  We study the design of truthful mechanisms for set systems, i.e., scenarios
where a customer needs to hire a team of agents to perform a complex task. In
this setting, frugality [Archer&Tardos'02] provides a measure to evaluate the
"cost of truthfulness", that is, the overpayment of a truthful mechanism
relative to the "fair" payment. We propose a uniform scheme for designing
frugal truthful mechanisms for general set systems. Our scheme is based on
scaling the agents' bids using the eigenvector of a matrix that encodes the
interdependencies between the agents. We demonstrate that the r-out-of-k-system
mechanism and the \sqrt-mechanism for buying a path in a graph [Karlin et.
al'05] can be viewed as instantiations of our scheme. We then apply our scheme
to two other classes of set systems, namely, vertex cover systems and k-path
systems, in which a customer needs to purchase k edge-disjoint source-sink
paths. For both settings, we bound the frugality of our mechanism in terms of
the largest eigenvalue of the respective interdependency matrix. We show that
our mechanism is optimal for a large subclass of vertex cover systems
satisfying a simple local sparsity condition. For k-path systems, while our
mechanism is within a factor of k + 1 from optimal, we show that it is, in
fact, optimal, when one uses a modified definition of frugality proposed in
[Elkind et al.'07]. Our lower bound argument combines spectral techniques and
Young's inequality, and is applicable to all set systems. As both r-out-of-k
systems and single path systems can be viewed as special cases of k-path
systems, our result improves the lower bounds of [Karlin et al.'05] and answers
several open questions proposed in that paper.
