Exhaustive Search, Combinatorial Optimization and Enumeration: Exploring the Potential of Raw Computing Power
For half a century since computers came into existence, the goal of finding elegant and efficient algorithms to solve “simple” (wellde fined and well-structured) problems has dominated algorithm design. Over the same time period, both processingan d storage capacity of computers have increased by roughly a factor of a million. The next few decades may well give us a similar rate of growth in raw computing power, due to various factors such as continuingmi niaturization, parallel and distributed computing. If a quantitative change of orders of magnitude leads to qualitative advances, where will the latter take place? Only empirical research can answer this question.
Asymptotic complexity theory has emerged as a surprisingly effective tool for predictingru n times of polynomial-time algorithms. For NPhard problems, on the other hand, it yields overly pessimistic bounds. It asserts the non-existence of algorithms that are efficient across an entire problem class, but ignores the fact that many instances, perhaps includingt hose of interest, can be solved efficiently. For such cases we need a complexity measure that applies to problem instances, rather than to over-sized problem classes.
Combinatorial optimization and enumeration problems are modeled by state spaces that usually lack any regular structure. Exhaustive search is often the only way to handle such “combinatorial chaos”. Several general purpose search algorithms are used under different circumstances. We describe reverse search and illustrate this technique on a case study of enumerative optimization: enumeratingt he k shortest Euclidean spanning trees.
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