[Paper Review] Improved Region-Growing and Combinatorial Algorithms for $k$-Route Cut Problems

Guru Guruganesh, Laura Sanità+1 more|arXiv (Cornell University)|Oct 19, 2014

Computational Geometry and Mesh Generation39 references3 citations

TL;DR

This paper presents improved combinatorial and region-growing algorithms for $k$-route cut problems, including $k$-route multicut, multiway cut, and $s$-$t$ cut. It introduces a novel recursive region-growing lemma that avoids the $O(\log^2 r)$ cost blowup in prior methods, achieving an $O(\ln r \ln \ln r)$-approximation for $k$-route multicut and significantly improving bicriteria approximation guarantees for $k$-route multiway cut.

ABSTRACT

We study the {\em $k$-route} generalizations of various cut problems, the most general of which is \emph{$k$-route multicut} ($k$-MC) problem, wherein we have $r$ source-sink pairs and the goal is to delete a minimum-cost set of edges to reduce the edge-connectivity of every source-sink pair to below $k$. The $k$-route extensions of multiway cut ($k$-MWC), and the minimum $s$-$t$ cut problem ($k$-$(s,t)$-cut), are similarly defined. We present various approximation and hardness results for these $k$-route cut problems that improve the state-of-the-art for these problems in several cases. (i) For {\em $k$-route multiway cut}, we devise simple, but surprisingly effective, combinatorial algorithms that yield bicriteria approximation guarantees that markedly improve upon the previous-best guarantees. (ii) For {\em $k$-route multicut}, we design algorithms that improve upon the previous-best approximation factors by roughly an $O(\sqrt{\log r})$-factor, when $k=2$, and for general $k$ and unit costs and any fixed violation of the connectivity threshold $k$. The main technical innovation is the definition of a new, powerful \emph{region growing} lemma that allows us to perform region-growing in a recursive fashion even though the LP solution yields a {\em different metric} for each source-sink pair. (iii) We complement these results by showing that the {\em $k$-route $s$-$t$ cut} problem is at least as hard to approximate as the {\em densest-$k$-subgraph} (DkS) problem on uniform hypergraphs.

Motivation & Objective

• To address the challenge of designing efficient approximation algorithms for $k$-route cut problems, which generalize classical multicut and multiway cut problems by requiring connectivity reduction below a threshold $k$.
• To overcome the high cost blowup in prior region-growing approaches due to differing metrics per source-sink pair in LP solutions.
• To improve bicriteria approximation guarantees for $k$-route multiway cut and reduce the approximation factor for $k$-route multicut by $O(\sqrt{\log r})$ when $k=2$.
• To establish a strong hardness result showing that $k$-route $s$-$t$ cut is at least as hard as densest $k$-subgraph on uniform hypergraphs.

Proposed method

• Introduces a new recursive region-growing lemma that allows efficient region expansion even when each source-sink pair has a different metric from the LP solution.
• Applies a bicriteria approach where the algorithm tolerates a slight violation of the $k$-connectivity threshold ($\gamma > 1$) to achieve better cost bounds.
• Uses a volume-based analysis with node and edge contributions to bound the cost of the cut set in terms of the optimal solution and logarithmic factors.
• Employs a recursive algorithm structure that partitions the graph and processes high-connectivity components separately, reducing depth and cost.
• Leverages a dual analysis framework combining LP relaxation with combinatorial rounding to maintain cost efficiency.
• Extends techniques to the node-cut variant ($k$-route node-multicut) by adapting the region-growing process to node-removal constraints.

Experimental results

Research questions

• RQ1 Can we design a region-growing algorithm for $k$-route multicut that avoids the $O(\log^2 r)$ cost blowup inherent in prior methods?
• RQ2 What is the best achievable bicriteria approximation ratio for $k$-route multiway cut, and can it be improved beyond previous bounds?
• RQ3 How does the complexity of $k$-route $s$-$t$ cut compare to other hard problems, and what are its implications for approximation algorithms?
• RQ4 Can we achieve a $\operatorname{poly}(k)$-approximation for $k$-route $s$-$t$ cut without improving the state-of-the-art for densest $k$-subgraph?

Key findings

• The paper achieves an $O(\ln r \ln \ln r)$-approximation for $k$-route multicut, improving upon the previous best factor by $O(\sqrt{\log r})$ when $k=2$.
• For $k$-route multiway cut, the proposed combinatorial algorithm yields a bicriteria approximation with a cost bound of $O((k-1)^2 \ln r \ln \ln r) \cdot \text{OPT}$, significantly improving prior guarantees.
• The new recursive region-growing lemma enables cost-efficient cuts without the $O(\log^2 r)$ blowup, by handling different metrics per pair via a volume-based radius selection.
• The algorithm for $k$-route node-multicut achieves the same asymptotic approximation ratio as the edge-cut version, extending the framework to node-based connectivity.
• A strong hardness result shows that $k$-route $s$-$t$ cut is at least as hard to approximate as the densest $k$-subgraph problem on uniform hypergraphs, implying that any $\operatorname{poly}(k)$-approximation without improving DkS is unlikely.
• The analysis proves that setting $\gamma = \frac{k}{k-1}$ yields a feasible solution with cost $O((k-1)^2 \ln r \ln \ln r) \cdot \text{OPT}$, matching the bicriteria bound for the multiway cut case.