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Understanding the Empirical Hardness of NP-Complete Problems

Understanding the Empirical Hardness of NP-Complete Problems, illustration

Credit: Eugene Kostsov

Problems are intractable when they "can be solved, but not fast enough for the solution to be usable."13 NP-complete problems are commonly said to be intractable; however, the reality is more complex. All known algorithms for solving NP-complete problems require exponential time in the worst case; however, these algorithms nevertheless solve many problems of practical importance astoundingly quickly, and are hence relied upon in a broad range of applications. The propositional satisfiability problem (SAT) serves as a good example. One of the most popular approaches for the formal verification of hardware and software relies on general-purpose SAT solvers and SAT encodings, typically with hundreds of thousands of variables. These instances can often be solved in seconds, even though the same solvers can be stymied by handcrafted instances involving only hundreds of variables.

Clearly, we could benefit from a more nuanced understanding of algorithm behavior than is offered by asymptotic, worst-case analysis. Our work asks the question most relevant to an end user: "How hard is it to solve a given family of problem instances, using the best available methods?" Formal, complexity-theoretic analysis of this question seems hopeless: the best available algorithms are highly complex (and, in some cases, only available in compiled form), and instance distributions representative of practical applications are heterogeneous and richly structured. For this reason, we turn to statistical, rather than combinatorial, analysis.


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