New lower bounds for the topological complexity of aspherical spaces

Date of Acceptance: 5/04/2015 15 pages, 4 figuresPeer reviewedPostprin

A mapping theorem for topological complexity

Peer reviewedPublisher PD

Bredon cohomology and robot motion planning

In this paper we study the topological invariant ${\sf {TC}}(X)$ reflecting the complexity of algorithms for autonomous robot motion. Here, $X$ stands for the configuration space of a system and ${\sf {TC}}(X)$ is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in $X$. We focus on the case when the space $X$ is aspherical; then the number ${\sf TC}(X)$ depends only on the fundamental group $\pi=\pi_1(X)$ and we denote it ${\sf TC}(\pi)$. We prove that ${\sf TC}(\pi)$ can be characterised as the smallest integer $k$ such that the canonical $\pi\times\pi$-equivariant map of classifying spaces $$E(\pi\times\pi) \to E_{\mathcal D}(\pi\times\pi)$$ can be equivariantly deformed into the $k$-dimensional skeleton of $E_{\mathcal D}(\pi\times\pi)$. The symbol $E(\pi\times\pi)$ denotes the classifying space for free actions and $E_{\mathcal D}(\pi\times\pi)$ denotes the classifying space for actions with isotropy in a certain family $\mathcal D$ of subgroups of $\pi\times\pi$. Using this result we show how one can estimate ${\sf TC}(\pi)$ in terms of the equivariant Bredon cohomology theory. We prove that ${\sf TC}(\pi) \le \max\{3, {\rm cd}_{\mathcal D}(\pi\times\pi)\},$ where ${\rm cd}_{\mathcal D}(\pi\times\pi)$ denotes the cohomological dimension of $\pi\times\pi$ with respect to the family of subgroups $\mathcal D$. We also introduce a Bredon cohomology refinement of the canonical class and prove its universality. Finally we show that for a large class of principal groups (which includes all torsion free hyperbolic groups as well as all torsion free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher are exactly the classes having Bredon cohomology extensions with respect to the family $\mathcal D$.Comment: This revision contains a few additional comments, among them is Corollary 3.5.

On the Complexity of Nash Equilibria of Action-Graph Games

We consider the problem of computing Nash Equilibria of action-graph games (AGGs). AGGs, introduced by Bhat and Leyton-Brown, is a succinct representation of games that encapsulates both "local" dependencies as in graphical games, and partial indifference to other agents' identities as in anonymous games, which occur in many natural settings. This is achieved by specifying a graph on the set of actions, so that the payoff of an agent for selecting a strategy depends only on the number of agents playing each of the neighboring strategies in the action graph. We present a Polynomial Time Approximation Scheme for computing mixed Nash equilibria of AGGs with constant treewidth and a constant number of agent types (and an arbitrary number of strategies), together with hardness results for the cases when either the treewidth or the number of agent types is unconstrained. In particular, we show that even if the action graph is a tree, but the number of agent-types is unconstrained, it is NP-complete to decide the existence of a pure-strategy Nash equilibrium and PPAD-complete to compute a mixed Nash equilibrium (even an approximate one); similarly for symmetric AGGs (all agents belong to a single type), if we allow arbitrary treewidth. These hardness results suggest that, in some sense, our PTAS is as strong of a positive result as one can expect

Benefits, Corporate Motives, and Communication Patterns in Strategic Philanthropic Relationships as Perceived by Nonprofit Partners

Businesses are increasingly held accountable both to their owners and to the larger society in which they operate. Accordingly, many companies are extending their resources to meet community needs through philanthropic partnerships with nonprofit organizations. Such ventures, however, have drawn close scrutiny of motives and benefits. For example, some consumers register skepticism when evaluating the sincerity of corporate intent in cause-related marketing arrangements. Attribution theory suggests that altruistic reasons for corporate good deeds may be discounted in the context of apparent self-interest. Likewise, a debate between shareholder and stakeholder theorists introduces questions about possibly conflicting obligations facing corporate managers. Some contend that good business and stakeholder accommodation do not mix. The emergence of strategic philanthropy potentially serves both interests, but little empirical study has been devoted to understanding the dynamics of such partnerships. Of particular interest is the perspective of nonprofit organizations who receive strategically motivated corporate gifts. This study used a grounded theory approach to tap the perceptions of nonprofit managers regarding these issues. Through in-depth interviews, the researcher learned that nonprofits commonly see in their partners a pattern of multiple corporate motives, with varying blends of altruism and self-interest. The largest donations were generally reported from companies expecting marketing benefits in return for their philanthropic investment. However, participants stressed that those expectations most often were unstated by the company. They described a negotiating environment in which nonprofits thoughtfully analyze potential corporate donors’ needs and then pitch mutual-benefit packages to engage them in partnership. In the most strategically driven alliances, relationships were characterized as interdependent, and benefits were viewed as approximately equal. Nonprofit managers reported that they work hard under the strategic model to obtain corporate gifts, but they also experience deeper, more satisfying relationships with their partners than in the past. Communal qualities were often described. In some partnerships, corporate motives were seen as evolving from a primarily marketing interest to an increasingly altruistic interest in the nonprofit mission. Theoretical implications and a proposed model are presented to guide further study. Observations and recommendations for corporate managers are also offered