Shortest Cycles with Monotone Submodular Costs

We introduce the following submodular generalization of the Shortest Cycle problem. For a nonnegative monotone submodular cost function f defined on the edges (or the vertices) of an undirected graph G, we seek for a cycle Cin Gof minimum cost OPT = f (C). We give an algorithm that given an n-vertex graph G, parameter e > 0, and the function f represented by an oracle, in time n O( log 1/e) finds a cycle Cin Gwith f (C) =(1 + e) center dot OPT. This is in sharp contrast with the non-approximability of the closely related Monotone Submodular Shortest (s, t)-Path problem, which requires exponentially many queries to the oracle for finding an n2/3- e -approximation Goel et al. [7], FOCS 2009. We complement our algorithm with a matching lower bound. We show that for every e > 0, obtaining a (1 + e)-approximation requires at least nO(log 1/e) queries to the oracle. When the function f is integer-valued, our algorithm yields that a cycle of cost OPT can be found in time n O( log OPT). In particular, for OPT = n O( 1) this gives a quasipolynomial-time algorithm computing a cycle of minimum submodular cost. Interestingly, while a quasipolynomial-time algorithm often serves as a good indication that a polynomial time complexity could be achieved, we show a lower bound that n O( log n) queries are required even when OPT = O( n). We also consider special cases of monotone submodular functions, corresponding to the number of different color classes needed to cover a cycle in an edge-colored multigraph G. For special cases of the corresponding minimization problem, we obtain fixed-parameter tractable algorithms and polynomial-time algorithms, when restricted to certain classes of inputs.