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

Degree-Constrained Orientation of Maximum Satisfaction: Graph Classes and Parameterized Complexity

The problem Max W-Light (Max W-Heavy) for an undirected graph is to assign a direction to each edge so that the number of vertices of outdegree at most W (resp. at least W) is maximized. It is known that these problems are NP-hard even for fixed W. For example, Max 0-Light is equivalent to the problem of finding a maximum independent set. In this paper, we show that for any fixed constant W, Max W-Heavy can be solved in linear time for hereditary graph classes for which treewidth is bounded by a function of degeneracy. We show that such graph classes include chordal graphs, circular-arc graphs, d-trapezoid graphs, chordal bipartite graphs, and graphs of bounded clique-width. To have a polynomial-time algorithm for Max W-Light, we need an additional condition of a polynomial upper bound on the number of potential maximal cliques to apply the metatheorem by Fomin, Todinca, and Villanger [SIAM J. Comput., 44(1):57-87, 2015]. The aforementioned graph classes, except bounded clique-width graphs, satisfy such a condition. For graphs of bounded clique-width, we present a dynamic programming approach not using the metatheorem to show that it is actually polynomial-time solvable for this graph class too. We also study the parameterized complexity of the problems and show some tractability and intractability results

Combining weak and strong lensing in cluster potential reconstruction

We propose a method for recovering the two-dimensional gravitational potential of galaxy clusters which combines data from weak and strong gravitational lensing. A first estimate of the potential from weak lensing is improved at the approximate locations of critical curves. The method can be fully linearised and does not rely on the existence and identification of multiple images. We use simulations to show that it recovers the surface-mass density profiles and distributions very accurately, even if critical curves are only partially known and if their location is realistically uncertain. We further describe how arcs at different redshifts can be combined, and how deviations from weak lensing can be included.Comment: 9 pages, 5 figures, A&A in press, changes to match the accepted versio

Constraints on Crustal Stress from Coseismic Slip Models and Focal Mechanisms.

Constraining crustal stress that leads to earthquakes is an active area of research with profound implications on understanding the forces that deform the surface of the earth and generate slip on faults. Surface deformation related to strain accumulation on faults prior to, during and following earthquakes are recorded geodetically (InSAR and GPS). These data are used to infer fault geometries and models of coseismic slip of an earthquake. Seismic energy radiated during earthquakes are used to produce focal mechanisms, which are geometric representations of faults, and provide insight on stress changes due to earthquakes. However, earthquakes are the response to stress accumulation on faults, but direct measurements of accumulated stress are difficult. In this dissertation, I develop, test, and apply a Bayesian Monte Carlo (BMC) estimation technique to infer crustal stress from both focal mechanisms and coseismic slip models, the latter of which has never been done prior to the work I present here. I apply the BMC method to investigate stresses leading to the 2008 Wenchuan, China, earthquake, and to the 1999 İzmit and Düzce, Turkey, earthquakes. I use various coseismic slip models from all three events, aftershock focal mechanisms of the Wenchuan earthquake, and seismicity recorded in the Sea of Marmara, adjacent to the İzmit earthquake. I find that a homogeneous stress is statistically consistent with slip during the Wenchuan earthquake, and that heterogeneous stresses along the trace of the mainshock, previously argued for based on aftershock focal mechanisms, may simply reflect ambiguities in the interpretation of stress from focal mechanisms. Coseismic slip models from the İzmit and Düzce earthquakes are also consistent with a homogeneous stress along all fault segments that slipped in those earthquakes, particularly if the coefficient of fault friction is about 0.2 or less. In the Sea of Marmara, inferred stresses from focal mechanisms indicate that stress differs from the eastern to the western segments of the Main Marmara fault. Additionally, results indicate a potential stress rotation along the western segment between about 1999 and 2003, towards a transform stress regime similar to the stress leading to the İzmit and Düzce earthquakes.PHDGeologyUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/111551/1/lmedina_1.pd