k-Trails: Recognition, Complexity, and Approximations

The notion of degree-constrained spanning hierarchies, also called k-trails, was recently introduced in the context of network routing problems. They describe graphs that are homomorphic images of connected graphs of degree at most k. First results highlight several interesting advantages of k-trails compared to previous routing approaches. However, so far, only little is known regarding computational aspects of k-trails. In this work we aim to fill this gap by presenting how k-trails can be analyzed using techniques from algorithmic matroid theory. Exploiting this connection, we resolve several open questions about k-trails. In particular, we show that one can recognize efficiently whether a graph is a k-trail. Furthermore, we show that deciding whether a graph contains a k-trail is NP-complete; however, every graph that contains a k-trail is a (k+1)-trail. Moreover, further leveraging the connection to matroids, we consider the problem of finding a minimum weight k-trail contained in a graph G. We show that one can efficiently find a (2k-1)-trail contained in G whose weight is no more than the cheapest k-trail contained in G, even when allowing negative weights. The above results settle several open questions raised by Molnar, Newman, and Sebo

Min-Max Theorems for Packing and Covering Odd (u,v)(u,v)-trails

We investigate the problem of packing and covering odd $(u,v)$-trails in a graph. A $(u,v)$-trail is a $(u,v)$-walk that is allowed to have repeated vertices but no repeated edges. We call a trail odd if the number of edges in the trail is odd. Let $\nu(u,v)$ denote the maximum number of edge-disjoint odd $(u,v)$-trails, and $\tau(u,v)$ denote the minimum size of an edge-set that intersects every odd $(u,v)$-trail. We prove that $\tau(u,v)\leq 2\nu(u,v)+1$. Our result is tight---there are examples showing that $\tau(u,v)=2\nu(u,v)+1$---and substantially improves upon the bound of $8$ obtained in [Churchley et al 2016] for $\tau(u,v)/\nu(u,v)$. Our proof also yields a polynomial-time algorithm for finding a cover and a collection of trails satisfying the above bounds. Our proof is simple and has two main ingredients. We show that (loosely speaking) the problem can be reduced to the problem of packing and covering odd $(uv,uv)$-trails losing a factor of 2 (either in the number of trails found, or the size of the cover). Complementing this, we show that the odd-$(uv,uv)$-trail packing and covering problems can be tackled by exploiting a powerful min-max result of [Chudnovsky et al 2006] for packing vertex-disjoint nonzero $A$-paths in group-labeled graphs

Value of Trail Access on Home Purchases

We use hedonic analysis of home sales data from the Twin Cities Metropolitan Area to estimate the effects of access of different types of trails on home value. Our model includes proximity to three distinct types bicycle facilities, controlling for local fixed effects and open space characteristics. Using interaction terms detect different preferences between city and suburban homebuyers. Regression results show that off-street bicycle trails situated alongside busy streets are negatively associated with home sale prices in both the city and suburbs. Proximity to off-street bicycle trails away from trafficked streets in the city are positively associated with home sale prices, with no significant result in the suburbs. On-street bicycle lanes have no effect in the city and are a disamenity in the suburbs. The following policy issues are relevant from this research. First, type of trail matters. On-street trails and road-side trails may not be as appreciated as many city planners or policy officials think. Second, city residents have different preferences than suburban residents. Third and as suspected, larger and more pressing factors likely influencing residential location decisions. The finding also suggest that urban planners and advocates need to be aware of the consequences of providing for bicycle facilities, as the change in welfare is not necessarily positive for all homeowners.

Symmetry reduction and heuristic search for error detection in model checking

The state explosion problem is the main limitation of model checking. Symmetries in the system being verified can be exploited in order to avoid this problem by defining an equivalence (symmetry) relation on the states of the system, which induces a semantically equivalent quotient system of smaller size. On the other hand, heuristic search algorithms can be applied to improve the bug finding capabilities of model checking. Such algorithms use heuristic functions to guide the exploration. Bestfirst is used for accelerating the search, while A* guarantees optimal error trails if combined with admissible estimates. We analyze some aspects of combining both approaches, concentrating on the problem of finding the optimal path to the equivalence class of a given error state. Experimental results evaluate our approach

Finding Differential Paths in ARX Ciphers through Nested Monte-Carlo Search

We propose the adaptation of Nested Monte-Carlo Search algorithm for finding differential trails in the class of ARX ciphers. The practical application of the algorithm is demonstrated on round-reduced variants of block ciphers from the SPECK family. More specifically, we report the best differential trails,up to 9 rounds, for SPECK32

Home on the Range: The Impact of the Cattle Trails on Indian Territory

This paper addresses how the cattle trails affected the Native American tribes in Indian Territory. There has been a plethora of research done on finding the cattle trails, life as a cowboy, and the economics of the cattle trade, but precious little is said about its impact on the Indians, whose land the trails went through. This paper examines the role of cattle in tribes prior to relocation, how the Five Civilized/Eastern Tribes utilized cattle and interacted with early cattle trails, and how the Western Plains Tribes reacted to cattle trails moving westward. The cattle trails provided for a new economy in Indian Territory that allowed the Native Americans to survive and thrive as they became involved in the cattle industry; however, that same economic development was misused to justify the elimination of reservations and open the land to white settlement

Diurnal variation of non-specular meteor trails

We present results of simulated radar observations of meteor trails in an effort to show how non-specular meteor trails are expected to vary as a function of a number of key atmospheric, ionospheric and meteoroid parameters. This paper identifies which geophysical sources effect the variability in non-specular trail radar observations, and provides an approach that uses some of these parameter dependencies to determine meteoroid and atmospheric properties based upon the radar meteor observations. The numerical model used follows meteor evolution from ablation and ionization to head echo plasma generation and through formation of field aligned irregularities (FAI). Our main finding is that non-specular meteor trail duration is highly sensitive to the presence of lower thermospheric winds or electric fields and the background ionospheric electron density. In an effort to make key predictions we present the first results of how the same meteoroid is expected to produce dramatically different meteor trails as a function of location and local time. For example, we show that mid-latitude trail durations are often shorter lasting than equatorial trail observations because of the difference in mid-latitude wind speed and equatorial drift speed. The simulated trails also account for observations showing that equatorial nighttime non-specular meteor trails last significantly longer and are observed more often than daytime trails