Theorem 1

In the last lecture, we studied the KNAPSACK problem. The KNAPSACK problem is an NP-hard problem but does admit a pseudo-polynomial time algorithm and can be solved efficiently if the input size is small. We used this pseudo-polynomial time algorithm to obtain an FPTAS for KNAPSACK. In this lecture, we study another class of problems, known as strongly NP-hard problems. Definition 1 (Strongly NP-hard Problems) An NPO problem π is said to be strongly NP-hard if it is NP-hard even if the inputs are polynomially bounded in combinatorial size of the problem 1. Many NP-hard problems are in fact strongly NP-hard. If a problem Π is strongly NP-hard, then Π does not admit a pseudo-polynomial time algorithm. For more discussion on this, refer to [1, Section 8.3] We study two such problems in this lecture, MULTIPROCESSOR SCHEDULING and BIN PACKING. A central problem in scheduling theory is to design a schedule such that the finishing time of the last jobs (also called makespan) is minimized. This problem is often referred to as the LOAD BALANCING , the MINIMUM MAKESPAN SCHEDULING or MULTIPROCESSOR SCHEDULING problem.