Induced cycle

Algorithms and complexity

The complexity of approximating the longest induced path or cycle problems can be related to that of finding large independent sets in graphs, by the following reduction. From any graph G with n vertices, form another graph H with twice as many vertices as G, by adding to Gn(n − 1)/2 vertices having two neighbors each, one for each pair of vertices in G. Then if G has an independent set of size k, H must have an induced path and an induced cycle of length 2k, formed by alternating vertices of the independent set in G with vertices of I. Conversely, if H has an induced path or cycle of length k, any maximal set of nonadjacent vertices in G from this path or cycle forms an independent set in G of size at least k/3. Thus, the size of the maximum independent set in G is within a constant factor of the size of the longest induced path and the longest induced cycle in H. Therefore, by the results of Håstad (1996) on inapproximability of independent sets, unless NP=ZPP, there does not exist a polynomial time algorithm for approximating the longest induced path or the longest induced cycle to within a factor of O(n) of the optimal solution.