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x <- choice(S) creates I S I copies ofthe algorithm, and assigns every memberof the set S to the variable x in one of the copies.
failure causes that copy of the algorithm to stop execution.
success causes all copies of the algorithm to stop execution and indicatesa "yes" answer to that instance of the problem.
A no_ndrterministic polynomial-time algorithm for the decision version ofthe CLIQUE problem is the following: Let G = (V, E) be an undirectedgraph and let k > 0.
procedure CLIQUE(C. k)rbegin
l . A*@:2. for all r e ,/do.,t € cholcr(i.{ + 1r.},,{}):i3. lf l,{ | < A' then frilurci4. foral l r , veA,t + t | do5. lf rx' + E thcn fsilure::6, success:
erd
The loop in line 2 nondeterministically selects a subset of vertices ?4 s y;lines 4-6 decide if I is a complete set. If success is reached in one of thecopies, then the final value ofl in that copy is a clique ofsize at least k. Usingthe above procedure we obtain a nondeterministic polynomial+ime algolrithm for th_e optimization version of rhe CLIeUE pioUiem as follows: LetG be an undirected graph with n vertices.procedure MAXCLIQUE(C):be8ln
fork-21elsaep- ldolf CLIQUE(G, ,t) ther rerurn k:
end
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