Approximation Algorithms for Correlated Knapsack Orienteering

TL;DR

This paper introduces the correlated knapsack orienteering problem, analyzes its adaptivity gap, and develops non-adaptive algorithms with provable approximation guarantees for various cases, including special distributions.

Contribution

It is the first to study CSKO, providing bounds on the adaptivity gap and designing approximation algorithms for general and special distribution cases.

Findings

The adaptivity gap of CSKO is at least ((igl(\, ext{max}\sqrt{ ext{} ext{log} ext{B}}, ext{sqrt} ext{} ext{} ext{log} ext{} ext{W}}igr))

Non-adaptive algorithms achieve an O( ext{log} ext{log W})-approximation in quasi-polytime

For weighted Bernoulli distributions, an O(1)-approximation is achievable in polynomial time.

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 $\rho\in V$, 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 $\rho$-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 $\Omega\bigl(\max\sqrt{\log{B}},\sqrt{\log\log{W}}\}\bigr)$}. 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 $\Omega(\sqrt{\log\log B})$ 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.