Finding a Path of Superlogarithmi Length

Andreas Björklund and Thore Husfeldt Department of Computer S ien e, Lund University Abstra t. We onsider the problem of nding a long, simple path in an undire ted graph. We present a polynomial-time algorithm that nds a path of length Ω (logL/ log logL) , where L denotes the length of the longest simple path in the graph. This establishes the performan e ratio O |V |(log log |V |/ log |V |) for the Longest Path problem, where V denotes the graph's verti es. 1 Introdu tion Given an unweighted, undire ted graphG = (V,E) the longest path problem is to nd the longest sequen e of distin t verti es v1 · · · vk su h that vivi+1 ∈ E. This is a lassi al NP-hard problem (number ND29 in Garey and Johnson [5℄) with a onsiderable body of resear h devoted to it, yet its approximability remains elusive: For most anoni al NP-hard problems, either dramati ally improved approximation algorithms have been devised, or strong negative results have been established, leading to a substantially improved understanding of the approximability of these problems. However, there is one problem whi h has resisted all attempts at devising either positive or negative results longest paths and y les in undire ted graphs. Essentially, there is no known algorithm whi h guarantees approximation ratio better than |V |/polylog|V | and there are no hardness of approximation results that explain this situation. [4℄ Indeed, the quoted ratio has been obtained only for spe ial lasses of graphs (for example, Hamiltonian graphs), while in the general ase the best known ratio prior to the present paper was of order |V |/ log |V |. We present a polynomial-time algorithm for the general ase that nds a path of length Ω((logL/ log logL)2) in a graph with longest path length L; the best previous bound was Ω(logL). This orresponds to a performan e ratio of order O ( |V | (