2,034 research outputs found
Boolean dimension and tree-width
The dimension is a key measure of complexity of partially ordered sets. Small
dimension allows succinct encoding. Indeed if has dimension , then to
know whether in it is enough to check whether in each
of the linear extensions of a witnessing realizer. Focusing on the encoding
aspect Ne\v{s}et\v{r}il and Pudl\'{a}k defined a more expressive version of
dimension. A poset has boolean dimension at most if it is possible to
decide whether in by looking at the relative position of and
in only permutations of the elements of . We prove that posets with
cover graphs of bounded tree-width have bounded boolean dimension. This stays
in contrast with the fact that there are posets with cover graphs of tree-width
three and arbitrarily large dimension. This result might be a step towards a
resolution of the long-standing open problem: Do planar posets have bounded
boolean dimension?Comment: one more reference added; paper revised along the suggestion of three
reviewer
Trading Determinism for Time in Space Bounded Computations
Savitch showed in that nondeterministic logspace (NL) is contained in
deterministic space but his algorithm requires
quasipolynomial time. The question whether we can have a deterministic
algorithm for every problem in NL that requires polylogarithmic space and
simultaneously runs in polynomial time was left open.
In this paper we give a partial solution to this problem and show that for
every language in NL there exists an unambiguous nondeterministic algorithm
that requires space and simultaneously runs in
polynomial time.Comment: Accepted in MFCS 201
Space Saving by Dynamic Algebraization
Dynamic programming is widely used for exact computations based on tree
decompositions of graphs. However, the space complexity is usually exponential
in the treewidth. We study the problem of designing efficient dynamic
programming algorithm based on tree decompositions in polynomial space. We show
how to construct a tree decomposition and extend the algebraic techniques of
Lokshtanov and Nederlof such that the dynamic programming algorithm runs in
time , where is the maximum number of vertices in the union of
bags on the root to leaf paths on a given tree decomposition, which is a
parameter closely related to the tree-depth of a graph. We apply our algorithm
to the problem of counting perfect matchings on grids and show that it
outperforms other polynomial-space solutions. We also apply the algorithm to
other set covering and partitioning problems.Comment: 14 pages, 1 figur
Solving weighted and counting variants of connectivity problems parameterized by treewidth deterministically in single exponential time
It is well known that many local graph problems, like Vertex Cover and
Dominating Set, can be solved in 2^{O(tw)}|V|^{O(1)} time for graphs G=(V,E)
with a given tree decomposition of width tw. However, for nonlocal problems,
like the fundamental class of connectivity problems, for a long time we did not
know how to do this faster than tw^{O(tw)}|V|^{O(1)}. Recently, Cygan et al.
(FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity
problems running in time $c^{tw}|V|^{O(1)} for a small constant c, e.g., for
Hamiltonian Cycle and Steiner tree. Naturally, this raises the question whether
randomization is necessary to achieve this runtime; furthermore, it is
desirable to also solve counting and weighted versions (the latter without
incurring a pseudo-polynomial cost in terms of the weights).
We present two new approaches rooted in linear algebra, based on matrix rank
and determinants, which provide deterministic c^{tw}|V|^{O(1)} time algorithms,
also for weighted and counting versions. For example, in this time we can solve
the traveling salesman problem or count the number of Hamiltonian cycles. The
rank-based ideas provide a rather general approach for speeding up even
straightforward dynamic programming formulations by identifying "small" sets of
representative partial solutions; we focus on the case of expressing
connectivity via sets of partitions, but the essential ideas should have
further applications. The determinant-based approach uses the matrix tree
theorem for deriving closed formulas for counting versions of connectivity
problems; we show how to evaluate those formulas via dynamic programming.Comment: 36 page
10481 Abstracts Collection -- Computational Counting
From November 28 to December 3 2010, the Dagstuhl Seminar 10481 ``Computational Counting\u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
On the performance of edge coloring algorithms for cubic graphs
This thesis visits the forefront of algorithmic research on edge coloring of cubic graphs. We select a set of algorithms that are among the asymptotically fastest known today. Each algorithm has exponential time complexity, owing to the NP-completeness of edge coloring, but their space complexities differ greatly. They are implemented in a popular high-level programming language to compare their performance on a set of real instances. We also explore ways to parallelize each of the algorithms and discuss what benefits and detriments those implementations hold
Search for the lepton-family-number nonconserving decay \mu -> e + \gamma
The MEGA experiment, which searched for the muon- and electron-number
violating decay \mu -> e + \gamma, is described. The spectrometer system, the
calibrations, the data taking procedures, the data analysis, and the
sensitivity of the experiment are discussed. The most stringent upper limit on
the branching ratio of \mu -> e + \gamma) < 1.2 x 10^{-11} was obtained
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