A polynomial algorithm for recognizing bounded cutwidth in hypergraphs

The Min Cut Linear Arrangement (Min Cut) or Backboard Permutation (BP) problem, where it is desired to minimize “backplane area” or “cutwidth” in hypergraphs, has a long history of interest. To determine, for given graphG and integerk, whetherG has cutwidth at mostk is known to beNP-complete even for planar graphs with maximum vertex degree 3. (As graphs are a special case of hypergraphs, it is alsoNP-complete for hypergraphs.) Recently, Cahoon and Sahni described O(n) and O(n3) algorithms for determining if a hypergraph had cutwidth 1 and 2, respectively. However, for any fixedk>2, it remained open whether determining if an arbitrary hypergraph has cutwidth at mostk was in the classP. We show a positive answer; specifically, we describe an O(nm) algorithm, withm=k2+3k+3, which determines if a hypergraph withn vertices has cutwidthk.

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