{ "source_file": "0930", "source_platform": "Medium", "author": "Maxime Labonne", "title_listed": "Constraint Programming in Python", "keywords": "约束规划,优化", "domain": "机器学习", "content_type": "深度技术解析", "is_paid_flag": "部分免费/会员墙", "url": "https://towardsdatascience.com/constraint-programming-67ac16fa0c81/", "publish_date": "2022-05", "corpus_stage": "预训练/SFT", "notes": "", "id": "100df50d0218", "collected_at": "2026-10-01T22:02:13", "title": "Constraint Programming in Python | Towards Data Science", "body_markdown": "![Revisit consent button](https://cdn-cookieyes.com/assets/images/revisit.svg)\nWe value your privacy\nWe use cookies to enhance your browsing experience, serve personalised ads or content, and analyse our traffic. By clicking \"Accept All\", you consent to our use of cookies.\nCustomise Reject All Accept All\nCustomise Consent Preferences ![](https://cdn-cookieyes.com/assets/images/close.svg)\nWe use cookies to help you navigate efficiently and perform certain functions. 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It also helps in adverstising on the website.\n\n\n * Cookie\nedgebucket\n * Duration\nsession\n * Description\nReddit sets this cookie to save the information about a log-on Reddit user, for the purpose of advertisement recommendations and updating the content.\n\n\n * Cookie\ndid\n * Duration\n1 year\n * Description\nArbor sets this cookie to show targeted ads to site visitors.This cookie expires after 2 months or 1 year.\n\n\nUncategorised\nOther uncategorised cookies are those that are being analysed and have not been classified into a category as yet.\nNo cookies to display.\nReject All Save My Preferences Accept All\nToggle Search[](https://towardsdatascience.com/)Toggle Mobile Navigation\n[](https://towardsdatascience.com/)\nPublish AI, ML & data-science insights to a global community of data professionals.\nSign in[Submit an Article](https://write.insightmediagroup.io)\n[Latest](https://towardsdatascience.com/latest/)[Editor's Picks](https://towardsdatascience.com/tag/editors-pick/)[Newsletter](https://towardsdatascience.com/tag/the-variable/)[ShipAI](https://towardsdatascience.com/shipai/)\n* * *\n[Write For TDS](https://towardsdatascience.com/submissions/)\n[LinkedIn](https://www.linkedin.com/company/towards-data-science/)[X](https://x.com/TDataScience)Toggle Search\n[Artificial Intelligence](https://towardsdatascience.com/category/artificial-intelligence/)\n# Constraint Programming in Python\nThe Programming Paradigm to Find One Solution Among 8,080,104 Candidates\n[Maxime Labonne](https://towardsdatascience.com/author/mlabonne/)\nMay 2, 20227 min read\n![Image by author, emojis by OpenMoji \\(CC BY-SA 4.0\\)](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/05/1siFUe3QoPF-jYlDJVttp5A.jpeg)\nImage by author, emojis by [OpenMoji](https://openmoji.org/) ([CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/))\nConstraint Programming is a technique to **find every solution** that respects a set of predefined constraints.\nIt is an invaluable tool for data scientists to solve a huge variety of problems, such as scheduling, timetabling, sequencing, etc. In this article, we'll see how to use CP in two different ways:\n 1. **Satisfiability** : the goal is to find one or multiple feasible solutions (_i.e._ , solutions that respect our constraints) by narrowing down a large set of potential solutions;\n 2. **Optimization** : the goal is to find the best feasible solution according to an objective function, just like Linear Programming (LP).\n\n\nWe'll use CP-SAT from [Google OR-Tools](https://developers.google.com/optimization), an excellent free and open source CP solver. Note that it is **different** from MPSolver, which is dedicated to Linear and Mixed Integer Programming. The difference between CP and LP is quite confusing, we'll touch on this topic at the end of the article.\nYou can run the code with the following [Google Colab notebook](https://colab.research.google.com/drive/1huTlPTaahdEEO29KKdlW9ic5zAwB3D58?usp=sharing).\nLearn this step by step with the interactive [AI Engineer](https://roadmap.sh/ai-engineer) and [Computer Science](https://roadmap.sh/computer-science) roadmaps.\n## **🪖 I. S** atisfiability with the 3 scouts problem\n![Image by author, emojis by OpenMoji \\(CC BY-SA 4.0\\)](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/05/0ecB7NyzmEihvfZ65.png)\nImage by author, emojis by [OpenMoji](https://openmoji.org/) ([CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/))\nIn the [previous article](https://towardsdatascience.com/integer-programming-vs-linear-programming-in-python-f1be5bb4e60e), we created an army to defeat our opponent. But there was one small problem: we had to guess how powerful his army was.\nThis time, let's send scouts to know the **exact number**. Our 3 scouts observed the enemy camp, and this is what they tell us:\n * **Scout 1** : \"_the number of soldiers is a multiple of 13_ \";\n * **Scout 2** : \"_the number of soldiers is a multiple of 19_ \";\n * **Scout 3** : \"_the number of soldiers is a multiple of 37_ \";\n * They all agree that the number of soldiers **doesn't exceed 10,000**.\n\n\nOur scouts have a personal way of counting soldiers, but we can **combine** these three observations to make a model.\nLet's call the number of soldiers _army_. We can translate our problem into the following congruence system:\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/05/14iLRmFqz9Lgh5jcS3RkbqA.png)\nIf you're not familiar with this notation, this is what it means in **programming terms** :\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/05/1fLKf0ZAlsIauO5oWCCSlIw.png)\nLet's implement it with OR-Tools. The first thing we need to do is to import and create the **CP-SAT model and solver**.\nThe **modeling process** is very similar to what we did in Linear Programming.\nThe first step to create our CP model is to declare the **variables**. In this example, we only have one: _army_ , the number of soldiers.\nWe have to give lower and upper bounds. The **lower bound** is 1 since we know there's an army, and the **upper bound** is 10,000 according to the scouts:\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/05/1chXXQplZZmimL8WbKnQoww.png)\nIn OR-Tools, we use the `NewIntVar` method to create this variable.\nThe second step is to declare the **constraints**.\nWe identified three constraints in this example. Modulo is a special operator, so we need a specific function to handle it with CP-SAT: `AddModuloEquality`. You can find a reference guide at [this address](https://developers.google.com/optimization/reference/python/sat/python/cp_model) if you need other methods.\nUnlike Linear Programming, we **don't have to define an objective function** here.\nThe reason is simple: there is nothing to optimize! We just want to find a **feasible solution** that satisfies our constraints, but there is no \"good\" or \"bad\" answers. This is a **key feature** of Constraint Programming.\nOur model is **complete** , we can now ask OR-Tools to solve it.\ntext\n\n```\n================= Solution =================\nSolved in 0.00 milliseconds\n```\n\ntext\n\n```\n🪖 Army = 9139\n```\n\ntext\n\n```\nCheck solution:\n - Constraint 1: 9139 % 13 = 0\n - Constraint 2: 9139 % 19 = 0\n - Constraint 3: 9139 % 37 = 0\n```\n\nWe obtained our solution in less than a millisecond: there are **9,139 soldiers** in the enemy army. Huzzah, we can now fire the scouts!\nWe limited the search space with an upper bound of 10,000, which gave us a **unique solution**. But is it still the case if we push this limit?\nAnother perk of CP is the ability to **find every possible solution** to a problem. This might take a long time when the search space is large because the solver has to brute force the entire space (instead of reducing it with heuristics). Let's explore this feature by printing every possible solution with a new upper bound of **100,000**.\nWith OR-Tools, we ask the solver to look for every possible solution thanks to the `enumerate_all_solutions` parameter. We then assign it a **callback** class that prints every solution the solver finds.\nWe found **10 solutions**! This was to be expected since we increased the upper bound tenfold: these solutions all are **multiples** of 9,139.\nAs you can see, this example has nothing to do with optimization: it's a pure **satisfiability problem**. On another note, this congruence system can be solved manually with the [Chinese remainder theorem](https://en.wikipedia.org/wiki/Chinese_remainder_theorem). But CP is not limited to that...\n## **🍻 II. Optimization and beer**\n![Image by author, emojis by OpenMoji \\(CC BY-SA 4.0\\)](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/05/1MC2frLKtps7KzfWm9bU9WA.gif)\nImage by author, emojis by [OpenMoji](https://openmoji.org/) ([CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/))\nLet's see another problem: our army will face the enemy in a few days. In the meantime, the quartermaster has to **prepare the rations** that will be used during the campaign.\nThe space in the supply wagons is **limited** and some rations are more **popular** than others. There are three possible rations:\n * 🥖 B**read:** it takes only 1 space but soldiers don't like it that much with a popularity of 3;\n * 🥩 M**eat:** it takes 3 spaces and has a popularity of 10;\n * 🍺 **Beer** : it takes 7 spaces but soldiers love it with a popularity of 26.\n\n\n![Image by author, emojis by OpenMoji \\(CC BY-SA 4.0\\)](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/05/0ohhu4xMwcjntxSt3.png)\nImage by author, emojis by [OpenMoji](https://openmoji.org/) ([CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/))\nThe supply wagons have a capacity of **19 spaces**. How to select the best rations to **maximize** the popularity?\nThis is an **optimization** problem we've already seen: actually, it is a variant of the famous [knapsack problem](https://en.wikipedia.org/wiki/Knapsack_problem). We could reuse the code from the previous article and just change the input parameters.\nThis time, we'll solve it using Constraint Programming. This paradigm is not limited to finding feasible solutions. It can also perform optimization using [different algorithms](https://en.wikipedia.org/wiki/Constrained_optimization) to handle this overhead.\nLet's create a model of the problem. First of all, we have to declare three variables: 🥖b**read,** 🥩me**at,** and 🍺 be**er.** It's possible to have 0 of them, but their number cannot exceed the maximal capacity.\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/05/1M2TB2GazZ_46yRhkbWOh9Q.png)\nThis time, we only have one constraint: the space occupied by the bread, the meat, and the beer **cannot exceed the wagons' capacity** (19).\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/05/1erkeHM-AVqXzt6rdX_lnAw.png)\nWe want to **maximize the total popularity** of the rations that are selected:\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/05/1mYmaK6fi8bQBiGBaaNH6HA.png)\nThe model is complete, CP-SAT can **solve the problem**!\ntext\n\n```\n================= Solution =================\nSolved in 0.00 milliseconds\n```\n\ntext\n\n```\nOptimal value = 68 popularity\nFood:\n - 🥖Bread = 2\n - 🥩Meat = 1\n - 🍺 Beer = 2\n```\n\nWe obtained the **highest popularity** (68) possible with a capacity of 19.\nIs the constraint respected? Let's quickly check it: 1×2 🥖 + 3×1 🥩 + 7×2 🍺 = 19, which is indeed ≤ 19.\nOkay, I'd like to ask another question: **how many solutions** to this problem are there? Once again, we can answer it with a specific callback to count them.\ntext\n\n```\n121\n```\n\nWe found **121 solutions** with a capacity of 19. But this number quickly increases: with a capacity of 1000, there are **8,080,104** possible solutions! And yet, CP-SAT finds the optimal solution in less than a second. How is it possible?\nCP solvers do not brute force the problem with an exhaustive search but **combine** heuristics and combinatorial search instead. More specifically, the three most popular techniques for constraint satisfaction problems are [**backtracking**](https://en.wikipedia.org/wiki/Backtracking), [**constraint propagation**](https://en.wikipedia.org/wiki/Local_consistency), and [**local search**](https://en.wikipedia.org/wiki/Local_search_\\(optimization\\)).\nCP-[SAT](https://en.wikipedia.org/wiki/Boolean_satisfiability_problem) is quite particular since it combines CP and **SAT** : it is part of a broader trend of merging CP, LP, SAT, and metaheuristics.\nWe said that the previous problem could be solved with Linear Programming, so let's compare the code of both solutions:\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/05/1yoDKrCAPjz81py6kOAjZYA.png)\n![Left: LP code, Right: CP code \\(image by author\\)](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/05/1K2RxthvVBImgb0lLYHnzZg.png)\nLeft: LP code, Right: CP code (image by author)\nAs you can see, the syntax is quite similar but it's not the same: model/solver vs. solver, `NewIntVar` instead of `IntVar`, etc. There's a bit of translation to do, but it's easily manageable.\nThese two techniques are **incredibly close to each other** : they both handle variables with constraints and perform optimization using math and heuristics. However, CP is limited to discrete parameters, while LP handles continuous ones. On the other hand, you can implement specialized constraints like [\"all different\"](https://www.ibm.com/docs/en/icos/12.9.0?topic=variables-all-different-constraint) in CP, but not in LP. Here is a summary of the main differences between these two technologies:\n![Image by author, emojis by OpenMoji \\(CC BY-SA 4.0\\)](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/05/0qy8YsO-Bsx9J_MdW.png)\nImage by author, emojis by [OpenMoji](https://openmoji.org/) ([CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/))\nIf you want to know more about this topic, I would recommend [this article](https://pubsonline.informs.org/doi/abs/10.1287/inte.31.6.29.9647) by Irvin J. Lustig and [Jean-François Puget](https://towardsdatascience.com/constraint-programming-67ac16fa0c81/None). CPLEX's documentation also details the differences [at this address](http://ibmdecisionoptimization.github.io/docplex-doc/mp_vs_cp.html), in terms of modeling and optimization.\n## Conclusion\n![Image by author](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/05/1nLLPbmxqGxwzPttCVO90IA.png)\nImage by author\nConstraint Programming is another incredible technique in the **mathematical optimization** toolbox. It is a radically different approach compared to traditional, declarative programming. In this article,\n * We saw **two applications** of CP with satisfiability and optimization;\n * We implemented **CP models** in OR-Tools and played with the callback function;\n * We highlighted the **differences** between CP and LP.\n\n\nWe limited ourselves to simple problems in this introduction, but CP has amazing applications in complex scheduling and routing problems. This is a topic I'd love to address in a future article.\nIf you're interested to know more about it, feel free to follow me on **Twitter** at [@maximelabonne](https://twitter.com/maximelabonne). Thanks for your attention!\n## Related articles\n> [**Introduction to Linear Programming in Python**](https://towardsdatascience.com/introduction-to-linear-programming-in-python-9261e7eb44b)\n> [**Integer vs. Linear Programming in Python**](https://towardsdatascience.com/integer-programming-vs-linear-programming-in-python-f1be5bb4e60e)\n## Register Now to Continue Reading\n### Unlock Towards Data Science with a Free Account\nAll articles on Towards Data Science are available to registered users. 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