  Inspired by equity considerations, we consider top-$k$ norm, ordered norm,
and symmetric monotonic norm objectives for various combinatorial optimization
problems. Top-$k$ norms and ordered norms have natural interpretations in terms
of minimizing the impact on individuals bearing largest costs. To model
decision-making with multiple equity criteria, we study the notion of
portfolios of solutions with the property that each norm or equity criteria has
an approximately optimal solution in this portfolio. We attempt to characterize
portfolios by their sizes and approximation factor guarantees for various
combinatorial problems. For a given problem, we investigate whether (1) there
exists a single solution that is approximately optimal for all norms, (2) there
exists a small approximately optimal portfolio of size larger than 1, (3) there
exist polynomial time algorithms to find these small portfolios. We study an
algorithmic framework to obtain single solutions that are approximately optimal
for all norms. We show the existence of such a solution for problems such as
$k$-clustering, ordered set cover, scheduling for job completion time
minimization, and scheduling for machine load minimization on identical
machines. We also give efficient algorithms to find these solutions in most
cases, except set cover where we show there is a gap in terms of computational
complexity. Our work improves upon the best-known approximation factor across
all norms for a single solution in $k$-clustering. For uncapacitated facility
location and scheduling for machine load minimization with identical jobs, we
obtain logarithmic sized portfolios, also providing a matching lower bound in
the latter case. Our work results in new open combinatorial questions, which
might be of independent interest.
