On Approximating Cutwidth and Pathwidth

TL;DR

We address the min-max graph layout problem of minimizing the maximum cut at any point (Minimum Cutwidth) and related measures such as Pathwidth via Vertex Separation Number. The authors introduce a novel metric-decomposition framework tailored to min-max objectives, enabling an LP-based approximation of $O(\beta(n)\log n)$ with $\beta(n)=\exp(O(\sqrt{\log\log n}))=\log^{o(1)}(n)$, and extend the technique to Pathwidth via VS. They also achieve a simultaneous $O(\beta(n)\log n)$-approximation for Cutwidth and MLA, and derive an $O(\beta(n)\log n)$-approximation for Vertex Separation Number and hence Pathwidth, all in a unified framework. The results surpass the long-standing recursive-partitioning limits and extend to weighted graphs; the paper also outlines direct LP/SDP relaxations and directed-min-max extensions as promising future directions. Overall, the work significantly advances our ability to approximate core min-max layout problems beyond classic partitioning methods and opens multiple avenues for tighter bounds and broader applicability.

Abstract

We study graph ordering problems with a min-max objective. A classical problem of this type is cutwidth, where given a graph we want to order its vertices such that the number of edges crossing any point is minimized. We give a $ \log^{1+o(1)}(n)$ approximation for the problem, substantially improving upon the previous poly-logarithmic guarantees based on the standard recursive balanced partitioning approach of Leighton and Rao (FOCS'88). Our key idea is a new metric decomposition procedure that is suitable for handling min-max objectives, which could be of independent interest. We also use this to show other results, including an improved $ \log^{1+o(1)}(n)$ approximation for computing the pathwidth of a graph.

Figures (3)

• Figure 1: Failure of recursive partitioning for cutwidth via Seymour's improved cutting lemma.
• Figure 2: Ordering of the pieces $S_v$ by Algorithm \ref{['alg:cutwidth']}. The edges passing over a piece $S\in \{S_v\}_{v\in N_{i,j}}$ must lie in some $\delta(S_v)$, for some $v\in N_{i',j'}$ and $j' \leq j$, or in $G[S]$.
• Figure 3: The order of pieces in Algorithm \ref{['alg:pathwidth']}. The vertical line depicts a cut; each edge crossing it belongs to $G[S]$ or touches some $w\in D_{j'}$ for $j'\leq j$.

Theorems & Definitions (41)

• Theorem 1
• Theorem 2
• Definition 1: Vertex Separation Number
• Theorem 3
• Theorem 4
• Lemma 5
• proof
• Lemma 6
• Definition 2: Radius Scale
• Definition 3: Size Class