$\ell_p$norm Multiway Cut
Abstract
We introduce and study $\ell_p$normmultiwaycut: the input here is an undirected graph with nonnegative edge weights along with $k$ terminals and the goal is to find a partition of the vertex set into $k$ parts each containing exactly one terminal so as to minimize the $\ell_p$norm of the cut values of the parts. This is a unified generalization of minsum multiway cut (when $p=1$) and minmax multiway cut (when $p=\infty$), both of which are wellstudied classic problems in the graph partitioning literature. We show that $\ell_p$normmultiwaycut is NPhard for constant number of terminals and is NPhard in planar graphs. On the algorithmic side, we design an $O(\log^2 n)$approximation for all $p\ge 1$. We also show an integrality gap of $\Omega(k^{11/p})$ for a natural convex program and an $O(k^{11/p\epsilon})$inapproximability for any constant $\epsilon>0$ assuming the small set expansion hypothesis.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.14840
 Bibcode:
 2021arXiv210614840C
 Keywords:

 Computer Science  Data Structures and Algorithms