Adiabatic Quantum Algorithms for the NP-Complete Maximum-Weight Independent Set, Exact Cover and 3SAT Problems

The problem Hamiltonian of the adiabatic quantum algorithm for the maximum-weight independent set problem (MIS) that is based on the reduction to the Ising problem (as described in [Choi08]) has flexible parameters. We show that by choosing the parameters appropriately in the problem Hamiltonian (without changing the problem to be solved) for MIS on CK graphs, we can prevent the first order quantum phase transition and significantly change the minimum spectral gap. We raise the basic question about what the appropriate formulation of adiabatic running time should be. We also describe adiabatic quantum algorithms for Exact Cover and 3SAT in which the problem Hamiltonians are based on the reduction to MIS. We point out that the argument in Altshuler et al.(arXiv:0908.2782 [quant-ph]) that their adiabatic quantum algorithm failed with high probability for randomly generated instances of Exact Cover does not carry over to this new algorithm.