Algorithms and Complexity Seminar
On Designing Approximation Algorithms with Guarantees Independent of the Graph Size
Ankur Moitra (MIT)
Leighton and Rao established an approximate min-cut max-flow theorem
for uniform multicommodity flows. Linial, London and Rabinovich and Aumann and
Rabani solved a major open question and proved that the min-cut max-flow ratio
for general maximum concurrent flow problems (when there are $k$ commodities)
is $O(\log k)$. Here we attempt to derive a more general theory of Steiner cut
and flow problems, and we prove bounds that are poly-logarithmic in $k$ for a
much broader class of multicommodity flow and cut problems. Our structural
results are motivated by the meta question: Suppose we are given a $poly(\log
n)$ approximation algorithm for a flow or cut problem - when can we give a
$poly(\log k)$ approximation algorithm for a generalization of this problem to
a Steiner cut or flow problem?
Thus we require that these approximation guarantees be independent of the size
of the graph, and only depend on the number of commodities (or the number of
Steiner nodes in a Steiner cut problem). For many natural applications of
multicommodity flows and cuts, we expect that the number of commodities $k$ is
much smaller than $n$, and for such problems we get approximation algorithms
that have much stronger guarantees.
Our approach fundamentally relies on metric geometry and oblivious routing. We
use the structural results we develop to constructively reduce a broad class of
Steiner cut and flow problems to a uniform case (on $k$ nodes) at the cost of a
loss of a $poly(\log k)$ in the approximation guarantee. We cannot concisely
define this class but we can use our results to give $poly(\log k)$
approximation algorithms for a number of problems for which such results were
previously unknown, such requirement cut, $l$-multicut, and natural
generalizations of oblivious routing, min-cut linear arrangement and minimum
linear arrangement.