Approximation Algorithms for Correlated Knapsack Orienteering

TL;DR

This work introduces correlated knapsack orienteering (CorrKO) and its variant CSKO, where two budgets govern travel and processing times and vertex rewards/sizes are random but revealed on completion. It proves a non-constant adaptivity gap for CorrKO/CSKO and provides non-adaptive, LP-based algorithms achieving $O(\log\log W)$- and $O(\log W)$-approximations in quasi-polytime and polynomial time, respectively, with extensions to cancellations. A central structural tool is a portal-based decomposition of an optimal adaptive policy, enabling a configuration LP formulation that can be rounded to non-adaptive solutions; the paper also develops a tight, fine-grained analysis around distributions supported on two points (2CorrKO) and relates these to orienteering with knapsack deadlines (OrientKD). The results place CorrKO in a nuanced position: harder than stochastic knapsack orienteering but easier than CorrO, with strong algorithmic guarantees in both quasi-polytime and polytime regimes, and practical reductions to well-studied subproblems like KnapOrient and OrientKD. These connections yield new LP-based approximation methods and sharpen the understanding of adaptivity gaps in stochastic routing with resource constraints.

Abstract

We consider the {\em correlated knapsack orienteering} (CSKO) problem: we are given a travel budget $B$, processing-time budget $W$, finite metric space $(V,d)$ with root $ρ\in V$, where each vertex is associated with a job with possibly correlated random size and random reward that become known only when the job completes. Random variables are independent across different vertices. The goal is to compute a $ρ$-rooted path of length at most $B$, in a possibly adaptive fashion, that maximizes the reward collected from jobs that are processed by time $W$. To our knowledge, CSKO has not been considered before, though prior work has considered the uncorrelated problem, {\em stochastic knapsack orienteering}, and {\em correlated orienteering}, which features only one budget constraint on the {\em sum} of travel-time and processing-times. We show that the {\em adaptivity gap of CSKO is not a constant, and is at least $Ω\bigl(\max\sqrt{\log{B}},\sqrt{\log\log{W}}\}\bigr)$}. Complementing this, we devise {\em non-adaptive} algorithms that obtain: (a) $O(\log\log W)$-approximation in quasi-polytime; and (b) $O(\log W)$-approximation in polytime. We obtain similar guarantees for CSKO with cancellations, wherein a job can be cancelled before its completion time, foregoing its reward. We also consider the special case of CSKO, wherein job sizes are weighted Bernoulli distributions, and more generally where the distributions are supported on at most two points (2-CSKO). Although weighted Bernoulli distributions suffice to yield an $Ω(\sqrt{\log\log B})$ adaptivity-gap lower bound for (uncorrelated) {\em stochastic orienteering}, we show that they are easy instances for CSKO. We develop non-adaptive algorithms that achieve $O(1)$-approximation in polytime for weighted Bernoulli distributions, and in $(n+\log B)^{O(\log W)}$-time for the more general case of 2-CSKO.

Figures (2)

• Figure 1: The ($S_{{v}}^{(2)}$, $R_{{v}}^{(2)}$), ($S_{{v}}^{(1)}$, $R_{{v}}^{(1)}$) pairs are shown respectively on the left and right of each highlighted vertex in the tree.
• Figure 2: The portal nodes $\mathsf{Por}\xspace$ and paths between consecutive portal nodes. The solid nodes depict $\mathsf{Por}\xspace_j$.

Theorems & Definitions (43)

• Theorem 4
• Corollary 5
• Theorem 6
• Theorem 7
• Theorem 8
• Claim 9
• Lemma 10
• Lemma 11
• Lemma 12
• Lemma 13