Fair Hitting Sequence Problem: Scheduling Activities with Varied Frequency Requirements

Abstract

Given a set $V=\{v_1,\ldots , v_n\}$ of n elements and a family $\mathcal{{S}} = \{ S_1, S_2, \ldots , S_m\}$ of (possibly intersecting) subsets of V, we consider a scheduling problem of perpetual monitoring (attending) these subsets. In each time step one element of V is visited, and all sets in $\mathcal{{S}}$ containing v are considered to be attended during this step. That is, we assume that it is enough to visit an arbitrary element in $S_j$ to attend to this whole set. Each set $S_j$ has an urgency factor $h_j$, which indicates how frequently this set should be attended relatively to other sets. Let $t_i^{(j)}$ denote the time slot when set $S_j$ is attended for the i-th time. The objective is to find a perpetual schedule of visiting the elements of V, so that the maximum value $h_j\left( t_{i+1}^{(j)}-t_i^{(j)}\right)$ is minimized. The value $h_j\left( t_{i+1}^{(j)}-t_i^{(j)}\right)$ indicates how urgent it was to attend to set $S_j$ at the time slot $t_{i+1}^{(j)}$. We call this problem the Fair Hitting Sequence (FHS) problem, as it is related to the minimum hitting set problem. In fact, the uniform FHS, when all urgency factors are equal, is equivalent to the minimum hitting set problem, implying that there is a constant $c_0>0$ such that it is NP-hard to compute $(c_0\log m)$-approximation schedules for FHS.

We demonstrate that scheduling based on one hitting set can give poor approximation ratios, even if an optimal hitting set is used. To counter this, we design a deterministic algorithm which partitions the family $\mathcal {S}$ into sub-families and combines hitting sets of those sub-families, giving $O(\log ^2 m)$-approximate schedules. Finally, we show an LP-based lower bound on the optimal objective value of FHS and use this bound to derive a randomized algorithm which with high probability computes $O(\log m)$-approximate schedules.

The work has been supported in part by the European project “Geospatial based Environment for Optimisation Systems Addressing Fire Emergencies” (GEO-SAFE), contract no. H2020-691161, by the Italian National Group for Scientific Computation GNCS-INdAM, by Networks Sciences and Technologies (NeST) initiative at University of Liverpool, and by the Polish National Science Center (NCN) grant 2017/25/B/ST6/02010.