3,474 research outputs found
Tetris is Hard, Even to Approximate
In the popular computer game of Tetris, the player is given a sequence of
tetromino pieces and must pack them into a rectangular gameboard initially
occupied by a given configuration of filled squares; any completely filled row
of the gameboard is cleared and all pieces above it drop by one row. We prove
that in the offline version of Tetris, it is NP-complete to maximize the number
of cleared rows, maximize the number of tetrises (quadruples of rows
simultaneously filled and cleared), minimize the maximum height of an occupied
square, or maximize the number of pieces placed before the game ends. We
furthermore show the extreme inapproximability of the first and last of these
objectives to within a factor of p^(1-epsilon), when given a sequence of p
pieces, and the inapproximability of the third objective to within a factor of
(2 - epsilon), for any epsilon>0. Our results hold under several variations on
the rules of Tetris, including different models of rotation, limitations on
player agility, and restricted piece sets.Comment: 56 pages, 11 figure
Quantum annealing with Jarzynski equality
We show a practical application of the Jarzynski equality in quantum
computation. Its implementation may open a way to solve combinatorial
optimization problems, minimization of a real single-valued function, cost
function, with many arguments. We consider to incorpolate the Jarzynski
equality into quantum annealing, which is one of the generic algorithms to
solve the combinatorial optimization problem. The ordinary quantum annealing
suffers from non-adiabatic transitions whose rate is characterized by the
minimum energy gap of the quantum system under
consideration. The quantum sweep speed is therefore restricted to be extremely
slow for the achievement to obtain a solution without relevant errors. However,
in our strategy shown in the present study, we find that such a difficulty
would not matter.Comment: 4 pages, to appear in Phys. Rev. Let
Low-Degree Spanning Trees of Small Weight
The degree-d spanning tree problem asks for a minimum-weight spanning tree in
which the degree of each vertex is at most d. When d=2 the problem is TSP, and
in this case, the well-known Christofides algorithm provides a
1.5-approximation algorithm (assuming the edge weights satisfy the triangle
inequality).
In 1984, Christos Papadimitriou and Umesh Vazirani posed the challenge of
finding an algorithm with performance guarantee less than 2 for Euclidean
graphs (points in R^n) and d > 2. This paper gives the first answer to that
challenge, presenting an algorithm to compute a degree-3 spanning tree of cost
at most 5/3 times the MST. For points in the plane, the ratio improves to 3/2
and the algorithm can also find a degree-4 spanning tree of cost at most 5/4
times the MST.Comment: conference version in Symposium on Theory of Computing (1994
Computational Difficulty of Global Variations in the Density Matrix Renormalization Group
The density matrix renormalization group (DMRG) approach is arguably the most
successful method to numerically find ground states of quantum spin chains. It
amounts to iteratively locally optimizing matrix-product states, aiming at
better and better approximating the true ground state. To date, both a proof of
convergence to the globally best approximation and an assessment of its
complexity are lacking. Here we establish a result on the computational
complexity of an approximation with matrix-product states: The surprising
result is that when one globally optimizes over several sites of local
Hamiltonians, avoiding local optima, one encounters in the worst case a
computationally difficult NP-hard problem (hard even in approximation). The
proof exploits a novel way of relating it to binary quadratic programming. We
discuss intriguing ramifications on the difficulty of describing quantum
many-body systems.Comment: 5 pages, 1 figure, RevTeX, final versio
NP-hardness of the cluster minimization problem revisited
The computational complexity of the "cluster minimization problem" is
revisited [L. T. Wille and J. Vennik, J. Phys. A 18, L419 (1985)]. It is argued
that the original NP-hardness proof does not apply to pairwise potentials of
physical interest, such as those that depend on the geometric distance between
the particles. A geometric analog of the original problem is formulated, and a
new proof for such potentials is provided by polynomial time transformation
from the independent set problem for unit disk graphs. Limitations of this
formulation are pointed out, and new subproblems that bear more direct
consequences to the numerical study of clusters are suggested.Comment: 8 pages, 2 figures, accepted to J. Phys. A: Math. and Ge
Quantum annealing with antiferromagnetic fluctuations
We introduce antiferromagnetic quantum fluctuations into quantum annealing in
addition to the conventional transverse-field term. We apply this method to the
infinite-range ferromagnetic p-spin model, for which the conventional quantum
annealing has been shown to have difficulties to find the ground state
efficiently due to a first-order transition. We study the phase diagram of this
system both analytically and numerically. Using the static approximation, we
find that there exists a quantum path to reach the final ground state from the
trivial initial state that avoids first-order transitions for intermediate
values of p. We also study numerically the energy gap between the ground state
and the first excited state and find evidence for intermediate values of p that
the time complexity scales polynomially with the system size at a second-order
transition point along the quantum path that avoids first-order transitions.
These results suggest that quantum annealing would be able to solve this
problem with intermediate values of p efficiently in contrast to the case with
only simple transverse-field fluctuations.Comment: 19 pages, 11 figures; Added references; To be published in Physical
Review
Edge Elimination in TSP Instances
The Traveling Salesman Problem is one of the best studied NP-hard problems in
combinatorial optimization. Powerful methods have been developed over the last
60 years to find optimum solutions to large TSP instances. The largest TSP
instance so far that has been solved optimally has 85,900 vertices. Its
solution required more than 136 years of total CPU time using the
branch-and-cut based Concorde TSP code [1]. In this paper we present graph
theoretic results that allow to prove that some edges of a TSP instance cannot
occur in any optimum TSP tour. Based on these results we propose a
combinatorial algorithm to identify such edges. The runtime of the main part of
our algorithm is for an n-vertex TSP instance. By combining our
approach with the Concorde TSP solver we are able to solve a large TSPLIB
instance more than 11 times faster than Concorde alone
Sequencing and analysis of the gastrula transcriptome of the brittle star Ophiocoma wendtii
Background
The gastrula stage represents the point in development at which the three primary germ layers diverge. At this point the gene regulatory networks that specify the germ layers are established and the genes that define the differentiated states of the tissues have begun to be activated. These networks have been well-characterized in sea urchins, but not in other echinoderms. Embryos of the brittle star Ophiocoma wendtii share a number of developmental features with sea urchin embryos, including the ingression of mesenchyme cells that give rise to an embryonic skeleton. Notable differences are that no micromeres are formed during cleavage divisions and no pigment cells are formed during development to the pluteus larval stage. More subtle changes in timing of developmental events also occur. To explore the molecular basis for the similarities and differences between these two echinoderms, we have sequenced and characterized the gastrula transcriptome of O. wendtii. Methods
Development of Ophiocoma wendtii embryos was characterized and RNA was isolated from the gastrula stage. A transcriptome data base was generated from this RNA and was analyzed using a variety of methods to identify transcripts expressed and to compare those transcripts to those expressed at the gastrula stage in other organisms. Results
Using existing databases, we identified brittle star transcripts that correspond to 3,385 genes, including 1,863 genes shared with the sea urchin Strongylocentrotus purpuratus gastrula transcriptome. We characterized the functional classes of genes present in the transcriptome and compared them to those found in this sea urchin. We then examined those members of the germ-layer specific gene regulatory networks (GRNs) of S. purpuratus that are expressed in the O. wendtii gastrula. Our results indicate that there is a shared ‘genetic toolkit’ central to the echinoderm gastrula, a key stage in embryonic development, though there are also differences that reflect changes in developmental processes. Conclusions
The brittle star expresses genes representing all functional classes at the gastrula stage. Brittle stars and sea urchins have comparable numbers of each class of genes and share many of the genes expressed at gastrulation. Examination of the brittle star genes in which sea urchin orthologs are utilized in germ layer specification reveals a relatively higher level of conservation of key regulatory components compared to the overall transcriptome. We also identify genes that were either lost or whose temporal expression has diverged from that of sea urchins
Quantum Algorithm to Solve Satisfiability Problems
A new quantum algorithm is proposed to solve Satisfiability(SAT) problems by
taking advantage of non-unitary transformation in ground state quantum
computer. The energy gap scale of the ground state quantum computer is analyzed
for 3-bit Exact Cover problems. The time cost of this algorithm on general SAT
problems is discussed.Comment: 5 pages, 3 figure
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