Nonconstructive Advances in Polynomial-Time Complexity

Tim field of computational complexity for concrete, practical combinatorial problems has developed in a remarkably smt~,.th fashi.rt One can point to several t'catures of the theory of polyoowaal-time computability which make it especially well-behaved, including: (1) the modelling of feasible computing by polynomial-time complexity is well-supported by the fact that Mmost all known polynomial-time algcrithm: for natural problems have running times bounded by polynomials of small degree; (2) problems are invariably known to be decidable in polynomial time by direct evidence in the form of efficient algorithms; (3) while the theory is formulated in terms of decision problems, almost all known algorithms proceed by actually constraeting a solution to the problem at hand. Herein we illustrate how recent advances in graph the~ry and graph algorithms dramatically alter this situation on all three counts. Powerful and e~y-to-apply tools are now available for classifying problems as decidable in polynomial time by nonconstructively proving only the existence of polynomial-time dceision algorithms. These tools neither specify the degree of the polynomial, nor produce the decision algorithm, nor guarantee that such an algorithm is ol any use in coustcecting a soh. ,ion. These developments present both pra'..tiuoners and theorists with novel challenges.