4 research outputs found
Worst-Case to Expander-Case Reductions
In recent years, the expander decomposition method was used to develop many graph algorithms, resulting in major improvements to longstanding complexity barriers. This powerful hammer has led the community to (1) believe that most problems are as easy on worst-case graphs as they are on expanders, and (2) suspect that expander decompositions are the key to breaking the remaining longstanding barriers in fine-grained complexity.
We set out to investigate the extent to which these two things are true (and for which problems). Towards this end, we put forth the concept of worst-case to expander-case self-reductions. We design a collection of such reductions for fundamental graph problems, verifying belief (1) for them. The list includes k-Clique, 4-Cycle, Maximum Cardinality Matching, Vertex-Cover, and Minimum Dominating Set. Interestingly, for most (but not all) of these problems the proof is via a simple gadget reduction, not via expander decompositions, showing that this hammer is effectively useless against the problem and contradicting (2)
Worst-Case to Expander-Case Reductions
In recent years, the expander decomposition method was used to develop many
graph algorithms, resulting in major improvements to longstanding complexity
barriers. This powerful hammer has led the community to (1) believe that most
problems are as easy on worst-case graphs as they are on expanders, and (2)
suspect that expander decompositions are the key to breaking the remaining
longstanding barriers in fine-grained complexity.
We set out to investigate the extent to which these two things are true (and
for which problems). Towards this end, we put forth the concept of worst-case
to expander-case self-reductions. We design a collection of such reductions for
fundamental graph problems, verifying belief (1) for them. The list includes
-Clique, -Cycle, Maximum Cardinality Matching, Vertex-Cover, and Minimum
Dominating Set. Interestingly, for most (but not all) of these problems the
proof is via a simple gadget reduction, not via expander decompositions,
showing that this hammer is effectively useless against the problem and
contradicting (2).Comment: ITCS 202
Improved Compression of the Okamura-Seymour Metric
Let be an undirected unweighted planar graph. Consider a vector
storing the distances from an arbitrary vertex to all vertices of a single face in their cyclic order. The pattern of
is obtained by taking the difference between every pair of consecutive
values of this vector. In STOC'19, Li and Parter used a VC-dimension argument
to show that in planar graphs, the number of distinct patterns, denoted , is
only . This resulted in a simple compression scheme requiring space to encode the distances between and
a subset of terminal vertices . This is known as the
Okamura-Seymour metric compression problem.
We give an alternative proof of the bound that exploits planarity
beyond the VC-dimension argument. Namely, our proof relies on cut-cycle
duality, as well as on the fact that distances among vertices of are
bounded by . Our method implies the following:
(1) An space compression of the Okamura-Seymour metric,
thus improving the compression of Li and Parter to .
(2) An optimal space compression of the Okamura-Seymour
metric, in the case where the vertices of induce a connected component in
.
(3) A tight bound of for the family of Halin graphs,
whereas the VC-dimension argument is limited to showing