2,899 research outputs found
Computing Tutte Paths
Tutte paths are one of the most successful tools for attacking problems on long cycles in planar graphs. Unfortunately, results based on them are non-constructive, as their proofs inherently use an induction on overlapping subgraphs and these overlaps prevent any attempt to bound the running time by a polynomial.
For special cases however, computational results of Tutte paths are known: For 4-connected planar graphs, Tutte paths are in fact Hamiltonian paths and Chiba and Nishizeki [N. Chiba and T. Nishizeki, 1989] showed how to compute such paths in linear time. For 3-connected planar graphs, Tutte paths have a significantly more complicated structure, and it has only recently been shown that they can be computed in polynomial time [A. Schmid and J. M. Schmidt, 2015]. However, Tutte paths are defined for general 2-connected planar graphs and this is what most applications need. In this unrestricted setting, no computational results for Tutte paths are known.
We give the first efficient algorithm that computes a Tutte path (in this unrestricted setting). One of the strongest existence results about such Tutte paths is due to Sanders [D. P. Sanders, 1997], which allows one to prescribe the end vertices and an intermediate edge of the desired path. Encompassing and strengthening all previous computational results on Tutte paths, we show how to compute such a special Tutte path efficiently. Our method refines both, the existence results of Thomassen [C. Thomassen, 1983] and Sanders [D. P. Sanders, 1997], and avoids that the subgraphs arising in the inductive proof intersect in more than one edge by using a novel iterative decomposition along 2-separators. Finally, we show that our algorithm runs in time O(n^2)
Lattice path matroids: enumerative aspects and Tutte polynomials
Fix two lattice paths P and Q from (0,0) to (m,r) that use East and North
steps with P never going above Q. We show that the lattice paths that go from
(0,0) to (m,r) and that remain in the region bounded by P and Q can be
identified with the bases of a particular type of transversal matroid, which we
call a lattice path matroid. We consider a variety of enumerative aspects of
these matroids and we study three important matroid invariants, namely the
Tutte polynomial and, for special types of lattice path matroids, the
characteristic polynomial and the beta invariant. In particular, we show that
the Tutte polynomial is the generating function for two basic lattice path
statistics and we show that certain sequences of lattice path matroids give
rise to sequences of Tutte polynomials for which there are relatively simple
generating functions. We show that Tutte polynomials of lattice path matroids
can be computed in polynomial time. Also, we obtain a new result about lattice
paths from an analysis of the beta invariant of certain lattice path matroids.Comment: 28 pages, 11 figure
A solution to the tennis ball problem
We present a complete solution to the so-called tennis ball problem, which is
equivalent to counting lattice paths in the plane that use North and East steps
and lie between certain boundaries. The solution takes the form of explicit
expressions for the corresponding generating functions. Our method is based on
the properties of Tutte polynomials of matroids associated to lattice paths. We
also show how the same method provides a solution to a wide generalization of
the problem.Comment: 9 pages, Late
Exponential Time Complexity of the Permanent and the Tutte Polynomial
We show conditional lower bounds for well-studied #P-hard problems:
(a) The number of satisfying assignments of a 2-CNF formula with n variables
cannot be counted in time exp(o(n)), and the same is true for computing the
number of all independent sets in an n-vertex graph.
(b) The permanent of an n x n matrix with entries 0 and 1 cannot be computed
in time exp(o(n)).
(c) The Tutte polynomial of an n-vertex multigraph cannot be computed in time
exp(o(n)) at most evaluation points (x,y) in the case of multigraphs, and it
cannot be computed in time exp(o(n/polylog n)) in the case of simple graphs.
Our lower bounds are relative to (variants of) the Exponential Time
Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF
formulas cannot be decided in time exp(o(n)). We relax this hypothesis by
introducing its counting version #ETH, namely that the satisfying assignments
cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds,
we transfer the sparsification lemma for d-CNF formulas to the counting
setting
Multi-core computation of transfer matrices for strip lattices in the Potts model
The transfer-matrix technique is a convenient way for studying strip lattices
in the Potts model since the compu- tational costs depend just on the periodic
part of the lattice and not on the whole. However, even when the cost is
reduced, the transfer-matrix technique is still an NP-hard problem since the
time T(|V|, |E|) needed to compute the matrix grows ex- ponentially as a
function of the graph width. In this work, we present a parallel
transfer-matrix implementation that scales performance under multi-core
architectures. The construction of the matrix is based on several repetitions
of the deletion- contraction technique, allowing parallelism suitable to
multi-core machines. Our experimental results show that the multi-core
implementation achieves speedups of 3.7X with p = 4 processors and 5.7X with p
= 8. The efficiency of the implementation lies between 60% and 95%, achieving
the best balance of speedup and efficiency at p = 4 processors for actual
multi-core architectures. The algorithm also takes advantage of the lattice
symmetry, making the transfer matrix computation to run up to 2X faster than
its non-symmetric counterpart and use up to a quarter of the original space
- …