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Learning Minimum Linear Arrangement of Cliques and Lines

Julien Dallot, Maciej Pacut, Marcin Bienkowski, Darya Melnyk, Stefan Schmid

TL;DR

The paper addresses online learning variants of the Minimum Linear Arrangement problem where the graph is revealed incrementally and updates to a MinLA of the revealed subgraph incur a Kendall's $\tau$ cost. It introduces a deterministic baseline and two optimal randomized algorithms for two restricted topologies: collections of cliques and collections of lines, achieving $2n-2$, $4\ln n$, and $8\ln n$ competitive ratios respectively, and proves an $\Omega(\log n)$ lower bound for randomized online algorithms. The methods hinge on careful probabilistic merging/orientation of components and harmonic-number analyses to bound moving and rearranging costs. The results provide asymptotically tight online guarantees in these restricted settings and offer insights for dynamic virtual network embedding scenarios, highlighting the potential and limits of online learning in MinLA.

Abstract

In the well-known Minimum Linear Arrangement problem (MinLA), the goal is to arrange the nodes of an undirected graph into a permutation so that the total stretch of the edges is minimized. This paper studies an online (learning) variant of MinLA where the graph is not given at the beginning, but rather revealed piece-by-piece. The algorithm starts in a fixed initial permutation, and after a piece of the graph is revealed, the algorithm must update its current permutation to be a MinLA of the subgraph revealed so far. The objective is to minimize the total number of swaps of adjacent nodes as the algorithm updates the permutation. The main result of this paper is an online randomized algorithm that solves this online variant for the restricted cases where the revealed graph is either a collection of cliques or a collection of lines. We show that the algorithm is $8 \ln n$-competitive, where $n$ is the number of nodes of the graph. We complement this result by constructing an asymptotically matching lower bound of $Ω(\ln n)$.

Learning Minimum Linear Arrangement of Cliques and Lines

TL;DR

The paper addresses online learning variants of the Minimum Linear Arrangement problem where the graph is revealed incrementally and updates to a MinLA of the revealed subgraph incur a Kendall's cost. It introduces a deterministic baseline and two optimal randomized algorithms for two restricted topologies: collections of cliques and collections of lines, achieving , , and competitive ratios respectively, and proves an lower bound for randomized online algorithms. The methods hinge on careful probabilistic merging/orientation of components and harmonic-number analyses to bound moving and rearranging costs. The results provide asymptotically tight online guarantees in these restricted settings and offer insights for dynamic virtual network embedding scenarios, highlighting the potential and limits of online learning in MinLA.

Abstract

In the well-known Minimum Linear Arrangement problem (MinLA), the goal is to arrange the nodes of an undirected graph into a permutation so that the total stretch of the edges is minimized. This paper studies an online (learning) variant of MinLA where the graph is not given at the beginning, but rather revealed piece-by-piece. The algorithm starts in a fixed initial permutation, and after a piece of the graph is revealed, the algorithm must update its current permutation to be a MinLA of the subgraph revealed so far. The objective is to minimize the total number of swaps of adjacent nodes as the algorithm updates the permutation. The main result of this paper is an online randomized algorithm that solves this online variant for the restricted cases where the revealed graph is either a collection of cliques or a collection of lines. We show that the algorithm is -competitive, where is the number of nodes of the graph. We complement this result by constructing an asymptotically matching lower bound of .
Paper Structure (19 sections, 15 theorems, 27 equations, 2 figures)

This paper contains 19 sections, 15 theorems, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Motivation: Dynamic Virtual Network Embedding
  4. Related Work
  5. Organization
  6. A Deterministic Algorithm
  7. An Optimal Randomized Algorithm for Cliques
  8. Description of the Algorithm
  9. Notation
  10. Upper Bound on the Expected Cost
  11. An Optimal Randomized Algorithm for Lines
  12. Description of the Algorithm
  13. Additional Notation for the Proofs
  14. Upper Bound on the Moving Costs
  15. Upper Bound on the Rearranging Costs
  16. ...and 4 more sections

Key Result

Theorem 1

Det is $(2n-2)$-competitive for the case where the graph is either a collection of lines or a collection of cliques.

Figures (2)

  • Figure 1: The two possible actions of Rand to move the merging components $X_{i}$ and $Z_{i}$ together when $G_{i+1}$ is revealed
  • Figure 2: The two possible actions of Rand to rearrange the components $X_{i}$ and $Z_{i}$ together for a particular configuration of the merging components

Theorems & Definitions (29)

  • Theorem 1
  • proof
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 4
  • proof
  • Theorem 5
  • ...and 19 more