3 The 1 – semi – unitary

We now present the algorithm to compute Ham(A) for arbitrary matrices over K of characteristic 3. ALGORITHM 4 Given a (n n)–matrix A over K, we perform the following steps. (i) We choose a skew–symmetric regular (n ? 2 n ? 2)–matrix R. (ii) We define B = I m + Kg(A) ^ R Kg(A) T which by proposition 4.5 is 2–semi–regular. (iii) We choose 2n?1 values for and compute by interpolation the coefficients of the hamiltonian polynomial p A. (iv) Using theorem 4.9 we get PHam(A) = c 2n?2. Next we present the randomized algorithm for determining whether a given undirected graph with n vertices has a hamiltonian path between two distinguished vertices labelled 1 and n. ALGORITHM 5 Let G be the adjacency matrix of a graph with n vertices and let K = GF(3 q) with q 2log 3 n. (i) Choose randomly an (n n)–matrix Z over K. (ii) Compute G Z. (iii) Using theorem 4.9, compute z = PHam(G Z). (iv) If z 6 = 0 then by 4.3(i) G has a hamiltonian path. (v) If z = 0 then by 4.3(ii) G has a hamiltonian path with high probability. We have presented a completely new approach to computing permanents of certain matrices, the 1–semi–unitary matrices, over fields of characteristic 3. We have shown that computing the permanent of 2–semi–unitary matrices is as difficult as the general case. Problem 5.1 What is the complexity of computing perma-nents of unitary matrices over some fixed characteristic p different from 2; 3? We have shown how computing permanents of 2–semi– unitary matrices can be used for a randomized algorithm which checks the existence of Hamiltonian paths in arbitrary graphs. Problem 5.2 Is there an interesting combinatorial problem P which can be shown to be in R using our algorithm for 1–semi–unitary matrices? We are currently investigating the satisfiability problem UNISAT of such a class of CNF-formulas, [5], but it is still not clear whether UNISAT is not polynomial for some trivial reason. (ii) Let m 2 be the number of hamiltonian paths from 1 to n in G. Then PHam(G) = m 2 (mod 3). (iii) G has a hamiltonian cycle iff Ham(G Z) is not identically 0 as a polynomial in the entries of Z. (iv) G has a hamiltonian path from 1 to n iff PHam(G Z) is not identically 0 as a polynomial in the …