In this lecture, we provide another class of relaxations, called Semidefinite Programming Relaxation. These serve as relaxations for several NP-hard problems, in particular, for problems that can be expressed as strict quadratic programs. The relaxed problems, together with techniques like randomized rounding, give good approximation algorithms to hard combinatorial problems. We will illustrate...

We develop new techniques for deriving strong computational lower bounds for a class of well-known NP-hard problems. This class includes weighted satisfiability, dominating set, hitting set, set cover, clique, and independent set. For example, although a trivial enumeration can easily test in time O(nk) if a given graph of n vertices has a clique of size k, we prove that unless an unlikely coll...

We report results from large-scale experiments in satissability testing. As has been observed by others, testing the satissability of random formulas often appears surprisingly easy. Here we show that by using the right distribution of instances, and appropriate parameter values, it is possible to generate random formulas that are hard, that is, for which satissability testing is quite diicult....

Aweak completeness phenomenon is investigated in the complexity class E = DTIME(2linear). According to standard terminology, a language H is m-hard for E if the set Pm(H), consisting of all languages A m H , contains the entire class E. A language C is P m-complete for E if it is m-hard for E and is also an element of E. Generalizing this, a language H is weakly m-hard for E if the set Pm(H) do...

We report results from large-scale experiments in satisfiability testing. As has been observed by others, testing the satisfiability of random formulas often appears surprisingly easy. Here we show that by using the right distribution of instances, and appropriate parameter values, it is possible to generate random formulas that are hard, that is, for which satisfiability testing is quite diffi...