All-Pairs Minimum Cuts in Near-Linear Time for Surface-Embedded Graphs

For an undirected $n$-vertex graph $G$ with non-negative edge-weights, we consider the following type of query: given two vertices $s$ and $t$ in $G$, what is the weight of a minimum $st$-cut in $G$? We solve this problem in preprocessing time $O(n\log^3 n)$ for graphs of bounded genus, giving the first sub-quadratic time algorithm for this class of graphs. Our result also improves by a logarithmic factor a previous algorithm by Borradaile, Sankowski and Wulff-Nilsen (FOCS 2010) that applied only to planar graphs. Our algorithm constructs a Gomory-Hu tree for the given graph, providing a data structure with space $O(n)$ that can answer minimum-cut queries in constant time. The dependence on the genus of the input graph in our preprocessing time is $2^{O(g^2)}$

Divide-and-conquer: Approaching the capacity of the two-pair bidirectional Gaussian relay network

The capacity region of multi-pair bidirectional relay networks, in which a relay node facilitates the communication between multiple pairs of users, is studied. This problem is first examined in the context of the linear shift deterministic channel model. The capacity region of this network when the relay is operating at either full-duplex mode or half-duplex mode for arbitrary number of pairs is characterized. It is shown that the cut-set upper-bound is tight and the capacity region is achieved by a so called divide-and-conquer relaying strategy. The insights gained from the deterministic network are then used for the Gaussian bidirectional relay network. The strategy in the deterministic channel translates to a specific superposition of lattice codes and random Gaussian codes at the source nodes and successive interference cancelation at the receiving nodes for the Gaussian network. The achievable rate of this scheme with two pairs is analyzed and it is shown that for all channel gains it achieves to within 3 bits/sec/Hz per user of the cut-set upper-bound. Hence, the capacity region of the two-pair bidirectional Gaussian relay network to within 3 bits/sec/Hz per user is characterized.Comment: IEEE Trans. on Information Theory, accepte

A JSJ-type decomposition theorem for symplectic fillings

Let $(M,\xi)$ be a contact 3-manifold and $T^2 \subset (M,\xi)$ a convex torus of a special type called a mixed torus. We prove a JSJ-type decomposition theorem for strong and exact symplectic fillings of $(M,\xi)$ when $(M,\xi)$ is cut along $T^2$. As an application we show the uniqueness of exact fillings when $(M,\xi)$ is obtained by Legendrian surgery on a knot in $(S^3,\xi_{std})$ when the knot is stabilized both positively and negatively.Comment: Version 2 clarifies some arguments and adds exposition. arXiv admin note: text overlap with arXiv:1204.3145 by other author

Accident Analysis and Prevention: Course Notes 1987/88

This report consists of the notes from a series of lectures given by the authors for a course entitled Accident Analysis and Prevention. The course took place during the second term of a one year Masters degree course in Transport Planning and Engineering run by the Institute for Transport Studies and the Department of Civil Engineering at the University of Leeds. The course consisted of 18 lectures of which 16 are reported on in this document (the remaining two, on Human Factors, are not reported on in this document as no notes were provided). Each lecture represents one chapter of this document, except in two instances where two lectures are covered in one chapter (Chapters 10 and 14). The course first took place in 1988, and at the date of publication has been run for a second time. This report contains the notes for the initial version of the course. A number of changes were made in the content and emphasis of the course during its second run, mainly due to a change of personnel, with different ideas and experiences in the field of accident analysis and prevention. It is likely that each time the course is run, there will be significant changes, but that the notes provided in this document can be considered to contain a number of the core elements of any future version of the course

Bell inequalities stronger than the CHSH inequality for 3-level isotropic states

We show that some two-party Bell inequalities with two-valued observables are stronger than the CHSH inequality for 3 \otimes 3 isotropic states in the sense that they are violated by some isotropic states in the 3 \otimes 3 system that do not violate the CHSH inequality. These Bell inequalities are obtained by applying triangular elimination to the list of known facet inequalities of the cut polytope on nine points. This gives a partial solution to an open problem posed by Collins and Gisin. The results of numerical optimization suggest that they are candidates for being stronger than the I_3322 Bell inequality for 3 \otimes 3 isotropic states. On the other hand, we found no Bell inequalities stronger than the CHSH inequality for 2 \otimes 2 isotropic states. In addition, we illustrate an inclusion relation among some Bell inequalities derived by triangular elimination.Comment: 9 pages, 1 figure. v2: organization improved; less references to the cut polytope to make the main results clear; references added; typos corrected; typesetting style change

Linear transformation distance for bichromatic matchings

Let $P=B\cup R$ be a set of $2n$ points in general position, where $B$ is a set of $n$ blue points and $R$ a set of $n$ red points. A \emph{$BR$-matching} is a plane geometric perfect matching on $P$ such that each edge has one red endpoint and one blue endpoint. Two $BR$-matchings are compatible if their union is also plane. The \emph{transformation graph of $BR$-matchings} contains one node for each $BR$-matching and an edge joining two such nodes if and only if the corresponding two $BR$-matchings are compatible. In SoCG 2013 it has been shown by Aloupis, Barba, Langerman, and Souvaine that this transformation graph is always connected, but its diameter remained an open question. In this paper we provide an alternative proof for the connectivity of the transformation graph and prove an upper bound of $2n$ for its diameter, which is asymptotically tight

Algorithm engineering for optimal alignment of protein structure distance matrices

Protein structural alignment is an important problem in computational biology. In this paper, we present first successes on provably optimal pairwise alignment of protein inter-residue distance matrices, using the popular Dali scoring function. We introduce the structural alignment problem formally, which enables us to express a variety of scoring functions used in previous work as special cases in a unified framework. Further, we propose the first mathematical model for computing optimal structural alignments based on dense inter-residue distance matrices. We therefore reformulate the problem as a special graph problem and give a tight integer linear programming model. We then present algorithm engineering techniques to handle the huge integer linear programs of real-life distance matrix alignment problems. Applying these techniques, we can compute provably optimal Dali alignments for the very first time

Linear Network Coding for Two-Unicast-ZZ Networks: A Commutative Algebraic Perspective and Fundamental Limits

We consider a two-unicast-$Z$ network over a directed acyclic graph of unit capacitated edges; the two-unicast-$Z$ network is a special case of two-unicast networks where one of the destinations has apriori side information of the unwanted (interfering) message. In this paper, we settle open questions on the limits of network coding for two-unicast-$Z$ networks by showing that the generalized network sharing bound is not tight, vector linear codes outperform scalar linear codes, and non-linear codes outperform linear codes in general. We also develop a commutative algebraic approach to deriving linear network coding achievability results, and demonstrate our approach by providing an alternate proof to the previous results of C. Wang et. al., I. Wang et. al. and Shenvi et. al. regarding feasibility of rate $(1,1)$ in the network.Comment: A short version of this paper is published in the Proceedings of The IEEE International Symposium on Information Theory (ISIT), June 201