7,724 research outputs found
Approximating Weighted Duo-Preservation in Comparative Genomics
Motivated by comparative genomics, Chen et al. [9] introduced the Maximum
Duo-preservation String Mapping (MDSM) problem in which we are given two
strings and from the same alphabet and the goal is to find a
mapping between them so as to maximize the number of duos preserved. A
duo is any two consecutive characters in a string and it is preserved in the
mapping if its two consecutive characters in are mapped to same two
consecutive characters in . The MDSM problem is known to be NP-hard and
there are approximation algorithms for this problem [3, 5, 13], but all of them
consider only the "unweighted" version of the problem in the sense that a duo
from is preserved by mapping to any same duo in regardless of their
positions in the respective strings. However, it is well-desired in comparative
genomics to find mappings that consider preserving duos that are "closer" to
each other under some distance measure [19]. In this paper, we introduce a
generalized version of the problem, called the Maximum-Weight Duo-preservation
String Mapping (MWDSM) problem that captures both duos-preservation and
duos-distance measures in the sense that mapping a duo from to each
preserved duo in has a weight, indicating the "closeness" of the two
duos. The objective of the MWDSM problem is to find a mapping so as to maximize
the total weight of preserved duos. In this paper, we give a polynomial-time
6-approximation algorithm for this problem.Comment: Appeared in proceedings of the 23rd International Computing and
Combinatorics Conference (COCOON 2017
Dynamics of quantum adiabatic evolution algorithm for Number Partitioning
We have developed a general technique to study the dynamics of the quantum
adiabatic evolution algorithm applied to random combinatorial optimization
problems in the asymptotic limit of large problem size . We use as an
example the NP-complete Number Partitioning problem and map the algorithm
dynamics to that of an auxilary quantum spin glass system with the slowly
varying Hamiltonian. We use a Green function method to obtain the adiabatic
eigenstates and the minimum excitation gap, ,
corresponding to the exponential complexity of the algorithm for Number
Partitioning. The key element of the analysis is the conditional energy
distribution computed for the set of all spin configurations generated from a
given (ancestor) configuration by simulteneous fipping of a fixed number of
spins. For the problem in question this distribution is shown to depend on the
ancestor spin configuration only via a certain parameter related to the energy
of the configuration. As the result, the algorithm dynamics can be described in
terms of one-dimenssional quantum diffusion in the energy space. This effect
provides a general limitation on the power of a quantum adiabatic computation
in random optimization problems. Analytical results are in agreement with the
numerical simulation of the algorithm.Comment: 32 pages, 5 figures, 3 Appendices; List of additions compare to v.3:
(i) numerical solution of the stationary Schroedinger equation for the
adiabatic eigenstates and eigenvalues; (ii) connection between the scaling
law of the minimum gap with the problem size and the shape of the
coarse-grained distribution of the adiabatic eigenvalues at the
avoided-crossing poin
Narrow sieves for parameterized paths and packings
We present randomized algorithms for some well-studied, hard combinatorial
problems: the k-path problem, the p-packing of q-sets problem, and the
q-dimensional p-matching problem. Our algorithms solve these problems with high
probability in time exponential only in the parameter (k, p, q) and using
polynomial space; the constant bases of the exponentials are significantly
smaller than in previous works. For example, for the k-path problem the
improvement is from 2 to 1.66. We also show how to detect if a d-regular graph
admits an edge coloring with colors in time within a polynomial factor of
O(2^{(d-1)n/2}).
Our techniques build upon and generalize some recently published ideas by I.
Koutis (ICALP 2009), R. Williams (IPL 2009), and A. Bj\"orklund (STACS 2010,
FOCS 2010)
Minimum Common String Partition: Exact Algorithms
In the minimum common string partition problem (MCSP), one gets two strings and is asked to find the minimum number of cuts in the first string such that the second string can be obtained by rearranging the resulting pieces. It is a difficult algorithmic problem having applications in computational biology, text processing, and data compression. MCSP has been studied extensively from various algorithmic angles: there are many papers studying approximation, heuristic, and parameterized algorithms. At the same time, almost nothing is known about its exact complexity. In this paper, we present new results in this direction. We improve the known 2? upper bound (where n is the length of input strings) to ?? where ? ? 1.618... is the golden ratio. The algorithm uses Fibonacci numbers to encode subsets as monomials of a certain implicit polynomial and extracts one of its coefficients using the fast Fourier transform. Then, we show that the case of constant size alphabet can be solved in subexponential time 2^{O(nlog log n/log n)} by a hybrid strategy: enumerate all long pieces and use dynamic programming over histograms of short pieces. Finally, we prove almost matching lower bounds assuming the Exponential Time Hypothesis
The Graph Motif problem parameterized by the structure of the input graph
The Graph Motif problem was introduced in 2006 in the context of biological
networks. It consists of deciding whether or not a multiset of colors occurs in
a connected subgraph of a vertex-colored graph. Graph Motif has been mostly
analyzed from the standpoint of parameterized complexity. The main parameters
which came into consideration were the size of the multiset and the number of
colors. Though, in the many applications of Graph Motif, the input graph
originates from real-life and has structure. Motivated by this prosaic
observation, we systematically study its complexity relatively to graph
structural parameters. For a wide range of parameters, we give new or improved
FPT algorithms, or show that the problem remains intractable. For the FPT
cases, we also give some kernelization lower bounds as well as some ETH-based
lower bounds on the worst case running time. Interestingly, we establish that
Graph Motif is W[1]-hard (while in W[P]) for parameter max leaf number, which
is, to the best of our knowledge, the first problem to behave this way.Comment: 24 pages, accepted in DAM, conference version in IPEC 201
Inapproximability of Combinatorial Optimization Problems
We survey results on the hardness of approximating combinatorial optimization
problems
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