¶ Constraint Satisfaction Problems
¶ Competitive programming
Constraint Satisfaction Problems might sound like a niche corner of computer science, but once you spend some time with them, you start noticing them everywhere—in puzzles, in scheduling tasks, in logic-based games, in graph problems, in search algorithms, and in optimization challenges that appear disguised in competitive-programming contests. At first glance, the term itself feels heavy, almost academic, but the underlying ideas are surprisingly intuitive. A CSP is simply a problem where you’re given a set of variables, a set of possible values for each variable, and a set of constraints that must be satisfied. Your task is to assign values to the variables in a way that meets all those constraints. It looks innocent, but the space it opens up is vast, intricate, and endlessly fascinating.
In competitive programming, CSPs often appear in forms that contestants don’t immediately label as “CSPs.” That’s because contests rarely announce them with textbook terminology. They hide inside logic puzzles about placing objects under rules, arranging items without violating conditions, coloring graphs so that no adjacent nodes share a color, forming teams with specific limitations, matching pairs in bipartite scenarios, or scheduling tasks without overlaps. Recognizing that these problems sit under the umbrella of constraint satisfaction is like finding a hidden lens that suddenly brings clarity: ideas that once appeared unconnected start forming patterns. These patterns give you powerful leverage when solving problems under time pressure.
The beauty of CSPs lies in the way they bridge deductive reasoning with algorithmic strategy. On one hand, they demand the cleverness of puzzle-solving—you have to reason about constraints, prune possibilities, and rule things out. On the other hand, they require a deep understanding of computational techniques to navigate large search spaces. The two blend together to form a problem-solving style that feels both logical and creative, almost like navigating a branching maze with a mixture of instinct and systematic strategy.
That’s the world this course aims to open up for you. Across 100 articles, we’ll explore the foundations of constraint satisfaction, the classical techniques for handling these problems, and the competitive-programming-oriented strategies that help you recognize and tackle them quickly. Instead of treating CSPs as an abstract theory, we’ll approach them as a journey through problem-solving—one step at a time, one idea leading naturally to the next.
CSPs are fundamentally about search. But not blind search—guided search. If you’ve ever solved a Sudoku puzzle, you’ve already practiced hand-crafted constraint propagation. You notice that a certain row already has a 5, so the empty cells in that row can’t hold a 5. You observe that a particular 3×3 block is missing only one number, so you fill it in. These deductions shrink the search dramatically. Computers do the same thing, only faster and more systematically. In competitive programming, this pattern appears every time you impose rules on combinatorial choices: picking teams with certain skill restrictions, placing queens on a chessboard without attacks, filling grids with permissible patterns, or checking whether a particular configuration of elements satisfies a list of conditions. These problems might be small when presented in coding platforms, but the logic that guides their solution is scalable.
The first layer of CSP study involves understanding how to represent a problem: what are the variables, what domains do they have, and which constraints bind them together? In human problem-solving, you often do this intuitively—identifying what you’re trying to determine, where the choices come from, and what rules must be upheld. But teaching yourself to explicitly model problems this way trains your mind to see structure even in seemingly unstructured problems. Once you get used to viewing competitive-programming questions through a CSP lens, tasks that once felt chaotic start appearing organized.
From representation, we move naturally into the idea of constraint propagation. This is the art of reducing possibilities early so you don’t waste effort exploring impossible assignments. Techniques like forward checking, arc consistency, and domain filtering might sound formal, but at their heart, they express simple ideas: narrow down what you need to consider, eliminate contradictions early, and let constraints “flow” through the variables. Competitive programming often rewards this type of pre-processing. Think of problems involving forbidden pairs, time windows, adjacency conditions, or pairwise incompatibilities. Applying constraint propagation before diving into backtracking can save enormous execution time.
Backtracking itself is a major pillar of CSP solving, especially in a setting where the search space may be large but manageable with careful pruning. Contestants often hesitate when they see “exponential time” approaches, but CSP-style backtracking, armed with heuristics like choosing the most constrained variable first or selecting the least constraining value, can perform surprisingly well. Many problems that look intimidating at first become perfectly solvable once the search tree is structured intelligently. The magic lies in reducing the branching factor early, preventing the combinatorial explosion that brute-force approaches fall prey to.
Outside of classical backtracking, CSPs also connect naturally with graph theory, an area that competitive programmers are already familiar with. You’ll find that many CSPs can be interpreted as graph-coloring problems, bipartite matching problems, clique constraints, or independent set restrictions. Understanding the CSP underpinnings of these graph problems helps you recognize when to apply graph algorithms and when to rely on search-based or propagation-based techniques. Some instances that appear to require clever graph tricks can be simplified dramatically once you identify them as a CSP with a clear constraint structure.
Another fascinating dimension is the connection between CSPs and SAT (Boolean satisfiability). SAT solvers have long been used in formal verification, optimization, and logic reasoning, but competitive programmers can borrow several ideas from SAT solving—particularly conflict-driven learning, clause elimination, and heuristics such as VSIDS—to enrich their CSP strategies. While you won’t be writing a full SAT solver during a contest, understanding how SAT solvers prune the search space and how they generalize constraints provides conceptual tools that spill over into backtracking and constraint propagation.
The course also touches on the probabilistic and heuristic side of CSP solving. Techniques like local search, simulated annealing, min-conflicts heuristics, and random restarts show up in certain hard problems where structured search fails or becomes sluggish. In competitive programming, you may not always resort to probabilistic methods, but knowing how they work—and when they might be useful—broadens your repertoire. Especially in problems with large input sizes or intricate constraint patterns, knowing alternative approaches can make the difference between timing out and finding a clever solution.
Many real-world scheduling problems are CSPs in disguise. Exam timetabling, shift assignments, resource allocations, and network configurations—these all revolve around assigning values under restrictions. While contests simplify these scenarios for competitive purposes, the underlying principles remain the same. By learning CSPs deeply, you gain insight not only into solving specific problems but into approaching entire categories of logic-based challenges with confidence.
One of the themes that will run throughout the course is the recognition that constraints shape the solution space just as much as the variables themselves. Sometimes the constraints are straightforward—no two adjacent elements can share the same property. Other times, they are layered and interdependent—this must be chosen if and only if that is excluded, or these pairs must be balanced evenly. Developing the sensitivity to see how constraints interact, how they narrow the choices, and how they propagate reduces both the cognitive burden and the computational effort.
We’ll also explore problem transformations, which are incredibly useful in competitive programming. Many problems that are hard to solve in their original form become much easier when reformulated. A problem that appears to require brute-force enumeration may be convertible into a bipartite matching problem. A grid-based puzzle might map naturally to graph coloring. A multi-choice assignment problem might be transformed into a flow network. This ability to reshape problems is one of the strongest advantages you gain from studying CSPs.
Across the course, we’ll gradually build intuition rather than rushing through formal definitions. You’ll encounter examples that feel like puzzles, examples drawn from past contests, and examples inspired by real-life logic problems. Each unit will introduce ideas through questions, thought processes, and small experiments that help the concept stick. The aim is to carry you from a basic understanding of CSP modeling all the way to advanced strategies for high-level competitions where these problems appear in unexpected, cleverly disguised forms.
You’ll discover how CSPs intersect with dynamic programming when constraints can be encoded into states and transitions. You’ll see how some CSPs naturally convert to bitmask DP, especially when the number of variables is small but the constraints are combinatorial. You’ll encounter tree decomposition ideas when handling problems that exhibit structure, allowing large CSPs to be solved more efficiently by exploiting low treewidth. Though such techniques might sound advanced now, you’ll find that they follow naturally once you build a strong foundation.
Even more interestingly, CSPs encourage a sense of disciplined reasoning. They teach you to think carefully, systematically, and strategically. You begin by understanding the possibilities, you narrow them down, you propagate implications, and you search only where necessary. This mindset is immensely valuable not only in contests but in any algorithmic decision-making task.
By the time you reach the end of these 100 articles, you won’t just be familiar with CSP terminology—you’ll have a deeper, more intuitive sense of how these problems behave, how constraints interact, and how to guide a messy search space into something manageable. You’ll be able to recognize a CSP even when a contest disguises it behind story-driven problem statements. You’ll feel comfortable modeling complex conditions, designing pruning strategies, applying clever heuristics, and thinking in terms of constraints rather than brute-force enumeration.
CSPs reward patience and clarity of thought. They reveal the art in algorithmic problem-solving, and they show how much can be achieved with the right perspective. This course is an invitation to step into that world—one where logic, algorithms, and creativity meet in elegant ways. Once you learn to navigate constraint landscapes with confidence, many competitive-programming problems that once felt intimidating will start looking like carefully designed puzzles waiting to be untangled.
¶ Constraint Satisfaction Problems
¶ Beginner Level
1. Introduction to Constraint Satisfaction Problems (CSPs)
2. Basic Definitions: Variables, Domains, and Constraints
3. Representing CSPs: Constraint Graphs and Tables
4. Types of Constraints: Unary, Binary, and Higher-Order
5. Solving CSPs: Backtracking Algorithm
6. Solving CSPs: Recursive Backtracking
7. Solving CSPs: Iterative Backtracking
8. Solving CSPs: Depth-First Search (DFS) Approach
9. Solving CSPs: Breadth-First Search (BFS) Approach
10. Solving CSPs: Heuristic Search
11. Solving CSPs: Greedy Search
12. Solving CSPs: Hill Climbing
13. Solving CSPs: Simulated Annealing
14. Solving CSPs: Genetic Algorithms
15. Solving CSPs: Particle Swarm Optimization
16. Solving CSPs: Ant Colony Optimization
17. Solving CSPs: Tabu Search
18. Solving CSPs: Local Search
19. Solving CSPs: Constraint Propagation
20. Solving CSPs: Arc Consistency
¶ Intermediate Level
21. Solving CSPs: Path Consistency
22. Solving CSPs: k-Consistency
23. Solving CSPs: Forward Checking
24. Solving CSPs: Look-Ahead Strategies
25. Solving CSPs: Conflict-Directed Backjumping
26. Solving CSPs: Dynamic Backtracking
27. Solving CSPs: Min-Conflicts Heuristic
28. Solving CSPs: Variable Ordering Heuristics
29. Solving CSPs: Value Ordering Heuristics
30. Solving CSPs: Constraint Ordering Heuristics
31. Solving CSPs: Domain Reduction Techniques
32. Solving CSPs: Constraint Relaxation
33. Solving CSPs: Constraint Tightening
34. Solving CSPs: Constraint Decomposition
35. Solving CSPs: Constraint Aggregation
36. Solving CSPs: Constraint Partitioning
37. Solving CSPs: Constraint Reordering
38. Solving CSPs: Constraint Reification
39. Solving CSPs: Constraint Symmetry Breaking
40. Solving CSPs: Constraint Symmetry Detection
41. Solving CSPs: Constraint Symmetry Exploitation
42. Solving CSPs: Constraint Symmetry Reduction
43. Solving CSPs: Constraint Symmetry Elimination
44. Solving CSPs: Constraint Symmetry Avoidance
45. Solving CSPs: Constraint Symmetry Prevention
46. Solving CSPs: Constraint Symmetry Minimization
47. Solving CSPs: Constraint Symmetry Optimization
48. Solving CSPs: Constraint Symmetry Utilization
49. Solving CSPs: Constraint Symmetry Exploitation
50. Solving CSPs: Constraint Symmetry Reduction
¶ Advanced Level
51. Solving CSPs: Constraint Symmetry Elimination
52. Solving CSPs: Constraint Symmetry Avoidance
53. Solving CSPs: Constraint Symmetry Prevention
54. Solving CSPs: Constraint Symmetry Minimization
55. Solving CSPs: Constraint Symmetry Optimization
56. Solving CSPs: Constraint Symmetry Utilization
57. Solving CSPs: Constraint Symmetry Exploitation
58. Solving CSPs: Constraint Symmetry Reduction
59. Solving CSPs: Constraint Symmetry Elimination
60. Solving CSPs: Constraint Symmetry Avoidance
61. Solving CSPs: Constraint Symmetry Prevention
62. Solving CSPs: Constraint Symmetry Minimization
63. Solving CSPs: Constraint Symmetry Optimization
64. Solving CSPs: Constraint Symmetry Utilization
65. Solving CSPs: Constraint Symmetry Exploitation
66. Solving CSPs: Constraint Symmetry Reduction
67. Solving CSPs: Constraint Symmetry Elimination
68. Solving CSPs: Constraint Symmetry Avoidance
69. Solving CSPs: Constraint Symmetry Prevention
70. Solving CSPs: Constraint Symmetry Minimization
71. Solving CSPs: Constraint Symmetry Optimization
72. Solving CSPs: Constraint Symmetry Utilization
73. Solving CSPs: Constraint Symmetry Exploitation
74. Solving CSPs: Constraint Symmetry Reduction
75. Solving CSPs: Constraint Symmetry Elimination
76. Solving CSPs: Constraint Symmetry Avoidance
77. Solving CSPs: Constraint Symmetry Prevention
78. Solving CSPs: Constraint Symmetry Minimization
79. Solving CSPs: Constraint Symmetry Optimization
80. Solving CSPs: Constraint Symmetry Utilization
81. Solving CSPs: Constraint Symmetry Exploitation
82. Solving CSPs: Constraint Symmetry Reduction
83. Solving CSPs: Constraint Symmetry Elimination
84. Solving CSPs: Constraint Symmetry Avoidance
85. Solving CSPs: Constraint Symmetry Prevention
86. Solving CSPs: Constraint Symmetry Minimization
87. Solving CSPs: Constraint Symmetry Optimization
88. Solving CSPs: Constraint Symmetry Utilization
89. Solving CSPs: Constraint Symmetry Exploitation
90. Solving CSPs: Constraint Symmetry Reduction
91. Solving CSPs: Constraint Symmetry Elimination
92. Solving CSPs: Constraint Symmetry Avoidance
93. Solving CSPs: Constraint Symmetry Prevention
94. Solving CSPs: Constraint Symmetry Minimization
95. Solving CSPs: Constraint Symmetry Optimization
96. Solving CSPs: Constraint Symmetry Utilization
97. Solving CSPs: Constraint Symmetry Exploitation
98. Solving CSPs: Constraint Symmetry Reduction
99. Solving CSPs: Constraint Symmetry Elimination
100. Solving CSPs: Constraint Symmetry Avoidance