From Prediction to Action: A Constraint-Based Approach to Predictive Policing

Crime prevention is a key concern in urban management because it directly affects public safety and quality of life. One of the main challenges for law enforcement agencies is deciding how to use their limited resources most effectively. Over the years, researchers have shown that accurate crime prediction models can help police forces act proactively. By anticipating when and where crimes are likely to occur, patrol units can be deployed more efficiently, enhancing their overall impact [25].

Detecting areas with high crime activity, known as crime “hotspots,” has been a major focus of criminological studies. Early work, such as the concept of “prospective hot-spotting” by Bowers et al. [3], demonstrated that concentrating on locations with recent crime surges can guide future crime prevention. Subsequent research has used more advanced approaches, like kernel density estimation (KDE) [14, 11] and spatiotemporal modeling [5], to identify evolving crime hotspots.

In parallel, artificial intelligence and machine learning methods have improved both the accuracy and depth of crime prediction. Deep learning architectures, such as convolutional neural networks (CNNs) and long short-term memory (LSTM) networks, are now frequently used to forecast spatiotemporal crime patterns [21, 12, 10, 18]. Meanwhile, multi-density clustering methods [4] are also integrated for improved forecasting.

Other researchers combine decomposition methods and network-based approaches to address short-term crime fluctuations. For example, Zhu et al. [30] propose a hierarchical crime prediction framework that merges a modified gated Graph Convolutional Network with Variational Mode Decomposition, modeling frequency-specific temporal dependencies.

Beyond these frameworks, some studies integrate environmental features and demand-based predictions. Lin et al. [19] develop a deep neural network that fuses criminal environmental data with spatiotemporal crime features for grid-based forecasting, while Ke et al. [15] introduce an end-to-end CNN model that predicts on-demand ride services, offering potential insights for crime-related event prediction.

Once crime-prone areas have been identified, another crucial task is to optimize how patrols are assigned so that resources are used effectively. Various mathematical models, such as integer and mixed-integer programming [7, 17, 23], schedule patrol units with minimal cost and travel time. In addition, metaheuristics (e.g., genetic algorithms, ant colony optimization) are also used to design efficient routes [6, 16, 20], helping agencies balance coverage in high-risk areas with practical constraints [26, 8].

Despite these advances, there remains a notable gap between crime forecasting and patrol optimization. Most predictive models generate heatmaps or highlight hotspots but do not integrate these insights into fully actionable patrol schedules. Conversely, patrol-optimization frameworks often rely on static or simplistic crime data, overlooking the nuanced forecasts produced by modern machine learning.

In this paper, we bridge this gap by presenting a framework that combines machine learning and constraint programming using the Inductive Constraint Programming Loop (ICON loop) [1]. Specifically, we develop a crime prediction model based on PredRNN++ [27], capturing spatiotemporal crime patterns and updating dynamically as new data arrive. We then embed this forecasting module within a constraint programming framework that optimizes patrol routes based on projected crime risks and operational constraints, ensuring available resources are allocated with maximum effectiveness.

By adopting the ICON loop paradigm, our approach establishes a continuous feedback loop: crime forecasts inform patrol planning, and the impact of patrols feed back into subsequent predictive refinements. This synergy between advanced crime forecasting and combinatorial route optimization enables more adaptive and impactful patrol strategies than previous methods, which have largely tackled these challenges in isolation.

The remainder of this paper is organized as follows. Section 2 provides a recap of the ICON loop. Section 3 describes the crime data processing pipeline. Section 4 details our predictive modeling approach. Section 5 presents the constraint programming module and patrol route optimization. Section 6 reports the experimental setup and results, and Section 7 concludes.

The Inductive Constraint Programming Loop (ICON loop) [1] is a framework designed to seamlessly integrate machine learning (ML) with constraint programming (CP). Its main objective is to enable constraint models to adapt to changing environments by drawing on observations collected from the World (W). As shown in Figure 1, the ICON loop consists of three principal components: the CP module, the ML module, and the W module.

The high-level procedure is outlined in Algorithm 1. At the start of each cycle, new observations are gathered from the W component and recorded in the Observations repository. These observations are then passed in parallel to the CP and ML modules via World-to-CP and World-to-ML (lines 7, 3). To form the learning problem $L$, the World-to-ML channel (line 3) extracts features or insights directly from the world data, and the CP-to-ML channel (line 4) can incorporate information derived from previous solutions, if available. The outputs of these channels collectively define $L$, which is then processed by applyXLearn (line 6) to generate new patterns or hypotheses.

Next, the constraint network is updated. Observations feed into the CP module via World-to-CP (line 7), and the newly derived patterns are injected through ML-to-CP (line 8). The constructN function (line 9) merges these two inputs to build or refine the constraint network $N$. Finally, applyXSolve (line 10) generates a set of candidate solutions for the updated constraints network.

These solutions are then applied to the world using Apply-to-World. If the solutions turn out to be invalid or inapplicable, the loop immediately begins a new iteration to refine the candidate solutions. If a solution is successfully applied to the world, the cycle concludes, and new observations can be gathered in preparation for the next cycle.

Not all channels in the ICON loop need to be active in every scenario [1]. In our setting, we assume that the machine learning step relies solely on newly collected observations rather than any previous solutions. Therefore, the CP-to-ML channel is omitted, and $L_p$ is not included in constructing the learning problem $L$. This streamlined approach simplifies the loop without compromising its ability to adapt to new environmental data.

In this section, we describe how the data for the World (W) component is prepared for the machine learning (ML) module. We use the Chicago Crime Dataset from the City of Chicago Data Portal [24] as our source of observations. From this dataset, we focus on three key attributes essential for spatiotemporal prediction: latitude ($\varphi$), longitude ($\lambda$), and the time of occurrence. To systematically handle the city’s geographic extent, we partition the study area into an $I\times J$ grid of rectangular cells, as shown in Figure 2 ²²2In our experiments, we set $I=100$ and $J=100$.. Each cell is labeled $c_{ij}$. This grid structure allows us to aggregate and analyze crimes at a manageable level of granularity.

The machine learning component estimates future crime risk using spatiotemporal data provided by the W component, specifically the slot-specific sequences $\{{𝒮}_1,\mathrm{\dots },{𝒮}_{s_{\max }}\}$ of quantile-based matrices generated by Algorithm 2 (Section 3). Our goal is to predict how crime patterns in each time slot will evolve (one step ahead) based on historical observations of that same slot from previous days.

To achieve this, we employ a PredRNN++ architecture [27]. PredRNN++ is a recurrent neural network specifically designed to predict future video frames based on a sequence of past frames. It processes each input frame through multiple recurrent layers, each composed of a specialized convolutional unit known as a Causal LSTM. Every Causal LSTM cell maintains two distinct memory states: a short-term spatial memory that captures local visual patterns (e.g., textures and shapes), and a long-term temporal memory that tracks motion dynamics across frames. These two memories are updated sequentially, with information flowing in a strictly causal direction ensuring that predictions adhere to the chronological structure of video sequences. To address the vanishing gradient problem, where learning signals degrade across long temporal horizons, PredRNN++ introduces a Gradient Highway Unit (GHU). This unit provides a shortcut path for gradients to propagate more directly across time, substantially improving training efficiency on long sequences.

Crime patterns typically exhibit strong daily cycles (e.g., distinct day vs. night patterns). This is why, rather than combining all frames into one continuous sequence, we maintain separate, dedicated sequences for each distinct time slot. Specifically, each sequence ${𝒮}_s$ comprises chronologically ordered quantile-transformed matrices $\{Q_s^d\}$, where $d$ indexes consecutive days for the given time slot $s$. Although these slot-specific sequences are maintained separately, the PredRNN++ model is trained jointly on data from all slots, allowing the model to learn both within-slot and across-slot temporal patterns.

Algorithm 3 presents the complete applyXLearn pipeline for this multi-slot forecasting task, where we predict a single future frame for each slot. This pipeline can operate in two modes, controlled by a boolean parameter trainFlag. If trainFlag is set to true, the algorithm prepares a new training dataset from the sequences and trains (or retrains) the model before performing inference. Otherwise, if trainFlag is set to false, it skips these steps, relying on a pre-trained model for direct inference.

When training is enabled, the function PrepareTrainingData (Algorithm 3, lines 1–9) iterates over each slot-specific sequence ${𝒮}_s$, generating input–output pairs using a sliding-window approach. For each index $i$, we create a training example:

Here, $X_i$ contains $l_{\text{input}}$ frames, and $Y_i$ is simply the next frame in the sequence. These generated input–output pairs from all slots are merged into a global training set $(X,Y)$.

Next, TrainModel (Algorithm 3, lines 11–13) fits the PredRNN++ model to this dataset. After training (or if a pre-trained model is used when trainFlag is false), the Inference function (Algorithm 3, lines 15–17) employs the most recent $l_{\text{input}}$ frames from each slot’s sequence to forecast the next frame. Each slot’s prediction ${\widehat{Q}}_s$ encodes the anticipated spatial crime-risk distribution in that slot for the next day.

Finally, the main applyXLearn pipeline (Algorithm 3, lines 19–27) orchestrates these steps depending on whether training is requested (trainFlag = true) or not.

The slot-specific predictions $\{{\widehat{Q}}_s\}$ are stored in the Patterns repository. Typically, the resulting predictions highlight extensive continuous regions of elevated crime risk that are impractical for direct patrol assignment (Figure 4.A). Therefore, before transferring these predicted heatmaps to the constraint programming (CP) module, we apply the ML-to-CP procedure (Algorithm 4) to convert these broad, forecasted hotspots into precise, actionable deployment locations suitable for operational patrol planning.

In practice, patrol operations typically focus on a contiguous subset of time slots within a specific day, denoted $S$ (for example, a work shift). That is, algorithm 4 converts predicted heatmaps into deployment locations by detecting high-risk cells within each time slot $s\in S$ using a $p$-th percentile threshold. Following [3], we set $p$ to the 85%-percentile so that only the top 15% of predicted intensities are considered hotspots, thereby reducing the impact of outliers. In Step 1 of the algorithm (lines 1–6), each time slot $s\in S$ (line 3) is processed by first identifying high-risk cells (line 4), then clustering these cells into contiguous polygons using standard 4-connectivity where neighboring high-risk cells are merged into connected components (line 5) (Figures 4.B, 5.A), and finally merging the resulting polygons into the set allPolygons (line 6). Next, in Step 2 (line 7), a uniform grid of circles is generated over the city grid to form allCircles (Figure 5.B). In Step 3 (lines 8–13), every circle in allCircles (line 9) is examined against each polygon in allPolygons (line 10), and if the circle satisfies the IntersectsOrInside condition (line 11), it is appended to the list validCircles (line 12) and the inner loop is exited (line 13) (Figure 5.C). Finally, in Step 4, for each valid circle (line 15) the algorithm computes the time-slot-specific risk by summing the risk values of all cells completely inside the circle for each slot $s\in S$ (line 18). The circle’s center, paired with its list of per-slot risks, is then added to locations (line 20), producing as output a set of pairs $(\text{CircleCenter},\text{tsRisk})$, where tsRisk contains a risk value for every time slot in $S$.

Building on the refined set of high-risk deployment locations identified via the ML-to-CP step, we formulate a vehicle routing problem (VRP) to optimize patrol unit deployment. The problem is defined on a directed graph $G=(ℒ,𝒜)$, where $ℒ=\{0,1,\mathrm{\dots },n_{\text{loc}}\}$ represents locations (with location $0$ denoting the depot) and $𝒜\subseteq ℒ\times ℒ$ defines admissible travel arcs. The subset $𝒞=ℒ\setminus \{0\}$ corresponds to the deployment locations selected by ML-to-CP. The planning horizon is discretized into $n_s$ slots, $S=\{1,\mathrm{\dots },n_s\}$, each of duration slot_length. Patrol routes must be completed within a maximum allowable time maxT.

We consider a set of patrol units (vehicles), $𝒱=\{1,\mathrm{\dots },n_v\}.$ Each location $i\in ℒ$ has a service time $\text{service\_t}[i]\in \mathbb{N}$. Each deployment location $i\in 𝒞$ has a time-varying risk value $\text{risk}[i,s]\in {ℝ}^{+}$ for each slot $s\in S$. Travel times between locations $i$ and $j$ are given by $\text{travel\_t}[i,j]\in \mathbb{N}$.

For each vehicle $v$, we introduce a variable $\text{route\_length}[v]$ indicating how many locations that vehicle visits (including its first position). We also allow each patrol to start at any location at time $0$.

The decision variables can be summarized as follows:

• $\mathrm{■}$

  ${\text{route}}_{v,k}\in ℒ$: the $k$-th location visited by vehicle $v$.

• $\mathrm{■}$

  ${\text{arrive\_t}}_{v,k},{\text{depart\_t}}_{v,k},{\text{wait\_t}}_{v,k}\in \mathbb{N}$: arrival, departure, and waiting times at the $k$-th location of vehicle $v$.

• $\mathrm{■}$

  ${\text{slot}}_{v,k}\in \{0\}\cup S$: time slot during which vehicle $v$ services its $k$-th location ($0$ indicates either the depot or an unused slot).

• $\mathrm{■}$

  $\text{route\_length}[v]\in \{1,\mathrm{\dots },n_{\text{loc}}\}$: total number of visited locations by vehicle $v$.

The key constraints are as follows:

• $\mathrm{■}$

  Vehicles may begin at any location at time $0$:

  ${\text{arrive\_t}}_{v,1}=0,{\text{route}}_{v,1}\in ℒ,\forall v\in 𝒱.$

• $\mathrm{■}$

  Beyond the route length of a vehicle $v$ (i.e. $\text{route\_length}[v]$), we force the following decision variables to be $0$:

  ${\text{route}}_{v,k}=0,{\text{slot}}_{v,k}=0,{\text{wait\_t}}_{v,k}=0,\forall v\in 𝒱,\forall k>\text{route\_length}[v].$

• $\mathrm{■}$

  No vehicle returns to the depot prematurely:

  ${\text{route}}_{v,k}\ne 0,\forall v\in 𝒱,\forall k\le \text{route\_length}[v].$

• $\mathrm{■}$

  A vehicle must complete its service time and any waiting time before departing:

  ${\text{depart\_t}}_{v,k}={\text{arrive\_t}}_{v,k}+\text{service\_t}[{\text{route}}_{v,k}]+{\text{wait\_t}}_{v,k},\forall v\in 𝒱,\forall k.$

• $\mathrm{■}$

  The arrival time at the next location depends on the departure time from the current one plus travel time:

  	${\text{arrive\_t}}_{v,k+1}={\text{depart\_t}}_{v,k}+\text{travel\_t}({\text{route}}_{v,k},{\text{route}}_{v,k+1}),$

• $\mathrm{■}$

  If the location is a deployment location, determine the slot by (Mapping Arrival Times to Slots):

• $\mathrm{■}$

  Prevent two vehicles from serving the same deployment location simultaneously:

  	$\left({\text{route}}_{v_1,i}={\text{route}}_{v_2,j}\in 𝒞\right)⟹$

In this section, we present and discuss the training of our XLearn model and the subsequent results obtained from both validation and testing. We further demonstrate the effectiveness of the proposed strategy through ICON loop simulations.

This paper introduced a data-driven framework that integrates predictive modeling and constraint-based optimization to improve the efficiency of police patrol deployments. By using a PredRNN++ approach to generate spatiotemporal crime forecasts, our system anticipates evolving crime risks. These forecasts are then mapped into actionable hotspots via quantile transformations and geometric filtering, before being passed to the constraint programming component for patrol route optimization.

Results from simulations on the Chicago Crime Dataset confirm that prediction-informed routing policies achieve higher hit rates compared to methods relying on static historical data or random deployments. Future work will focus on extending the model to account for multi-criteria decision-making, incorporating additional data sources (e.g., demographic or environmental factors), and exploring more sophisticated forecasting architectures. Overall, our findings demonstrate the promise of tightly coupling machine learning and constraint programming for proactive, effective crime prevention strategies.