The design of approximation algorithms has reached a period of relative maturity as a discipline. We hope that the wealth of results presented herein makes a strong case that this is so. However, we also believe that there is much work remaining, with many more fundamental contributions yet to be discovered.
We will outline a few questions to highlight some of the research that we speculate might have the potential to surprise us with new directions for this area. Since “top 10” lists are the norm not just for year-end film critics and late-night show hosts, we will structure these thoughts in that format.
For many optimization problems, the ultimate result is a performance guarantee with a matching lower bound (based on a complexity-theoretic assumption, at least until the time that questions such as P vs. NP are resolved). For a significant fraction of this book, we have been concerned with designing α-approximation algorithms for some constant α – polynomial-time algorithms that find solutions of objective function value within a factor of α of optimal. Implicitly, we were seeking the best value α that is achievable for the given problem at hand.
One of the significant developments of the past decade is the introduction of the unique games conjecture, as discussed in Section 16.5. This conjecture provides a stronger complexity-theoretic hypothesis on which to base lower bounds for performance guarantees, and recent work has shown that tight bounds on performance guarantees follow for a wide swath of optimization problems.