Approximating Upper Degree-Constrained Partial Orientations

In the Upper Degree-Constrained Partial Orientation problem we are given an undirected graph $G=(V,E)$, together with two degree constraint functions $d^-,d^+ : V \to \mathbb{N}$. The goal is to orient as many edges as possible, in such a way that for each vertex $v \in V$ the number of arcs entering $v$ is at most $d^-(v)$, whereas the number of arcs leaving $v$ is at most $d^+(v)$. This problem was introduced by Gabow [SODA'06], who proved it to be MAXSNP-hard (and thus APX-hard). In the same paper Gabow presented an LP-based iterative rounding $4/3$-approximation algorithm. Since the problem in question is a special case of the classic 3-Dimensional Matching, which in turn is a special case of the $k$-Set Packing problem, it is reasonable to ask whether recent improvements in approximation algorithms for the latter two problems [Cygan, FOCS'13; Sviridenko & Ward, ICALP'13] allow for an improved approximation for Upper Degree-Constrained Partial Orientation. We follow this line of reasoning and present a polynomial-time local search algorithm with approximation ratio $5/4+\varepsilon$. Our algorithm uses a combination of two types of rules: improving sets of bounded pathwidth from the recent $4/3+\varepsilon$-approximation algorithm for 3-Set Packing [Cygan, FOCS'13], and a simple rule tailor-made for the setting of partial orientations. In particular, we exploit the fact that one can check in polynomial time whether it is possible to orient all the edges of a given graph [Gy\'arf\'as & Frank, Combinatorics'76].Comment: 12 pages, 1 figur

Orientation-Constrained Rectangular Layouts

We construct partitions of rectangles into smaller rectangles from an input consisting of a planar dual graph of the layout together with restrictions on the orientations of edges and junctions of the layout. Such an orientation-constrained layout, if it exists, may be constructed in polynomial time, and all orientation-constrained layouts may be listed in polynomial time per layout.Comment: To appear at Algorithms and Data Structures Symposium, Banff, Canada, August 2009. 12 pages, 5 figure

A Family of Quasisymmetry Models

We present a one-parameter family of models for square contingency tables that interpolates between the classical quasisymmetry model and its Pearsonian analogue. Algebraically, this corresponds to deformations of toric ideals associated with graphs. Our discussion of the statistical issues centers around maximum likelihood estimation.Comment: 17 pages, 10 figure

One-dimensional domain walls in thin ferromagnetic films with fourfold anisotropy

We study the properties of domain walls and domain patterns in ultrathin epitaxial magnetic films with two orthogonal in-plane easy axes, which we call fourfold materials. In these materials, the magnetization vector is constrained to lie entirely in the film plane and has four preferred directions dictated by the easy axes. We prove the existence of $90^\circ$ and $180^\circ$ domain walls in these materials as minimizers of a nonlocal one-dimensional energy functional. Further, we investigate numerically the role of the considered domain wall solutions for pattern formation in a rectangular sample.Comment: 20 pages, 4 figure

A Fast BCS/FCS Algorithm for Image Segmentation

A fast and efficient segmentation algorithm based on the Boundary Contour System/Feature Contour System (BCS/FCS) of Grossberg and Mingolla [3] is presented. This implementation is based on the FFT algorithm and the parallelism of the system.Consejo Nacional de Ciencia y Tecnología (63l462); Defense Advanced Research Projects Agency (AFOSR 90-0083); Office of Naval Research (N00014-92-J-l309