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Cutwidth Bounds via Vertex Partitions

Antoine Amarilli, Benoît Groz

TL;DR

This work analyzes how to bound a graph's cutwidth by partitioning its vertices into classes and examining (i) the quotient multigraph $G/P$ with cutwidth $x$ and (ii) the maximum cutwidth $y$ within any class. The authors first establish a simple bound $2x+y$ and then derive a tighter bound $ ext{cw}(G) \,\le\, 1.5x+y$ by carefully choosing, for each class, between a local order and its reverse to minimize cross-class edges. They prove this constant factor is tight via a lower-bound construction with parameters $x$ and $y$ (even $x$ and $y\ge 1.5x$), showing that the $x+y$ bound claimed in prior work is not valid in general. The results extend to the SCC condensation of directed graphs, where $x= ext{cw}$ of the condensation and $y$ bounds the cw of SCCs, yielding a general upper bound of $1.5x+y$ and highlighting the limits of partition-based strategies for cutwidth.

Abstract

We study the cutwidth measure on graphs and ways to bound the cutwidth of a graph by partitioning its vertices. We consider bounds expressed as a function of two quantities: on the one hand, the maximal cutwidth y of the subgraphs induced by the classes of the partition, and on the other hand, the cutwidth x of the quotient multigraph obtained by merging each class to a single vertex. We consider in particular the decomposition of directed graphs into strongly connected components (SCCs): in this case, y is the maximal cutwidth of an SCC, and x is the cutwidth of the directed acyclic condensation multigraph. We show that the cutwidth of a graph is always in O(x + y), specifically it can be upper bounded by 1.5x + y. We also show a lower bound justifying that the constant 1.5 cannot be improved in general

Cutwidth Bounds via Vertex Partitions

TL;DR

This work analyzes how to bound a graph's cutwidth by partitioning its vertices into classes and examining (i) the quotient multigraph with cutwidth and (ii) the maximum cutwidth within any class. The authors first establish a simple bound and then derive a tighter bound by carefully choosing, for each class, between a local order and its reverse to minimize cross-class edges. They prove this constant factor is tight via a lower-bound construction with parameters and (even and ), showing that the bound claimed in prior work is not valid in general. The results extend to the SCC condensation of directed graphs, where of the condensation and bounds the cw of SCCs, yielding a general upper bound of and highlighting the limits of partition-based strategies for cutwidth.

Abstract

We study the cutwidth measure on graphs and ways to bound the cutwidth of a graph by partitioning its vertices. We consider bounds expressed as a function of two quantities: on the one hand, the maximal cutwidth y of the subgraphs induced by the classes of the partition, and on the other hand, the cutwidth x of the quotient multigraph obtained by merging each class to a single vertex. We consider in particular the decomposition of directed graphs into strongly connected components (SCCs): in this case, y is the maximal cutwidth of an SCC, and x is the cutwidth of the directed acyclic condensation multigraph. We show that the cutwidth of a graph is always in O(x + y), specifically it can be upper bounded by 1.5x + y. We also show a lower bound justifying that the constant 1.5 cannot be improved in general

Paper Structure

This paper contains 12 sections, 4 theorems, 8 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cutwidth.
  4. Subdivisions.
  5. Proof of \ref{['prp:subdiv']}
  6. Vertex partitionings.
  7. SCCs.
  8. Upper bound
  9. Simple bound in $\bm{2x+y}$.
  10. More elaborate bound in $\bm{1.5x+y}$.
  11. Lower bound
  12. Conclusion and Future Work

Key Result

theorem 1

Let $G = (V,E)$ be an undirected multigraph, $P$ a partition of $G$, and $G/P$ be the quotient multigraph of $G$ under $P$. If the cutwidth of $G/P$ is $x$ and the cutwidth of every class of $P$ is at most $y$, then the cutwidth of $G$ is at most $1.5x + y$.

Figures (4)

  • Figure 1: Partitioning edges that may contribute to the cuts before or after reversing $<_C$.
  • Figure 2: Example of the directed graph $H$ obtained in the proof of Proposition \ref{['prp:lower-bound-condensation']} (left), together with its condensation multigraph (right). For this graph, the maximum cutwidth of an SCC is $y=3$ (indeed the only nontrivial SCC is formed by the filled brown nodes ($\{2,3,4\}$ and the nodes connecting them), connected by dashed brown edges), and the cutwidth of the condensation multigraph is $x = 2$, however it can be checked that the cutwidth of $H$ is $1.5x + y = 6$ and not $x + y = 5$.
  • Figure 3: Multigraph $G$ used in the proof of Proposition \ref{['prp:lower-bound-condensation']} (left) and its condensation multigraph (right).
  • Figure 4: Graph $G_n$ used in the proof of Proposition \ref{['prp:nolow']} (left), and its condensation multigraph (right).

Theorems & Definitions (8)

  • proof
  • theorem 1
  • corollary 1
  • proof : Proof of Claim \ref{['clm:choice']}
  • proposition 1
  • proof : Proof of Claim \ref{['clm:cwlow']}
  • proposition 2
  • proof