The Cycling Property for the Clutter of Odd st-Walks

A binary clutter is cycling if its packing and covering linear program have integral optimal solutions for all Eulerian edge capacities. We prove that the clutter of odd st- walks of a signed graph is cycling if and only if it does not contain as a minor the clutter of odd circuits of K5 nor the clutter of lines of the Fano matroid. Corollaries of this result include, of many, the characterization for weakly bipartite signed graphs, packing two- commodity paths, packing T-joins with small |T|, a new result on covering odd circuits of a signed graph, as well as a new result on covering odd circuits and odd T-joins of a signed graft

Packing odd TT-joins with at most two terminals

Take a graph $G$, an edge subset $\Sigma\subseteq E(G)$, and a set of terminals $T\subseteq V(G)$ where $|T|$ is even. The triple $(G,\Sigma,T)$ is called a signed graft. A $T$-join is odd if it contains an odd number of edges from $\Sigma$. Let $\nu$ be the maximum number of edge-disjoint odd $T$-joins. A signature is a set of the form $\Sigma\triangle \delta(U)$ where $U\subseteq V(G)$ and $|U\cap T)$ is even. Let $\tau$ be the minimum cardinality a $T$-cut or a signature can achieve. Then $\nu\leq \tau$ and we say that $(G,\Sigma,T)$ packs if equality holds here. We prove that $(G,\Sigma,T)$ packs if the signed graft is Eulerian and it excludes two special non-packing minors. Our result confirms the Cycling Conjecture for the class of clutters of odd $T$-joins with at most two terminals. Corollaries of this result include, the characterizations of weakly and evenly bipartite graphs, packing two-commodity paths, packing $T$-joins with at most four terminals, and a new result on covering edges with cuts.Comment: extended abstract appeared in IPCO 2014 (under the different title "the cycling property for the clutter of odd st-walks"

Single Commodity Flow Algorithms for Lifts of Graphic and Cographic Matroids

Consider a binary matroid M given by its matrix representation. We show that if M is a lift of a graphic or a cographic matroid, then in polynomial time we can either solve the single commodity flow problem for M or find an obstruction for which the Max-Flow Min-Cut relation does not hold. The key tool is an algorithmic version of Lehman's Theorem for the set covering polyhedron

Clean clutters and dyadic fractional packings

A vector is dyadic if each of its entries is a dyadic rational number, i.e., an integer multiple of 1 2k for some nonnegative integer k. We prove that every clean clutter with a covering number of at least two has a dyadic fractional packing of value two. This result is best possible for there exist clean clutters with a covering number of three and no dyadic fractional packing of value three. Examples of clean clutters include ideal clutters, binary clutters, and clutters without an intersecting minor. Our proof is constructive and leads naturally to an albeit exponential algorithm. We improve the running time to quasi-polynomial in the rank of the input, and to polynomial in the binary cas

Evaluation of Living Streets’ Fitter for Walking project

Evaluation of Living Streets’ Fitter for Walking projec

Odd Paths, Cycles and TT-joins: Connections and Algorithms

Minimizing the weight of an edge set satisfying parity constraints is a challenging branch of combinatorial optimization as witnessed by the binary hypergraph chapter of Alexander Schrijver's book ``Combinatorial Optimization'' (Chapter 80). This area contains relevant graph theory problems including open cases of the Max Cut problem, or some multiflow problems. We clarify the interconnections of some problems and establish three levels of difficulties. On the one hand, we prove that the Shortest Odd Path problem in an undirected graph without cycles of negative total weight and several related problems are NP-hard, settling a long-standing open question asked by Lov\'asz (Open Problem 27 in Schrijver's book ``Combinatorial Optimization''. On the other hand, we provide a polynomial-time algorithm to the closely related and well-studied Minimum-weight Odd $\{s,t\}$-Join problem for non-negative weights, whose complexity, however, was not known; more generally, we solve the Minimum-weight Odd $T$-Join problem in FPT time when parameterized by $|T|$. If negative weights are also allowed, then finding a minimum-weight odd $\{s,t\}$-join is equivalent to the Minimum-weight Odd $T$-Join problem for arbitrary weights, whose complexity is only conjectured to be polynomially solvable. The analogous problems for digraphs are also considered.Comment: 24 pages, 2 figure

The Trail, 1996-02-22

https://soundideas.pugetsound.edu/thetrail_all/2686/thumbnail.jp

Casco Bay Weekly : 27 March 2003

https://digitalcommons.portlandlibrary.com/cbw_2003/1011/thumbnail.jp

Recycling Edwardian Geology Excursions for Five Modern Cyclists’ Geotrails in England

Around the 1900s, the Geologists’ Association offered a few excursions, although most were for pedestrians, for cyclists. At that time the London-based Geologists’ Association was well placed to employ the railway network to transport its members to and from excursion areas; locations immediately north of London (especially in the counties of Bedfordshire, Buckinghamshire and Hertfordshire) were particularly favored due to their regular trains. Its excursions were advertised in its Monthly Circular and usually later reported in its Proceedings; both now provide sufficient information to recreate and contextualize the cycling and other excursions as modern cyclists’ geotrails. The original excursions were offered at a time when leisure cycling was booming, the ‘Edwardian period’ which is noteworthy as one of major political, social and technological changes. The impact of these changes is evident today, and linking modern geotrails to the Edwardian background can make them attractive to non-specialist users and encourage their engagement with geology, particularly economic geology. Five new cyclists’ geotrails, based upon ‘Edwardian period’ excursions and their socio-political and technological background, have been published and these are contextualized, analyzed and examined as models for similar future provision