# The *T*-join Problem in Sparse Graphs: Applications to Phase Assignment Problem in VLSI Mask Layout

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## Abstract

Given a graph *G* with weighted edges, and a subset of nodes *T*, the *T*-join problem asks for a minimum weight edge set A such that a node u is incident to an odd number of edges of *A* iff *u* ∈ *T*. We describe the applications of the *T*-join problem in sparse graphs to the phase assignment problem in VLSI mask layout and to conformal refinement of finite element meshes. We suggest a practical algorithm for the Tjoin problem. In sparse graphs, this algorithm is faster than previously known methods. Computational experience with industrial VLSI layout benchmarks shows the advantages of the new algorithm.

## Keywords

Planar Graph Perfect Match Dual Graph Sparse Graph Conflict Graph
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