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, Tetali, and Tripathi [ITT2012], using Lov\'asz 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 $1 \leq \ell_f \leq |E|$. 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 (MLVC) 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.

翻译：最小线性排序问题（MLOP）推广了诸如最小线性排列和最小集合覆盖等经典组合优化问题。MLOP旨在通过物品（设为 $[n]$）的排序 $\sigma$ 最小化聚合成本 $f(\cdot)$，即 $\min_{\sigma} \sum_{i\in [n]} f(E_{i,\sigma})$，其中 $E_{i,\sigma}$ 是 $\sigma$ 映射到索引 $[i]$ 的物品集合。尽管已有大量关于MLOP变体及其近似的研究文献，但图拟阵MLOP是否为NP-hard问题仍不清楚。我们通过最小延迟顶点覆盖和最小和顶点覆盖问题的非平凡归约解决了这一疑问。进一步，我们利用主划分理论提出了一种新的组合算法来近似单调子模MLOP，这与Iwata、Tetali和Tripathi [ITT2012] 使用子模函数Lovász延拓的舍入算法形成对比。我们展示了单调子模MLOP的一个 $(2-\frac{1+\ell_{f}}{1+|E|})$ 近似，其中 $\ell_{f}=\frac{f(E)}{\max_{x\in E}f(\{x\})}$ 满足 $1 \leq \ell_f \leq |E|$。我们的理论为该问题的特例提供了新的近似界，特别地，对于拟阵MLOP（其中 $f = r$ 是拟阵的秩函数），得到了一个 $(2-\frac{1+r(E)}{1+|E|})$ 近似。我们进一步证明了最小延迟顶点覆盖（MLVC）是 $\frac{4}{3}$ 可近似的，由此也给出了其自然线性规划松弛的积分间隙的下界，这可能具有独立意义。