Numerical evidence for phase transitions of NP-complete problems for instances drawn from Lévy-stable distributions

Abstract

Random NP-Complete problems have come under study as an important tool used in the analysis of optimization algorithms and help in our understanding of how to properly address issues of computational intractability. In this thesis, the Number Partition Problem and the Hamiltonian Cycle Problem are taken as representative NP-Complete classes. Numerical evidence is presented for a phase transition in the probability of solution when a modified Lévy-Stable distribution is used in instance creation for each. Numerical evidence is presented that show hard random instances exist near the critical threshold for the Hamiltonian Cycle problem. A choice of order parameter for the Number Partition Problem’s phase transition is also given. Finding Hamiltonian Cycles in Erdös-Rényi random graphs is well known to have almost sure polynomial time algorithms, even near the critical threshold. To the author’s knowledge, the graph ensemble presented is the first candidate, without specific graph structure built in, to generate graphs whose Hamiltonicity is intrinsically hard to determine. Random graphs are chosen via their degree sequence generated from a discretized form of Lévy-Stable distributions. Graphs chosen from this distribution still show a phase transition and appear to have a pickup in search cost for the algorithms considered. Search cost is highly dependent on the particular algorithm used and the graph ensemble is presented only as a potential graph ensemble to generate intrinsically hard graphs that are difficult to test for Hamiltonicity. Number Partition Problem instances are created by choosing each element in the list from a modified Lévy-Stable distribution. The Number Partition Problem has no known good approximation algorithms and so only numerical evidence to show the phase transition is provided without considerable focus on pickup in search cost for the solvers used. The failure of current approximation algorithms and potential candidate approximation algorithms are discussed.