Minimum d-dimensional arrangement with fixed points

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 Zd (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(k1/2 · 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(k1/4). We also show that it is NP-hard to approximate 2-dimAP+ within a factor better than Ω(k1/4-e). We also consider a (conceptually harder, but practically even more interesting) variant of 2-dimAP+, where the target space is the grid Z√n x Z√n, instead of the entire integer lattice Z2. 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(k1/2), and an Ω(k1/2-e)-inapproximability result. Our results naturally extend to the case of arbitrary fixed target dimension d ≥ 1.