Minimum d-dimensional arrangement with fixed points

TL;DR

This paper studies the d-dimensional arrangement problem with fixed points, providing approximation algorithms, integrality gap bounds, and hardness results, especially for the 2D case and its variants relevant to VLSI design.

Contribution

It introduces approximation algorithms and hardness bounds for the generalized d-dimAP+ problem, extending previous work to fixed points and grid constraints.

Findings

O(k^{1/2} log n)-approximation for 2D d-dimAP+

NP-hardness of approximation within Omega(k^{1/4- ext{epsilon}})

Approximation and integrality gap bounds for grid-constrained variants

Abstract

In the Minimum $d$-Dimensional Arrangement Problem (d-dimAP) we are given a graph with edge weights, and the goal is to find a 1-1 map of the vertices into $\mathbb{Z}^d$ (for some fixed dimension $d\geq 1$) minimizing the total weighted stretch of the edges. This problem arises in VLSI placement and chip design. Motivated by these applications, we consider a generalization of d-dimAP, where the positions of some of the vertices (pins) is fixed and specified as part of the input. We are asked to extend this partial map to a map of all the vertices, again minimizing the weighted stretch of edges. This generalization, which we refer to as d-dimAP+, arises naturally in these application domains (since it can capture blocked-off parts of the board, or the requirement of power-carrying pins to be in certain locations, etc.). Perhaps surprisingly, very little is known about this problem from an approximation viewpoint. For dimension $d=2$, we obtain an $O(k^{1/2} \cdot \log n)$-approximation algorithm, based on a strengthening of the spreading-metric LP for 2-dimAP. The integrality gap for this LP is shown to be $\Omega(k^{1/4})$. We also show that it is NP-hard to approximate 2-dimAP+ within a factor better than $\Omega(k^{1/4-\eps})$. We also consider a (conceptually harder, but practically even more interesting) variant of 2-dimAP+, where the target space is the grid $\mathbb{Z}_{\sqrt{n}} \times \mathbb{Z}_{\sqrt{n}}$, instead of the entire integer lattice $\mathbb{Z}^2$. For this problem, we obtain a $O(k \cdot \log^2{n})$-approximation using the same LP relaxation. We complement this upper bound by showing an integrality gap of $\Omega(k^{1/2})$, and an $\Omega(k^{1/2-\eps})$-inapproximability result. Our results naturally extend to the case of arbitrary fixed target dimension $d\geq 1$.