Online Combinatorial Optimization with Graphical Dependencies

TL;DR

The paper addresses online combinatorial optimization when inputs exhibit structured correlations captured by Δ-MRFs, interpolating between independence and strong dependence. It provides two unified approaches: (i) a minimization framework that reduces Δ-MRF arrivals to p-sample independent models, achieving O(Δ)-competitive results for Facility Location, Steiner Tree, and related covering problems, and (ii) a maximization framework that extends balanced prices to MRFs, yielding O(Δ)-competitive posted-pricing for XOS-valued buyers and O(k^2(Δ+log k)) for k-uniform hypergraph matching. A key insight is that a single n-dimensional sample from the MRF suffices for the minimization reductions, and the analyses leverage tail-core decompositions and coupling arguments to manage correlated tails. The results significantly extend prior MRF work, enabling nontrivial, linear-in-Δ guarantees for broad online allocation and coverage problems, with hardness matching up to polylog factors in several cases, and implications for truthful mechanism design via posted pricing.

Abstract

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.

Figures (2)

• Figure 1: Outline of the sequence of reductions for the minimization case. An arrow $A \rightarrow B$ means problem A reduces to problem B.
• Figure 2: MRF $\mathcal{M} = (\Omega, E, \{\psi_i\}_i,\{\psi_e\}_e\}$ with random variables $Y_1, \dots, Y_n$.

Theorems & Definitions (58)

• Theorem 1.1: Informal \ref{['thm:full_reduction_minimization']}
• Corollary 1.2
• Theorem 1.3: Informal Theorem \ref{['thm:max_main']}
• Definition 2.1: Markov Random Field
• Definition 2.2: Weighted maximum degree
• Definition 3.1: Generalized coverage constraints ArgyFrieGuptSeil2022
• Definition 3.2: (Online) Subadditive Coverage
• Definition 3.3: $\Delta$-MRF prophet model
• Definition 3.4: $p$-sample independent model
• Theorem 3.5