Hardness and approximation of submodular minimum linear ordering problems

The minimum linear ordering problem (MLOP) generalizes well-known combinatorial optimization problems such as minimum linear arrangement and minimum sum set cover. MLOP seeks to minimize an aggregated cost \(f(\cdot )\) due to an ordering \(\sigma \) of the items (say [n]), i.e., \(\min _{\sigma } \sum _{i\in [n]} f(E_{i,\sigma })\), where \(E_{i,\sigma }\) is the set of items mapped by \(\sigma \) to indices [i]. Despite an extensive literature on MLOP variants and approximations for these, it was unclear whether the graphic matroid MLOP was NP-hard. We settle this question through non-trivial reductions from mininimum latency vertex cover and minimum sum vertex cover problems. We further propose a new combinatorial algorithm for approximating monotone submodular MLOP, using the theory of principal partitions. This is in contrast to the rounding algorithm by Iwata et al. (in: APPROX, 2012), using Lovász extension of submodular functions. We show a \((2-\frac{1+\ell _{f}}{1+|E|})\)-approximation for monotone submodular MLOP where \(\ell _{f}=\frac{f(E)}{\max _{x\in E}f(\{x\})}\) satisfies \(1 \le \ell _f \le |E|\). Our theory provides new approximation bounds for special cases of the problem, in particular a \((2-\frac{1+r(E)}{1+|E|})\)-approximation for the matroid MLOP, where \(f = r\) is the rank function of a matroid. We further show that minimum latency vertex cover is \(\frac{4}{3}\)-approximable, by which we also lower bound the integrality gap of its natural LP relaxation, which might be of independent interest.