Limitations of Stochastic Selection with Pairwise Independent Priors

TL;DR

This work considers the instructive case of pairwise-independent priors and matroid constraints, and examines the class of matroids which satisfy the so-called partition property -- these include most common matroids encountered in optimization.

Abstract

Motivated by the growing interest in correlation-robust stochastic optimization, we investigate stochastic selection problems beyond independence. Specifically, we consider the instructive case of pairwise-independent priors and matroid constraints. We obtain essentially-optimal bounds for contention resolution and prophet inequalities. The impetus for our work comes from the recent work of Caragiannis et al., who derived a constant-approximation for the single-choice prophet inequality with pairwise-independent priors. For general matroids, our results are tight and largely negative. For both contention resolution and prophet inequalities, our impossibility results hold for the full linear matroid over a finite field. We explicitly construct pairwise-independent distributions which rule out an omega(1/Rank)-balanced offline CRS and an omega(1/log Rank)-competitive prophet inequality against the (usual) oblivious adversary. For both results, we employ a generic approach for constructing pairwise-independent random vectors -- one which unifies and generalizes existing pairwise-independence constructions from the literature on universal hash functions and pseudorandomness. Specifically, our approach is based on our observation that random linear maps turn linear independence into stochastic independence. We then examine the class of matroids which satisfy the so-called partition property -- these include most common matroids encountered in optimization. We obtain positive results for both online contention resolution and prophet inequalities with pairwise-independent priors on such matroids, approximately matching the corresponding guarantees for fully independent priors. These algorithmic results hold against the almighty adversary for both problems.

Figures (2)

• Figure 1: A snapshot of the construction of $B_\ell$ and partition $\mathbf P_\ell$ for the initial $3$ levels. In this illustration, vertically aligned dots in groups of three represent the same basis vector. At any given level $\ell$, vectors indicated in solid colors comprise the alive vectors $B_\ell$ of level $\ell$. Furthermore, the colored boxes at each level $\ell$, along with their constituent alive basis vectors, represent the parts $P_\ell(i)$ of the partition of $\mathbf P_\ell$. Random parts $\mathbf {\widetilde{P}_\ell}$ that survive through to the next level are indicated by upward arrows.
• Figure 2: Construction of $\Sigma_3(j)$ matrix.

Theorems & Definitions (45)

• Theorem 2.1: Theorem 3.6 from dughmi20
• Lemma 2.2: Lemma 1 from pi-uniform-prophet
• Lemma 4.1
• proof
• Lemma 4.1
• Lemma 4.1
• Theorem 5.1
• Claim 5.2
• proof
• Lemma 5.3