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

The Complexity of Scheduling for p-norms of Flow and Stretch

We consider computing optimal k-norm preemptive schedules of jobs that arrive over time. In particular, we show that computing the optimal k-norm of flow schedule, is strongly NP-hard for k in (0, 1) and integers k in (1, infinity). Further we show that computing the optimal k-norm of stretch schedule, is strongly NP-hard for k in (0, 1) and integers k in (1, infinity).Comment: Conference version accepted to IPCO 201

Growth kinetics effects on self-assembled InAs/InP quantum dots

A systematic manipulation of the morphology and the optical emission properties of MOVPE grown ensembles of InAs/InP quantum dots is demonstrated by changing the growth kinetics parameters. Under non-equilibrium conditions of a comparatively higher growth rate and low growth temperature, the quantum dot density, their average size and hence the peak emission wavelength can be tuned by changing efficiency of the surface diffusion (determined by the growth temperature) relative to the growth flux. We further observe that the distribution of quantum dot heights, for samples grown under varying conditions, if normalized to the mean height, can be nearly collapsed onto a single Gaussian curve.Comment: 2 figure

Improved Approximation Algorithms for Stochastic Matching

In this paper we consider the Stochastic Matching problem, which is motivated by applications in kidney exchange and online dating. We are given an undirected graph in which every edge is assigned a probability of existence and a positive profit, and each node is assigned a positive integer called timeout. We know whether an edge exists or not only after probing it. On this random graph we are executing a process, which one-by-one probes the edges and gradually constructs a matching. The process is constrained in two ways: once an edge is taken it cannot be removed from the matching, and the timeout of node $v$ upper-bounds the number of edges incident to $v$ that can be probed. The goal is to maximize the expected profit of the constructed matching. For this problem Bansal et al. (Algorithmica 2012) provided a $3$-approximation algorithm for bipartite graphs, and a $4$-approximation for general graphs. In this work we improve the approximation factors to $2.845$ and $3.709$, respectively. We also consider an online version of the bipartite case, where one side of the partition arrives node by node, and each time a node $b$ arrives we have to decide which edges incident to $b$ we want to probe, and in which order. Here we present a $4.07$-approximation, improving on the $7.92$-approximation of Bansal et al. The main technical ingredient in our result is a novel way of probing edges according to a random but non-uniform permutation. Patching this method with an algorithm that works best for large probability edges (plus some additional ideas) leads to our improved approximation factors