Influence-aware Task Assignment in Spatial Crowdsourcing (Technical Report)

A spatial task $s$ can be completed only if a worker arrives at its location before the expiration deadline $s.p+s.\varphi$. With single-task assignment mode, the SC server assigns each task to one worker at a time.

We use $|A|$ to denote the total number of assigned tasks in task assignment $A$. The problem investigated is stated as follows:

ITA Problem Statement. Given a set of workers and a set of tasks at the current time in an SC platform, our problem is to find a task assignment $\mathit{A_{\mathit{opt}}}$ that achieves the following goals:

1) primary optimization goal: maximize the total number of assigned tasks (i.e., $\forall$ $\mathit{A_{i}\in\mathbb{A}}$ ($|\mathit{A_{i}}|$$\leq$$|\mathit{\mathit{A_{opt}}}|$)), where $\mathbb{A}$ denotes all possible assignments; and

2) secondary optimization goal: maximize worker-task influence of assignments.

We propose a framework, Data-driven Influence-aware Task Assignment (DITA), to solve the ITA problem. The framework has two components: worker-task influence modeling and task assignment, as shown in Figure 2.

The first component aims to calculate worker-task influence. Specifically, we employ Latent Dirichlet Allocation (LDA) to measure workers’ affinities (i.e., worker-task affinity) towards tasks, where we treat the categories of tasks that workers have already completed as documents to train the LDA model, and then the categories of tasks and workers at the current time are input into the trained LDA model to compute the worker-task affinity. For worker willingness calculation, we propose a Historical Acceptance (HA) algorithm to measure the probability of a worker visiting the location of a task based on the task-performing history of the worker and the real-time locations of the worker and task. For worker propagation, we first exploit an Independent Cascade (IC) model to simulate information propagation process of tasks in a given social network, and then we propose a Random reverse reachable-based Propagation Optimization (RPO) algorithm to calculate worker propagation based on IC and social networks.

In the task assignment component, considering the spatio-temporal constraints (i.e., the reachable regions of workers and expiration times of tasks) of workers and tasks, we optimize the task assignment based on worker-task influence at each time instance and propose a basic Influence-aware Assignment (IA) method. Taking worker-task influence and location entropy into account, we propose an Entropy-based Influence-aware Assignment (EIA) method. Moreover, a Distance-based Influence-aware Assignment (DIA) method which considers worker-task influence and workers’ travel costs is developed.

In SC, different workers exhibit different affinities (i.e., preferences) for the same categories of tasks, leading to different task-performing behaviors. For example, a worker may like to report a hot spot, while another worker may prefer to monitor traffic conditions. Since task categories contain semantic information (e.g., restaurant) and the Latent Dirichlet Allocation (LDA) model [21] performs well at modeling semantic affinity (i.e., semantic matching) between text documents by learning topics, we employ it to quantify worker-task affinity.

In LDA, a document is regarded as a set of words generated by several topics, where each topic is described by terms following a probability distribution. The modeling process can be formalized as follows:

Here $P(v_{i}|d)$ is the probability of term $v_{i}$ for a document $d$ and $|\mathit{Top}|$ is the number of topics. Next, $P(v_{i}|t_{j})$ is the probability of $v_{i}$ within topic $t_{j}$, and $P(t_{j}|d)$ is the probability of picking a term from $t_{j}$ in document $d$. LDA estimates the topic-term distribution, $P(v_{i}|t_{j})$, and the document-topic distribution, $P(t_{j}|d)$, using Dirichlet priors. It iterates multiple times over each term in $d$ until the parameters in LDA converge. This way, we get the topic distribution of each document. Each topic is a probability distribution over a set of words. In the LDA model, words that are related semantically have high probability of belonging to the same topic.

In order to adapt LDA to model worker-task affinity, we treat each task category as a word, and we treat the categories of tasks in the historical task-performing records of worker $w_{i}$ as a document, denoted by $\mathit{dc}_{w_{i}}$. The documents across all workers on an SC platform form a set of documents that is used to train the LDA model, cf. Figure 3. Based on the documents, LDA can learn topics. Each topic is represented by a probability distribution over categories. For worker $w_{i}$ and task $s_{i}$ at the current time, we can use the trained LDA model to calculate the topic distribution, where the topic distribution of $w_{i}$ is calculated from the historical task-performing records that reflect the preferred category distribution of $w_{i}$, and the topic distribution of $s_{i}$ is calculated based on its categories. Then the learned topics and the document $\mathit{dc}_{s_{i}}$ formed by the categories of the location of $s_{i}$ are used to estimate the worker-task affinity, $P_{\mathit{aff}}$, as follows:

where $t$ denotes a topic and $\mathit{Top}$ is a set of learned topics. Further, $P(w_{i}|t)$ and $P(s_{i}|t)$ quantify how well topic $t$ matches the topic distribution of $w_{i}$’s historical task-performing records and the topic distribution of task $s_{i}$, respectively. A larger $P_{\mathit{aff}}(w_{i},s_{i})$ value indicates that $w_{i}$ is more likely to perform $s_{i}$, since the preferred category distribution of $w_{i}$ and that of the task $s_{i}$ are correlated better.

In general, different workers exhibit different willingness to visit the location of a task. Previous studies only consider real-time locations of workers and tasks [22, 23] when assigning tasks. However, measuring a worker’s willingness to visit the location of a task according to the distance between the worker’s real-time location and the task location represents an incomplete picture. The real-time location is temporary, and this ignores the worker’s historical task-performing patters.

To tackle this issue, we propose a Historical Acceptance (HA) approach to measure the willingness of a worker $w$ to visit the locations of particular tasks based on the worker’s historical task-performing records (denoted as $S_{w}$) and the real-time locations of workers and tasks, where $S_{w}=\{(s_{1},t^{a}_{s_{1}},t^{l}_{s_{1}}),(s_{2},t^{a}_{s_{2}},t^{l}_{s_{2}}),\ldots,(s_{n},t^{a}_{s_{n}},t^{l}_{s_{n}})\}$ and each triplet $(s_{i},t^{a}_{s_{i}},t^{l}_{s_{i}})$ consists of task $s_{i}$, a task arrival time $t^{a}_{s_{i}}$, and a task completion time $t^{l}_{s_{i}}$. The worker willingness is measured as the probability that a worker moves from the locations of the tasks they have performed to the location of the current task.

In particular, HA computes worker willingness in terms of stationary distribution modeling of workers’ historical mobility and movement probability density calculation.

When knowing information of a task, a worker has the potential to propagate the task’s information to other workers independently through social media. We propose worker propagation to measure the probability of workers being known tasks.

There are two main challenges when computing worker propagation. First, since the information propagation in a social network is complex, it is important to simulate the propagation process reasonably. Second, based on the propagation process, the computation of worker propagation should be completed in limited time since we aim to assign tasks to workers online. To address these challenges, we propose an approximation method, called Random reverse reachable-based Propagation Optimization (RPO), for calculating the worker propagation.

We combine worker-task affinity $P_{\mathit{aff}}$, worker willingness $P_{\mathit{wil}}$, and worker propagation $WP_{w_{s}}$ to calculate the worker-task influence, $\mathit{if}(w_{s},s)$, of worker $w_{s}$ and task $s$, as follows:

where $w_{s}$ is a worker who knows the information of $s$, $W$ denotes the worker set in the social network, and $P_{\mathit{pro}}(w_{s},w_{i})$ is the $i$-th value in $WP_{w_{s}}$.

In order to guarantee an approximation ratio of computing worker propagation based on Random Reverse Reachable (RRR) sets, we present a feasibility analysis of computing a suitable number $N$ of RRR sets. First, we introduce the notions of informed range and martingale and then use these to propose lemmas that facilitate the computation of the number of RRR sets. Then corresponding proofs are provided to guarantee a high approximation ratio between the estimated worker propagation and the worker propagation computed based on RRR sets. The notions of informed range and martingale [34] are defined as follows:

where $v_{j}=0$ if $\{w_{s}\}\cap R_{j}=\emptyset$; otherwise, $v_{j}=1$.

An important property of martingales [34] is shown as follows:

where $\mathit{Var}[\cdot]$ is the variance of a random variable.

Given a worker $w_{s}$, since each worker $w_{i}$ is selected uniformly at random to generate $R_{i}$ and the generation of $R_{i}$ is independent of $R_{1},R_{2},\ldots,R_{i-1}$, we have $\mathbb{E}[v_{i}|v_{1},v_{2},\ldots,v_{i-1}]=\mathbb{E}[v_{i}]=\sigma(w_{s})/|W|$. Let $\alpha=\sigma(w_{s})/|W|$ and $x_{i}=\sum_{j=1}^{i}(v_{j}-\alpha)$. It is clear that $\mathbb{E}[x_{i}]=0$ and $\mathbb{E}[x_{i}|x_{1},x_{2},\ldots,x_{i-1}]=x_{i-1}$, which indicates that $x_{1},x_{2},\ldots,x_{N}$ is a martingale. Moreover, based on $x_{i}=\sum_{j=1}^{i}(v_{j}-\alpha)$, it is clear that $|x_{1}|\leq 1$ and $|x_{i}-x_{i-1}|\leq 1$ for any $i\in[2,N]$. Combining this with the independence of $R_{i}$, we have:

Based on Lemma 3 and Equation 4, the following corollary is derived.

We obtain the corollary by applying Lemma 3 to $-x_{1},-x_{2},\ldots,-x_{N}$.

Given a set $\mathbb{R}=\{R_{1},R_{2},\ldots,R_{N}\}$ of RRR sets, let $f_{R}(w_{s})$ be the fraction of RRR sets in $\mathbb{R}$ that cover $w_{s}$. It is clear that the number $N_{p}(w_{s})$ of workers who can be informed by $w_{s}$ is $|W|\cdot f_{R}(w_{s})$. Based on Corollary 2, we have following lemma.

Lemma 4 shows that when $N^{\prime}$ is sizable, the calculation of worker propagation based on Equation 5 guarantees a $(1-\epsilon)$-approximate solution. However, the value of $\sigma$ is different for different workers, which makes it hard to derive a suitable value of $N^{\prime}$. Moreover, $N^{\prime}$ should be as smaller as possible to reduce the computation time. To address this issue, we derive a lower bound on $N^{{}^{\prime}}$. Let $w_{s}^{\tau}$ be a worker with maximum informed range, i.e., $\sigma(w_{s}^{\tau})\geq\sigma(w_{s})$, for any $w_{s}$ in $G$. We can rewrite Lemma 4 as follows:

The $\gamma$ in Lemma 5 is a threshold, to be computed in Lemma 6. Since we aim to guarantee an approximation that is as high as possible, the setting of $\lambda$ should be as low as possible (e.g., $\lambda=1/|W|^{o}$, where $o\geq 1$).

However, in real cases, $\sigma(w_{s}^{\tau})$ is unknown in advance. To address this problem, we derive a lower bound on $\sigma(w_{s}^{\tau})$ with the help of a so-called greedy informed worker that can be calculated in advance. A greedy informed worker is defined as follows:

In order to use $w_{s}^{\theta}$ to derive the lower bound on $\sigma(w_{s}^{\tau})$, we construct a test $T(\cdot)$ related to $w_{s}^{\theta}$ on a set of values (denoted as $K=\{k_{1},k_{2},\ldots\}$) and run the test on $K$. If $k_{i}>\sigma(w_{s}^{\tau})$ then $T(k_{i})=\mathit{false}$ with a high probability, and $k_{i}$ can be considered as a lower bound of $\sigma(w_{i}^{\tau})$. Let $N_{p}^{\mathit{opt}}=|W|\cdot f_{R}(w_{s}^{\theta})$. Based on Corollary 1, we can construct $T(\cdot)$ based on following lemma:

Based on Lemma 6, it is easy to see that if $N_{p}^{\mathit{opt}}\geq\gamma$, then $\sigma(w_{s}^{\tau})\geq k_{i}$ holds at least with probability $1-\lambda^{*}$, and $\sigma(w_{s}^{\tau})$ can be set to $N_{p}^{\mathit{opt}}\cdot k_{i}/\gamma$. We can set $K=\{|W|/2,|W|/4,|W|/8,\ldots,2\}$ and then run the test $T(k_{i}):N_{p}^{\mathit{opt}}\geq\gamma$ on $O(\log_{2}|W|)$ values of $K$ to compute the lower of $\sigma(w_{s}^{\tau})$. Moreover, since we need to guarantee the approximation with high probability (e.g., $1-1/|W|^{o}$), the setting of $\lambda^{*}$ can be $\frac{1}{|W|^{o}\cdot\log_{2}|W|}$.

Based on Lemmas 5 and 6, $\mathit{N}$ should be set to $\max\{N_{R}^{\prime}(\gamma),N_{R}(k_{i})\}$ to guarantee the approximation. However, it is difficult to choose suitable settings for $\epsilon$ and $\epsilon^{*}$ that minimize $\max\{N_{R}^{\prime}(\gamma),N_{R}(k_{i})\}$. To address that problem, $\max\{N_{R}^{\prime}(\gamma),N_{R}(k_{i})\}$ can be approximated with a simple function of $\epsilon$ and $\epsilon^{*}$, and then $\epsilon^{*}=\sqrt{2}\epsilon$ is derived as the minimizer of the function.

The proofs of above lemmas are shown as follows:

Proof of Lemma 4. Given any worker $w_{s}$, let $\alpha=\mathbb{E}[f_{R}(w_{s})]$, by Lemma 2, $\alpha=\mathbb{E}[f_{R}(w_{s})]=\sigma(w_{s})/|W|$.

Based on Corollary 2, we have

The proof of Lemma 5 is similar to that of Lemma 4 when setting $\alpha=\mathbb{E}[f_{R}(w_{s}^{\tau})]$.

Proof of Lemma 6. Given any worker $w_{s}$, let $\alpha=\mathbb{E}[f_{R}(w_{s})]$. The following equation can be derived.

Let $\beta=\frac{(1+\epsilon^{*})\cdot k_{i}}{|W|\cdot\alpha}-1$. By Equation III-E, we can get $\beta>\frac{\epsilon^{*}\cdot k_{i}}{|W|\cdot\alpha}>\epsilon^{*}$. Based on Corollary 1, we have

According to the union bound and $N_{p}(w_{s})\leq N_{p}^{\mathit{opt}}$, we have $N_{p}^{\mathit{opt}}<\gamma$ with at least $1-\lambda^{*}$ probability.

Taking worker-task influence as the priority of task assignment, we propose a basic Influence-aware Assignment (IA) algorithm to solve the ITA problem by transforming it to a Minimum Cost Maximum Flow (MCMF) [11] problem.

To adapt MCMF to the ITA problem, we first construct a task assignment graph based on the available workers and tasks. Specifically, given a set of workers $W=\{w_{1},w_{2},\ldots\}$, and a set of tasks $S=\{s_{1},s_{2},\ldots\}$ at time $t$, we construct a graph $G=(N,E)$, where $N$ and $E$ denote sets of nodes and edges, respectively. Let $|N|=|W|+|S|+2$ and $|E|=|W|+|S|+m$, where $m$ is the number of available assignments for all workers. Since tasks expire at their deadlines and workers only accept tasks in their reachable range, the available assignments for worker $w$, denoted as $w.A$, should satisfy the following conditions:

i) task $s$ is located in the reachable circular range of worker $w$, i.e., $d(w.l,s.l)\leq w.r$.

ii) worker $w$ has enough time to reach the location of $s$ before it expires, i.e., $t+t(w.l,s.l)\leq s.p+s.\varphi$.

We use $d(w.l,s.l)$ to denote the Euclidean distance between $w.l$ and $s.l$, and use $t(w.l,s.l)$ to denote the travel time from $w.l$ to $s.l$. For the sake of simplicity, we assume all the workers share the same travel speed, meaning that the travel time and distance are equivalent. However, the proposed algorithms can also address the cases where workers are moving at different speeds. Let $|w.A|$ be the number of available assignments for worker $w$, and thus $m$ can be derived by summing $|w.A|$ for all workers: $m=\sum_{w\in W}|w.A|$.

In graph $G$, nodes $n_{i}$ and $n_{|W|+j}$ correspond to a worker $w_{i}$ and a task $s_{j}$, respectively. Moreover, we add two new nodes (denoted as $n_{0}$ and $n_{|W|+|S|+1}$) as the source ($N_{s}$) and destination ($N_{d}$), respectively. An example graph $G$ for four workers and four tasks at the same time is illustrated in Figure 4. The graph is generated by following steps:

i) $N_{s}$ connects all worker nodes, and the capacities of the corresponding edges are set to 1, i.e., $c=1$, since each worker can perform only one task at a time. The costs of these edges are set to 0.

ii) Each task node connects with $N_{d}$, and the capacities of the corresponding edges are set to 1, indicating that each task can be assigned to at most 1 worker. The costs of these edges are set to 0.

iii) If the assignment $(s_{j},w_{i})$ is available, i.e., $(s_{j},w_{i})\in w.A$, we add an edge from worker node $n_{i}$ to task node $n_{|W|+j}$. The capacities of the corresponding edges are set to 1, and the cost (denoted as $w(n_{i},n_{|W|+j})$) is the ratio between 1 and the worker-task influence, $\mathit{if}(w_{i},s_{j})$, of $w_{i}$ and $s_{j}$, i.e., $w(n_{i},n_{|W|+j})=\frac{1}{\mathit{if}(w_{i},s_{j})+1}$.