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 C in G of minimum cost 𝖮𝖯𝖳 = f(C). We give an algorithm that given an n-vertex graph G, parameter ɛ > 0, and the function f represented by an oracle, in time n^𝒪(log 1/ɛ) finds a cycle C in G with f(C)≤ (1+ɛ). 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 n^2/3-ɛ-approximation Goel et al. [7], FOCS 2009. We complement our algorithm with a matching lower bound. We show that for every ɛ > 0, obtaining a (1+ɛ)-approximation requires at least n^{Ω (log 1/ ɛ)} 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.

我们引入以下最短周期问题的子模推广。对于在无向图G的边（或顶点）上定义的非负单调子模成本函数f ，我们在G中寻找最小成本 𝖮𝖯𝖳 = f(C)的循环C。我们给出一个算法，给定一个n顶点图G ，参数 ɛ > 0，以及由预言机表示的函数f ，在时间n ^{𝒪 (log 1/ɛ) 中找到}G中的循环C，且f(C) ≤ ( 1+ε）。𝖮𝖯𝖳。这与密切相关的单调子模最短 ( s,t -路径问题)的不可近似性形成鲜明对比，该问题需要对预言机进行指数级的多次查询才能找到n ^2/3-ɛ近似 Goel 等人[7] ]，FOCS 2009。我们用匹配的下界补充了我们的算法。我们表明，对于每个 ɛ > 0，获得 (1+ɛ) 近似值需要至少 n Ω (log 1/ ɛ) 次向预言机^查询。

当函数f为整数值时，我们的算法得出可以在时间n ^{𝒪(log 𝖮𝖯𝖳)}内找到成本周期 𝖮𝖯𝖳 。特别是，对于 𝖮𝖯𝖳 = n ^𝒪(1)，这给出了计算最小子模成本循环的拟多项式时间算法。有趣的是，虽然拟多项式时间算法通常可以很好地表明可以实现多项式时间复杂度，但我们显示了一个下限，即即使当 𝖮𝖯𝖳= 𝒪( ⁿ )时也需要n ^𝒪(log n ⁾查询。

我们还考虑单调子模函数的特殊情况，对应于覆盖边缘彩色多重图G中的循环所需的不同颜色类的数量。对于相应的最小化问题的特殊情况，当限制于某些类别的输入时，我们获得固定参数易处理算法和多项式时间算法。