Public Event Scheduling with Busy Agents

TL;DR

This work addresses Public Event Scheduling with Busy Agents (PESBA), where the goal is to maximize aggregate agent attendance by jointly scheduling events and agents’ preemptive tasks. The authors prove NP-hardness even in a restricted setting and develop a spectrum of algorithms that achieve provable guarantees under different timeline regimes, leveraging submodular optimization and matroid structure. For polynomially bounded timelines, a greedy approach yields a $\frac{1}{2}$-approximation, with a $1-\frac{1}{e}$-approximation achievable through submodular maximization with matroids; for arbitrary timelines, they extend a one-event approximation to a $\frac{1}{\alpha+1}$-approximation for multiple events, attaining $\frac{1}{2}$ in the end via an optimal one-event solver. A key contribution is the polynomial-time computation of the agreement function via min-cost max-flow and the identification of a scheduling matroid underpinning submodularity, enabling scalable, principled approximation frameworks with clear guarantees.

Abstract

We study a public event scheduling problem, where multiple public events are scheduled to coordinate the availability of multiple agents. The availability of each agent is determined by solving a separate flexible interval job scheduling problem, where the jobs are required to be preemptively processed. The agents want to attend as many events as possible, and their agreements are considered to be the total length of time during which they can attend these events. The goal is to find a schedule for events as well as the job schedule for each agent such that the total agreement is maximized. We first show that the problem is NP-hard, and then prove that a simple greedy algorithm achieves $\frac{1}{2}$-approximation when the whole timeline is polynomially bounded. Our method also implies a $(1-\frac{1}{e})$-approximate algorithm for this case. Subsequently, for the general timeline case, we present an algorithmic framework that extends a $\frac{1}α$-approximate algorithm for the one-event instance to the general case that achieves $\frac{1}{α+1}$-approximation. Finally, we give a polynomial time algorithm that solves the one-event instance, and this implies a $\frac{1}{2}$-approximate algorithm for the general case.

Figures (7)

• Figure 1: Illustration of the model of PSBA problem. There are two public events $E=\set{e_1,e_2}$ with length $l(e_1)=2$ and $l(e_2)=3$ represented by shaded rectangles, and two agents $A=\set{1,2}$. Agent $1$'s job set is $J_1$ containing two jobs $[1,3]$ with processing time $2$ and $[2,7]$ with processing time $3$. Agent $2$'s job set is $J_2$ containing two jobs $[7,11]$ with processing time $3$ and $[5,8]$ with processing time $2$. The time span is $T=[1,11]$ containing $11$ time slots. The figure shows one optimal event schedule ${\cal S}=\set{(e_1,3),(e_2,8)}$ maximizing $\sum_{i\in A}\mathsf{agr}_i({\cal S})$ and a corresponding optimal job schedule where the black disk indicates job's processing. Agent $1$ can attend the whole course of $e_1$ and $e_2$ under schedule ${\cal S}$ and thus get agreement $5$; agent $2$ can attend whole $e_1$ but can only attend $e_2$ for $2$ times and thus get agreement $4$ from ${\cal S}$.
• Figure 2: Originally $e_1$ has overlap with $e_2$ which starts before it and thus we delay $e_1$ to $[11,13)$ and then shift $e_1$ and $e_2$ together to $[6,12)$. The black disks indicate jobs' processing. We can see the shift increases the number of agreement time slots of $J_1$'s schedule from $4$ to $6$ while keeping the number of agreement time slots of $J_2$'s schedule unchanged.
• Figure 3: An example for min-cost max-flow construction. The left part of the figure shows the original job set and event schedule with $\Psi=\set{[1,2),[3,4),[4,5),[6,7),[7,8)}$. We cut the whole timeline into a set of segments $\Phi=\set{[1,3),[3,5),[5,7),[7,8)}$ with polynomial size according to $\Psi$. The constructed flow network is shown in the right part of the figure. For each job, we have a vertex in $A$, and for each segment, we have a vertex in $B$. The capacity and the cost of each edge are designed to ensure that a flow assignment corresponds to a job schedule, e.g., $(2,0)$ of the edge in $E_3$ connecting $\phi_1$ and $t$ indicates the capacity of this edge is $2$ and cost is $0$.
• Figure 4: Illustration of Case (III). The subfigure (i) shows the original bipartite graph $G:=(A\cup B, E)$, where $A$ corresponds to the job set and $B$ corresponds to the time slot set. The large vertex set $L=\set{b_2,b_3,b_4}$ and the small vertex set $S=\set{b_6,b_7}$. The subfigure (ii) shows a matching $M_S$ that does not use any vertices from $S$. The subfigure (iii) shows a matching $M_L$ that does not use any vertices from $L$. We can prove that $\lvert L\rvert+\lvert\mathsf{right}(M_L)\setminus S\rvert>\lvert A\rvert$. In the example, $\mathsf{right}(M_L)\setminus S=\set{b_1,b_5}$ which is shown in the subfigure (iii). This inequality implies that a vertex exists in $A$ that forms an alternative path. In the example, such a vertex is $a_1$, where $a_1$ is matched with $b_3\in L$ in $M_S$ and $a_1$ is matched with $b_1\in \mathsf{right}(M_L)\setminus S$. Thus, the path $(b_1,a_1,b_3)$ forms an alternative path shown in the subfigure (iv). By rematching along the alternative path, we match $a_1$ with other vertices that are not in $S$ and thus release a vertex from $L$.
• Figure 5: Illustration of time slots aggregation (line \ref{['line:aggregate']} of \ref{['alg:poly:position']}). The subfigure (i) presents an optimal job schedule computed by the min-cost max-flow algorithm. If we do not gather the agreement time slots, it is possible that the agreement time slots are separated by the job's processing. In subfigure (ii), such time slots producing agreements are aggregated to form agreement segments. Each segment can have at most one agreement segment.

Theorems & Definitions (38)

• proof
• Theorem 1
• proof
• Theorem 2
• Definition 1: Submodular Maximization with Group Constraints and Imperfect Oracle (SMGC)
• proof : Proof of \ref{['thm:pseudo']}
• Claim 1: DBLP:journals/siamcomp/CalinescuCPV11
• Lemma 4
• proof
• proof : Proof of \ref{['lem:pseudo:agr']}