In previous chapters we have explored constructions that optimize the three main design objectives of wirelength, skew, and delay. However, in practice we often seek to optimize multiple objectives simultaneously. This chapter explores ways of representing and addressing multiple competing objectives. We begin with a minimum density formulation for balancing the utilization of horizontal and vertical routing resources and describe heuristics with expected performance bounded by constants times optimal. This enables the simultaneous optimization of up to three objectives (e.g., radius/density/wirelength, or skew/density/wirelength at once), without degrading solution quality with respect to any of the objectives. We also discuss a non-uniform lower bound schema that affords tighter estimates of solution quality for a given problem instance.
Next, we develop a general framework of multiple-objective optimization, based on multi-weighted graphs (i.e., where edge weights are vectors rather than scalars). This formulation captures distinct criteria such as wirelength, jogs and congestion, and enables effective routing in graph-based regimes (i.e., routing in building-block designs, field-programmable gate arrays, and where obstacles are present). Finally, we discuss a network-flow based approach to prescribed-width routing where multiple objectives induce an arbitrarily costed region; applications of this include, e.g., circuit-board routing, and routing with respect to reliability or thermal considerations. This methodology departs from conventional shortest-path or graph-search based methods in that it applies to routing regions with a continuous cost function, as well as to regions containing solid polygonal obstacles. Extensions address the minimum-surface problem of Plateau, which is of independent interest.
KeywordsSpan Tree Multiple Objective Minimum Span Tree Steiner Tree Minimum Density
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