Bicriteria Network Design Problems

We study a general class of bicriteria network design problems. A generic problem in this class is as follows: Given an undirected graph and two minimization objectives (under different cost functions), with a budget specified on the first, find a <subgraph \from a given subgraph-class that minimizes the second objective subject to the budget on the first. We consider three different criteria - the total edge cost, the diameter and the maximum degree of the network. Here, we present the first polynomial-time approximation algorithms for a large class of bicriteria network design problems for the above mentioned criteria. The following general types of results are presented. First, we develop a framework for bicriteria problems and their approximations. Second, when the two criteria are the same %(note that the cost functions continue to be different) we present a ``black box'' parametric search technique. This black box takes in as input an (approximation) algorithm for the unicriterion situation and generates an approximation algorithm for the bicriteria case with only a constant factor loss in the performance guarantee. Third, when the two criteria are the diameter and the total edge costs we use a cluster-based approach to devise a approximation algorithms --- the solutions output violate both the criteria by a logarithmic factor. Finally, for the class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms for a number of bicriteria problems using dynamic programming. We show how these pseudopolynomial-time algorithms can be converted to fully polynomial-time approximation schemes using a scaling technique.Comment: 24 pages 1 figur

Does Interacting with Trustworthy People Enhance Mindfulness? An Experience Sampling Study of Mindfulness in Everyday Situations

Mindfulness is known to increase after meditation interventions. But might features of our everyday situations outside of meditation not also influence our mindfulness from moment-to-moment? Drawing from psychological research on interpersonal trust, we suggest that interacting with trustworthy people could influence the expression of mindfulness. And, extending this research on trust, we further suggest that the influence of trustworthy social interactions on mindfulness could proceed through two pathways: a particularized pathway (where specific interactions that are especially high (or low) in trustworthiness have an immediate influence on mindfulness) or a generalized pathway (where the typical level of trustworthiness a person perceives across all their interactions exerts a more stable influence on their mindfulness). To explore these two pathways, study participants (N = 201) repeatedly reported their current levels of mindfulness and their prior interactions with trustworthy leaders and teammates during their everyday situations using an experience sampling protocol ( = 3,605 reports). Results from mixed-effects models provide little support for the particularized pathway: specific interactions with trustworthy leaders and teammates had little immediate association with mindfulness. The generalized pathway, however, was strongly associated with mindfulness—and remained incrementally predictive beyond relevant individual differences and features of situations. In sum, people who typically interact with more trustworthy partners may become more mindful

Spanning trees short or small

We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number $k$ of nodes are required to be connected in the solution. A prototypical example is the $k$MST problem in which we require a tree of minimum weight spanning at least $k$ nodes in an edge-weighted graph. We show that the $k$MST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio $2\sqrt{k}$ for the general edge-weighted case and $O(k^{1/4})$ for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of decomposable graphs which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees, and more generally, that of finding networks with minimum diameter. A simple technique is used to provide a polynomial-time solution for finding $k$-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu.Comment: 27 page