Constant factor approximation of vertex-cuts in planar graphs

Abstract

We devise the first constant factor approximation algorithm for minimum quotient vertex-cuts in planar graphs. Our algorithm achieves approximation ratio 1+ (1 + + o(1)) if there is an optimal quotient vertex-cut (A * , B * , C * ) where the weight of C * is of low order compared to those of A * and B * ; this holds, for example, when the input graph has uniform weights and costs. The ratio further im-W . We use our algorithm for quotient vertex-cuts to achieve the first constant-factor pseudo-approximation for vertex separators in planar graphs.Our technical contribution is two-fold. First, we prove a structural theorem for planar graphs, showing the existence of a nearoptimal quotient vertex-cut whose high-level structure is that of a bounded-depth tree. Second, we develop an algorithm that optimizes over such complex structures in running time that depends (exponentially) not on the size of the structure, but rather only on its depth. These techniques may be applicable in other problems.

Multicommodity flow, well-linked terminals, and routing problems

“…Our result on the flow-cut gap in node capacitated planar graphs gives an alternate O(1) approximation, albeit with a much larger constant, for the minimum quotient node cut problem in planar graphs [3]. Although our proof is similar in outline to that for the edge capacitated problem that we presented in [9], there is a crucial technical difference.…”

“…However, in the node capacitated case, there is no such reduction which preserves planarity. This was the technical difficulty in generalizing the O(1) gap proof of Klein, Plotkin and Rao [21] to the node problem (see [3] for some additional discussion). Thus we need to work directly with node capacities.…”

“…Given a weight function on the nodes w : V → R + , the minimum quotient cut is a cut that minimizes the ratio of the capacity of the cut to the node weight of the smaller side in the partition produced by the cut. The sparsity of the product multicommodity flow instance induced by w is within a factor of 2 of the minimum quotient cut value for w. Approximation algorithms for computing sparsest cuts and minimum quotient cuts can be obtained via flow-cut gaps [28,21,29,6] or via other methods that can give better bounds than the flow-cut gap [5,31,3]. …”

“…The second issue is whether to use the flow-linked decomposition from Theorem 2.1 or the cut-linked decomposition from Theorem 2.3. The reason to distinguish between the two is that in node capacitated planar graphs, the previously known bound on the flow-cut gap is O(log n) [28] while the approximation ratio for finding minimum quotient cuts is O(1) [3]. In this paper we are able to improve the flow-cut gap to O(1), however, as we do not provide all the details for this improvement, we use the decomposition guaranteed by Theorem 2.3.…”

“…Then we apply the decomposition algorithm given in Theorem 2.3 to to obtain a collection of cut-linked instances. In the decomposition we use an O(1)-approximation algorithm for minimum quotient node cuts [3]. From Theorem 2.3 and Remark 2.4, with α = Θ(1/(log n log k)) and β(G) = O(1), it follows that we find a collection of α-node-cut-linked instances such that there remain Ω(opt log n) pairs amongst these instances.…”