In the NODE-DISJOINT PATHS problem, the input is an undirected graph in which some vertices have been specially marked: a certain number of “sources”
s1; s2,...sk and an equal number of “destinations” t1; t2,...tk. The goal is to find k node-disjoint paths (that is, paths which have no nodes in common) where the ith path goes from si to ti. Show that this problem is NP-complete.
Here is a sequence of progressively stronger hints.
(i) Reduce from 3SAT.
(ii) For a 3SAT formula with m clauses and n variables, use k = m + n sources and destinations. Introduce one source/destination pair (sx, tx) for each variable x, and one source/destination pair (sc, tc) for each clause c.
(iii) For each 3SAT clause, introduce 6 new intermediate vertices, one for each literal occurring in that clause and one for its complement.
(iv) Notice that if the path from sc to tc goes through some intermediate vertex representing, say, an occurrence of variable x, then no other path can go through that vertex. What vertex would you like the other path to be forced to go through instead?
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