A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 17.2 Hard Problems 5693EA8224B3 611 1DCFigure 17.4 An illustration of the TRAVELING SALESMAN problem. Fivevertices are shown, with edges between each pair of cities. The problem is to visitall of the cities exactly once, returning to the start city, with the least total cost.route would be ABDCEA, with cost 11. The best route for this particular graphwould be ABEDCA, with cost 9.We cannot solve this problem in polynomial time with a guess-and-test nondeterministiccomputer. The problem is that, given a candidate cycle, while we canquickly check that the answer is indeed a cycle of the appropriate form, and whilewe can quickly calculate the length of the cycle, we have no easy way of knowingif it is in fact the shortest such cycle. However, we can solve a variant of thisproblem cast in the form of a decision problem. A decision problem is simply onewhose answer is either YES or NO. The decision problem form of TRAVELINGSALESMAN is as follows:TRAVELING SALESMAN (2)Input: A complete, directed graph G with positive distances assigned toeach edge in the graph, and an integer k.Output: YES if there is a simple cycle with total distance ≤ k containingevery vertex in G, and NO otherwise.We can solve this version of the problem in polynomial time with a non-deterministiccomputer. The non-deterministic algorithm simply checks all of the possiblesubsets of edges in the graph, in parallel. If any subset of the edges is anappropriate cycle of total length less than or equal to k, the answer is YES; otherwisethe answer is NO. Note that it is only necessary that some subset meet therequirement; it does not matter how many subsets fail. Checking a particular subsetis done in polynomial time by adding the distances of the edges and verifyingthat the edges form a cycle that visits each vertex exactly once. Thus, the checkingalgorithm runs in polynomial time. Unfortunately, there are 2 |E| subsets to check,so this algorithm cannot be converted to a polynomial time algorithm on a regu-