Rigorous analysis of heuristics for NP-hard problems

NP-hard optimization problems have to be solved in practice. Since it is our belief that there are no efficient algorithms that always solve such problems, we seek for heuristic algorithms that perform well ”most of the time”. This brings up the important issue of how to evaluate the quality of heuristics. One common criteria is the approximation ratio which gives a worst case guarantee for the quality of the heuristic with respect to the optimal solution. However, there are problems such as max 3-SAT for which this measure cannot differentiate between a most simple heuristic and a sophisticated one. This issue motivated the rise of other models for evaluating heuristics such as the pure random model, the planted random model, the semi-random model and the smoothed analysis model. These models give a rigorous framework for evaluating heuristics. This thesis deals mainly with problems under the framework of the pure and the planted random models. Within these models, our work is focused on sparser instances than were previously studied. The results are divided into three parts. The first part (Chapters 2, 3, 4) deals with refutation of kCNF formulas. The density of a kCNF formula is the ratio between the number of clauses and the number of variables. We give a polynomial time algorithm that refutes almost all 3CNF formulas with n variables and at least cn3/2 clauses. For densities below n1/2 we show a weaker result: almost all 3CNF formulas with at least cn1+2/5 clauses have a polynomial size witness for unsatisfiability. Furthermore, when such a refutation witness exists it can be found in time exp(O(n1/5 log n)). We also give a result of a negative type: a certain semidefinite program is unlikely to refute a random 3CNF formula with less than n3/2−o(1) clauses. The second part (Chapter 5) deals with the eigenvalue gap (of the adjacency matrix) of sparse random graphs. Consider a random graph G taken from Gn,p where the expected degree d ' np is a large enough constant. It is well known that the largest eigenvalue of such a graph is likely to be (1 + o(1))d. Using an idea of Alon and Kahale [3], we show that by removing from G a few vertices with exceptionally high degree, one gets a new graph in which the seconds largest eigenvalue (in absolute value) is at most O( √ d). This result is then used to extend the analysis of certain heuristics to sparser (random) instances of NP-hard problems. We demonstrate it on the algorithm of Goerdt and Krivelevich (from STACS 2001) for refuting random 2kCNF formulas. We also give heuristics for refuting the existence of large cuts and large independent sets in such random graphs. In the third part we give a heuristic for finding a maximum independent set in a random sparse graph. We use the following planted model for producing a random instance. The random graph G contains n vertices, every edge is included independently with probability p, where pn is larger than some fixed constant d0. Thereafter, for some constant α, a subset I of αn vertices is chosen at random, and all edges within this subset are removed. As α becomes smaller, the problem is more similar to the problem of finding a maximum independent set in a random Gn,p graph and thus it is considered harder. This model was previously studied in [4] for p = 1/2 and in [18] for