  We study mechanism design for combinatorial cost sharing. Imagine that
multiple items or services are available to be shared among a set of interested
agents. The outcome of a mechanism in this setting consists of an assignment,
determining for each item the set of players who are granted service, together
with respective payments. Although there are several works studying specialized
versions of such problems, there has been almost no progress for general
combinatorial cost sharing domains until recently \cite{DobzinskiO17}. The main
goal of our work is to further understand this interplay in terms of budget
balance and social cost approximation. Towards this, we provide a refinement of
cross-monotonicity (trace-monotonicity) that is applicable to iterative
mechanisms. The trace here refers to the order in which players become
finalized. On top of this, we also provide two parameterizations of cost
functions which capture the behavior of their average cost-shares. Based on our
trace-monotonicity property, we design a scheme of ascending cost sharing
mechanisms which is applicable to the combinatorial cost sharing setting with
symmetric submodular valuations. Using our first cost function
parameterization, we identify conditions under which our mechanism is weakly
group-strategyproof, $O(1)$-budget-balanced and $O(H_n)$-approximate with
respect to the social cost. Finally, we consider general valuation functions
and exploit our second parameterization to derive a more fine-grained analysis
of the Sequential Mechanism introduced by Moulin. This mechanism is budget
balanced by construction, but in general only guarantees a poor social cost
approximation of $n$. We identify conditions under which the mechanism achieves
improved social cost approximation guarantees.
