The Computational Complexity of the Traveling Salesman Problem

In this note, we show that the Traveling Salesman Problem cannot be solved in polynomial-time on a classical computer. Disclaimer: This article was authored by Craig Alan Feinstein in his private capacity. No official support or endorsement by the U.S. Government is intended or should be inferred. Consider the following well-known NP-hard problem: Traveling Salesman Problem A traveling salesman starts at city 1, travels to cities 2, . . . , n− 1 in any order that the salesman chooses, and then ends his trip in city n. Let us denote δ(i, j) to be the distance from city i to city j. The goal of the Traveling Salesman Problem is to find the minimum total distance possible for the traveling salesman to travel. There are no restrictions on the possible distances δ(i, j) between each of the cities other than the requirement that each δ(i, j) is a positive integer and δ(i, j) = δ(j, i) [1, 3, 4]. We give a simple proof that no deterministic and ex-act algorithm can solve the Traveling Salesman Problemin o(2n) time:For any nonempty subset S ⊆ {2, . . . , n} and for anycity i ∈ S, let us define ∆(S, i) to be the length of theshortest path that starts at city 1, visits all cities in theset S − {i}, and finally stops at city i. Then the Trav-eling Salesman Problem is equivalent to the problem ofcomputing ∆({2, . . . , n}, n). Clearly, ∆({i}, i) = δ(1, i)and∆(S, i) = min{∆(S − {i}, j) + δ(j, i) | j ∈ S − {i}},when |S| ≥ 2.This recursive formula cannot be simplified, so thefastest way to compute ∆({2, . . . , n}, n) is to apply thisrecursive formula to ∆({2, . . . , n}, n). Since this involvescomputing ∆(S, i) for all Θ(2n) nonempty subsets S ⊆{2, . . . , n− 1} and each i ∈ S, we obtain a lower boundof Θ(2n) for the worst-case running-time of any deter-ministic and exact algorithm that solves the TravelingSalesman Problem.This lower bound is confirmed by the fact that thefastest known deterministic and exact algorithm whichsolves the Traveling Salesman Problem was first pub-lished in 1962 and has a running-time of Θ∗(2n) [2, 4].References [1] T.H. Cormen, C.E. Leiserson, and R.L. Rivest, Intro-duction to Algorithms, McGraw-Hill, 1990. [2] M. Held and R.M. Karp, “A Dynamic ProgrammingApproach to Sequencing Problems”, Journal of SIAM10, pp. 196-210, 1962. [3] C.H. Papadimitriou and K. Steiglitz, Combinatorial Op-timization: Algorithms and Complexity, Prentice-Hall,Englewood Cliffs, NJ, 1982. [4] G.J. Woeginger, “Open problems around exact algo-rithms”, Discrete Applied Mathematics, 156(3): 397-405, 2008.