CHAPTER 10 P and NP

Algorithm Crossword Puzzle This algorithm solves a crossword puzzle, represented by a Boolean matrix D[i,j], 1 = i,j = n, and a finite set of words W ⊆ Σ *. Σ is the alphabet, and D[i,j] is true if the square is blank and false if it is blocked. We construct the solution in a new matrix S[i,j], 1 = i,j = n. The algorithm returns true if the crossword can be solved. Input Parameters: D, W Output Parameters: None puzzle(D,W) { for i = 1 to n for j = 1 to n if (D[i,j]) S[i,j] = guess(Σ) else S[i,j] = blocked for each word w in S if (w  W) return false return true }

Algorithm Graph k-coloring This algorithm finds a k-coloring of G = (V,E), if there is one, and stores it in the array c. The algorithm returns true if a coloring is found. Input Parameters: G = (V,E), k Output Parameters: None graph_coloring(G,k) { for each v in V c[v] = guess({1,2,...,k}) for each v in V for each w in N(v) if (c[w] == c[v]) return false return true }

Algorithm Hamiltonian Cycle This algorithm finds a Hamiltonian cycle in G = (V,E) if there is one and returns true in that case.

Input Parameter: G = (V,E) Output Parameters: None hamiltonian_cycle(G) { n = |V| for i = 1 to n visited[i] = false for i = 1 to n { c[i] = guess(V) visited[c[i]] = true } c[0] = c[n] // first node is the same as last // check that only edges of G are used for i = 0 to n - 1 if ((c[i],c[i + 1])  E) return false // check that all vertices have been visited for i = 1 to n if (!(visited[i])) return false return true }

Algorithm TSP This algorithm finds a Hamiltonian cycle in G = (V, E, weight) of total weight at most w if there is one and returns true in that case. Input Parameters: G = (V, E, weight), w Output Parameters: None tsp(G,w) { n = |V| for i = 1 to n c[i] = guess(V) c[0] = c[n]...

// check that only edges of G are used, // and compute the total weight of the tour totalweight = 0 for i = 1 to n - 1 if ((c[i],c[i + 1])  E) return false else totalweight = totalweight + weight((c[i],c[i + 1])) // reject tours whose total weight is too large if (totalweight > w) return false // check that all vertices are visited for i = 1 to n visited[i] = false for i = 1 to n visited[c[i]] = true for i = 1 to n if (!(visited[i])) return false return true }

Example graph_coloring(G,k) { for each v in V c[v] = 0 for each v in V { c[v] = guess({1,2,...,k}) for each w in N(v) if (c[w] == c[v]) return false } return true }

Algorithm Satisfiability Witness This algorithm takes as input a CNF formula ϕ on variables x 1,...,x n, and either returns a satisfying assignment for ϕ in the array x or false if there is no such assignment. It assumes that we have an algorithm A that decides whether a formula is satisfiable or not.

Input Parameter: ϕ Output Parameter: x satisfiability_witness( ϕ, x) { if (!(A( ϕ )) return false for i = 1 to n { ψ = ϕ [x i → true] if (A( ϕ )) { x[i] = true ϕ = ψ } else { x[i] = false ϕ = ϕ [x i → false] } return true }