3,107 research outputs found
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
Approximately Sampling Elements with Fixed Rank in Graded Posets
Graded posets frequently arise throughout combinatorics, where it is natural
to try to count the number of elements of a fixed rank. These counting problems
are often -complete, so we consider approximation algorithms for
counting and uniform sampling. We show that for certain classes of posets,
biased Markov chains that walk along edges of their Hasse diagrams allow us to
approximately generate samples with any fixed rank in expected polynomial time.
Our arguments do not rely on the typical proofs of log-concavity, which are
used to construct a stationary distribution with a specific mode in order to
give a lower bound on the probability of outputting an element of the desired
rank. Instead, we infer this directly from bounds on the mixing time of the
chains through a method we call .
A noteworthy application of our method is sampling restricted classes of
integer partitions of . We give the first provably efficient Markov chain
algorithm to uniformly sample integer partitions of from general restricted
classes. Several observations allow us to improve the efficiency of this chain
to require space, and for unrestricted integer partitions,
expected time. Related applications include sampling permutations
with a fixed number of inversions and lozenge tilings on the triangular lattice
with a fixed average height.Comment: 23 pages, 12 figure
Instanton Expansion of Noncommutative Gauge Theory in Two Dimensions
We show that noncommutative gauge theory in two dimensions is an exactly
solvable model. A cohomological formulation of gauge theory defined on the
noncommutative torus is used to show that its quantum partition function can be
written as a sum over contributions from classical solutions. We derive an
explicit formula for the partition function of Yang-Mills theory defined on a
projective module for arbitrary noncommutativity parameter \theta which is
manifestly invariant under gauge Morita equivalence. The energy observables are
shown to be smooth functions of \theta. The construction of noncommutative
instanton contributions to the path integral is described in some detail. In
general, there are infinitely many gauge inequivalent contributions of fixed
topological charge, along with a finite number of quantum fluctuations about
each instanton. The associated moduli spaces are combinations of symmetric
products of an ordinary two-torus whose orbifold singularities are not resolved
by noncommutativity. In particular, the weak coupling limit of the gauge theory
is independent of \theta and computes the symplectic volume of the moduli space
of constant curvature connections on the noncommutative torus.Comment: 52 pages LaTeX, 1 eps figure, uses espf. V2: References added and
repaired; V3: Typos corrected, some clarifying explanations added; version to
be published in Communications in Mathematical Physic
Convex Rank Tests and Semigraphoids
Convex rank tests are partitions of the symmetric group which have desirable
geometric properties. The statistical tests defined by such partitions involve
counting all permutations in the equivalence classes. Each class consists of
the linear extensions of a partially ordered set specified by data. Our methods
refine existing rank tests of non-parametric statistics, such as the sign test
and the runs test, and are useful for exploratory analysis of ordinal data. We
establish a bijection between convex rank tests and probabilistic conditional
independence structures known as semigraphoids. The subclass of submodular rank
tests is derived from faces of the cone of submodular functions, or from
Minkowski summands of the permutohedron. We enumerate all small instances of
such rank tests. Of particular interest are graphical tests, which correspond
to both graphical models and to graph associahedra
On the diagram of 132-avoiding permutations
The diagram of a 132-avoiding permutation can easily be characterized: it is
simply the diagram of a partition. Based on this fact, we present a new
bijection between 132-avoiding and 321-avoiding permutations. We will show that
this bijection translates the correspondences between these permutations and
Dyck paths given by Krattenthaler and by Billey-Jockusch-Stanley, respectively,
to each other. Moreover, the diagram approach yields simple proofs for some
enumerative results concerning forbidden patterns in 132-avoiding permutations.Comment: 20 pages; additional reference is adde
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