CS 261 : A Second Course in Algorithms Lecture # 15 : Introduction to Approximation Algorithms ∗

All of CS161 and the first half of CS261 focus on problems that can be solved in polynomial time. A sad fact is that many practically important and frequently occurring problems do not seem to be polynomial-time solvable, that is, are NP -hard. As an algorithm designer, what does it mean if a problem is NP -hard? After all, a real-world problem doesn’t just go away after you realize that it’s NP -hard. The good news is that NP -hardness is not a death sentence — it doesn’t mean that you can’t do anything practically useful. But NP -hardness does throw the gauntlet to the algorithm designer, and suggests that compromises may be necessary. Generally, more effort (computational and human) will lead to better solutions to NP -hard problems. The right effort vs. solution quality trade-off depends on the context, as well as the relevant problem size. We’ll discuss algorithmic techniques across the spectrum — from low-effort decent-quality approaches to high-effort high-quality approaches. So what are some possible compromises? First, you can restrict attention to a relevant special case of an NP -hard problem. In some cases, the special case will be polynomialtime solvable. (Example: the Vertex Cover problem is NP -hard in general graphs, but on Problem Set #2 you proved that, in bipartite graphs, the problem reduces to max flow/min cut.) In other cases, the special case remains NP -hard but is still easier than the general case. (Example: the Traveling Salesman Problem in Lecture #16.) Note that this approach requires non-trivial human effort — implementing it requires understanding and articulating