Most existing work in online stochastic combinatorial optimization assumes that inputs are drawn from independent distributions -- a strong assumption that often fails in practice. At the other extreme, arbitrary correlations are equivalent to worst-case inputs via Yao's minimax principle, making good algorithms often impossible. This motivates the study of intermediate models that capture mild correlations while still permitting non-trivial algorithms. In this paper, we study online combinatorial optimization under Markov Random Fields (MRFs), a well-established graphical model for structured dependencies. MRFs parameterize correlation strength via the maximum weighted degree $Δ$, smoothly interpolating between independence ($Δ= 0$) and full correlation ($Δ\to \infty$). While naïvely this yields $e^{O(Δ)}$-competitive algorithms and $Ω(Δ)$ hardness, we ask: when can we design tight $Θ(Δ)$-competitive algorithms? We present general techniques achieving $O(Δ)$-competitive algorithms for both minimization and maximization problems under MRF-distributed inputs. For minimization problems with coverage constraints (e.g., Facility Location and Steiner Tree), we reduce to the well-studied $p$-sample model. For maximization problems (e.g., matchings and combinatorial auctions with XOS buyers), we extend the "balanced prices" framework for online allocation problems to MRFs.

翻译：现有在线随机组合优化研究大多假设输入数据来自独立分布——这一强假设在实践中常不成立。另一极端情况下，任意相关性通过姚氏极小极大原理等价于最坏情况输入，使得设计有效算法通常不可行。这促使我们研究能捕捉适度相关性、同时仍允许非平凡算法的中间模型。本文研究基于马尔可夫随机场（MRF）的在线组合优化问题，MRF是刻画结构化依赖关系的成熟图模型。MRF通过最大加权度$Δ$参数化相关性强弱，平滑地插值于独立性（$Δ=0$）与完全相关性（$Δ\\to\\infty$）之间。虽然朴素方法会得到$e^{O(Δ)}$竞争比算法和$Ω(Δ)$硬度下界，但我们探究：何时能设计紧致的$Θ(Δ)$竞争比算法？我们提出通用技术，为MRF分布输入下的最小化与最大化问题实现$O(Δ)$竞争比算法。对于具有覆盖约束的最小化问题（如设施选址与斯坦纳树），我们将其归约至广泛研究的$p$样本模型。对于最大化问题（如匹配问题与含XOS买家的组合拍卖），我们将在线分配问题的“平衡定价”框架扩展至MRF场景。