## Minimum d-dimensional arrangement with fixed points

#### journal contribution

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n 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 ℤ^{d} (for some fixed dimension *d* ≥ 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 *k* 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} · 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 Ω(*k*^{1/4}). We also show that it is NP-hard to approximate 2-DIMAP+ within a factor better than Ω(*k*^{1/4–∊}). We also consider a (conceptually harder, but practically even more interesting) variant of 2-dimAP+, where the target space is the grid , instead of the entire integer lattice ℤ^{2}. For this problem, we obtain a *O*(*k*log*k*log*n*)-approximation using the same LP relaxation. We complement this upper bound by showing an integrality gap of Ω(*k*^{1/2}), and an Ω(*k*^{1/2–∊})-inapproximability result.

Our results naturally extend to the case of arbitrary fixed target dimension *d* ≥ 1.