Exponential-Time Approximation (Schemes) for Vertex-Ordering Problems

TL;DR

This work initiates a program of exponential-time approximation schemes for vertex-ordering problems by introducing a balanced-cut framework that yields runtimes below $O^*(2^n)$ for several problems and enables $(1+\varepsilon)$- and constant-factor approximations. Central to the methods is solving Directed Minimum $(k,n-k)$-Cut as a building block to partition the vertex set and combine partial solutions, with a self-improvement bootstrapping technique producing a $1+\varepsilon$-approximation for Feedback Arc Set in $O((2-\delta)^n)$ time. The authors obtain a spectrum of results: unweighted and weighted variants for Feedback Arc Set, Cutwidth, Optimal Linear Arrangement, and Pathwidth, including $2$- and $(3/2)$-approximation guarantees and subexponential-time algorithms (e.g., $O^*(2^{\omega n/3})$, $O(1.66^n)$). These findings suggest a fruitful link between exponential-time exact algorithms and approximation schemes for vertex-ordering problems, with potential extensions to other orderings like Treewidth or Bandwidth. The work also highlights open questions on lower-bounds, space optimization, and broader applicability of the balanced-cut paradigm.

Abstract

In this paper, we begin the exploration of vertex-ordering problems through the lens of exponential-time approximation algorithms. In particular, we ask the following question: Can we simultaneously beat the running times of the fastest known (exponential-time) exact algorithms and the best known approximation factors that can be achieved in polynomial time? Following the recent research initiated by Esmer et al. (ESA 2022, IPEC 2023, SODA 2024) on vertex-subset problems, and by Inamdar et al. (ITCS 2024) on graph-partitioning problems, we focus on vertex-ordering problems. In particular, we give positive results for Feedback Arc Set, Optimal Linear Arrangement, Cutwidth, and Pathwidth. Most of our algorithms build upon a novel ``balanced-cut'' approach, which is our main conceptual contribution. This allows us to solve various problems in very general settings allowing for directed and arc-weighted input graphs. Our main technical contribution is a (1+ε)-approximation for any ε > 0 for (weighted) Feedback Arc Set in O*((2-δ)^n) time, where δ > 0 is a constant only depending on ε.

Figures (4)

• Figure 1: An example for $\mathop{\mathrm{cut}}\nolimits_{\pi}$ in the directed setting. The vertices are ordered by some ordering $\pi$ from left to right. The cuts $\mathop{\mathrm{cut}}\nolimits_{\pi}^1$ and $\mathop{\mathrm{cut}}\nolimits_{\pi}^3$ are depicted by dashed lines. The cut $\mathop{\mathrm{cut}}\nolimits_{\pi}^1$ is empty and the cut $\mathop{\mathrm{cut}}\nolimits_{\pi}^3$ consists of all the red arcs. The stretch of both red arcs is 3.
• Figure 2: An example of Feedback Arc Set. The input graph is given on the left and the middle/right show two optimal solutions for Directed Minimum $(k,n-k)$-Cut with $k=3$ (red arcs). The optimal solution to Feedback Arc Set is shown in blue (and coincides with the red arc on the right). If we use the cut in the middle picture, then we can only find a 2-approximation while the cut on the right leads to an optimal solution.
• Figure 3: An example of Directed Cutwidth. The input instance is depicted on the left with the exception that all vertices in the left triangle have arcs towards all vertices in the right triangle unless the respective reverse arc is present. We decided to not show these arcs for the sake of readibility. On the right, two optimal solutions to Directed Minimum $(k,n-k)$-Cut with $k=3$ are shown (red arcs) with optimal orderings for the two respective subproblems of Directed Cutwidth are depicted. The ordering shown on the top has a directed cutwidth of 3 (e.g. at positions 4 and 5) and the ordering on the bottom has a directed cutwidth of $2$.
• Figure 4: An example of Optimal Linear Arrangement where the vertices are divided into two parts $L'$ and $R'$ with two edges between the two parts and some ordering within the two parts is fixed. For the sake of simplicity, we do not show edges within the two parts. The two orderings only differ in that we reverse the ordering of the vertices in $R'$. The average length within $L'$ of the two edges between $L'$ and $R'$ is 1.5 in both orderings. The average length within $R'$ is 3 in the above ordering and $2'$ in the below ordering.

Theorems & Definitions (24)

• Proposition 1
• proof
• Lemma 1
• proof
• Theorem 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof