs. The obtained experimental results are in accordance
with the theoretical expected results.Comment: Working pape
Solving the Shortest Vector Problem in Lattices Faster Using Quantum Search
By applying Grover's quantum search algorithm to the lattice algorithms of
Micciancio and Voulgaris, Nguyen and Vidick, Wang et al., and Pujol and
Stehl\'{e}, we obtain improved asymptotic quantum results for solving the
shortest vector problem. With quantum computers we can provably find a shortest
vector in time , improving upon the classical time
complexity of of Pujol and Stehl\'{e} and the of Micciancio and Voulgaris, while heuristically we expect to find a
shortest vector in time , improving upon the classical time
complexity of of Wang et al. These quantum complexities
will be an important guide for the selection of parameters for post-quantum
cryptosystems based on the hardness of the shortest vector problem.Comment: 19 page
A Simple Model to Generate Hard Satisfiable Instances
In this paper, we try to further demonstrate that the models of random CSP
instances proposed by [Xu and Li, 2000; 2003] are of theoretical and practical
interest. Indeed, these models, called RB and RD, present several nice
features. First, it is quite easy to generate random instances of any arity
since no particular structure has to be integrated, or property enforced, in
such instances. Then, the existence of an asymptotic phase transition can be
guaranteed while applying a limited restriction on domain size and on
constraint tightness. In that case, a threshold point can be precisely located
and all instances have the guarantee to be hard at the threshold, i.e., to have
an exponential tree-resolution complexity. Next, a formal analysis shows that
it is possible to generate forced satisfiable instances whose hardness is
similar to unforced satisfiable ones. This analysis is supported by some
representative results taken from an intensive experimentation that we have
carried out, using complete and incomplete search methods.Comment: Proc. of 19th IJCAI, pp.337-342, Edinburgh, Scotland, 2005. For more
information, please click
http://www.nlsde.buaa.edu.cn/~kexu/papers/ijcai05-abstract.ht
Probing nonlinear adiabatic paths with a universal integrator
We apply a flexible numerical integrator to the simulation of adiabatic
quantum computation with nonlinear paths. We find that a nonlinear path may
significantly improve the performance of adiabatic algorithms versus the
conventional straight-line interpolations. The employed integrator is suitable
for solving the time-dependent Schr\"odinger equation for any qubit
Hamiltonian. Its flexible storage format significantly reduces cost for storage
and matrix-vector multiplication in comparison to common sparse matrix schemes.Comment: 8 pages, 6 figure
On the Lattice Isomorphism Problem
We study the Lattice Isomorphism Problem (LIP), in which given two lattices
L_1 and L_2 the goal is to decide whether there exists an orthogonal linear
transformation mapping L_1 to L_2. Our main result is an algorithm for this
problem running in time n^{O(n)} times a polynomial in the input size, where n
is the rank of the input lattices. A crucial component is a new generalized
isolation lemma, which can isolate n linearly independent vectors in a given
subset of Z^n and might be useful elsewhere. We also prove that LIP lies in the
complexity class SZK.Comment: 23 pages, SODA 201
Heuristic average-case analysis of the backtrack resolution of random 3-Satisfiability instances
An analysis of the average-case complexity of solving random 3-Satisfiability
(SAT) instances with backtrack algorithms is presented. We first interpret
previous rigorous works in a unifying framework based on the statistical
physics notions of dynamical trajectories, phase diagram and growth process. It
is argued that, under the action of the Davis--Putnam--Loveland--Logemann
(DPLL) algorithm, 3-SAT instances are turned into 2+p-SAT instances whose
characteristic parameters (ratio alpha of clauses per variable, fraction p of
3-clauses) can be followed during the operation, and define resolution
trajectories. Depending on the location of trajectories in the phase diagram of
the 2+p-SAT model, easy (polynomial) or hard (exponential) resolutions are
generated. Three regimes are identified, depending on the ratio alpha of the
3-SAT instance to be solved. Lower sat phase: for small ratios, DPLL almost
surely finds a solution in a time growing linearly with the number N of
variables. Upper sat phase: for intermediate ratios, instances are almost
surely satisfiable but finding a solution requires exponential time (2 ^ (N
omega) with omega>0) with high probability. Unsat phase: for large ratios,
there is almost always no solution and proofs of refutation are exponential. An
analysis of the growth of the search tree in both upper sat and unsat regimes
is presented, and allows us to estimate omega as a function of alpha. This
analysis is based on an exact relationship between the average size of the
search tree and the powers of the evolution operator encoding the elementary
steps of the search heuristic.Comment: to appear in Theoretical Computer Scienc
On optimizing over lift-and-project closures
The lift-and-project closure is the relaxation obtained by computing all
lift-and-project cuts from the initial formulation of a mixed integer linear
program or equivalently by computing all mixed integer Gomory cuts read from
all tableau's corresponding to feasible and infeasible bases. In this paper, we
present an algorithm for approximating the value of the lift-and-project
closure. The originality of our method is that it is based on a very simple cut
generation linear programming problem which is obtained from the original
linear relaxation by simply modifying the bounds on the variables and
constraints. This separation LP can also be seen as the dual of the cut
generation LP used in disjunctive programming procedures with a particular
normalization. We study some properties of this separation LP in particular
relating it to the equivalence between lift-and-project cuts and Gomory cuts
shown by Balas and Perregaard. Finally, we present some computational
experiments and comparisons with recent related works
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