assignment problem in combinatorics

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Assignment problem for forming pairs of real numbers

Suppose I have two sets of real numbers, $X$ and $Y$ , each of cardinality $N$ . I would like to assign these points to pairs $(X_i, Y_j)$ such that the sum of squared intra-pair distances is minimized. One option is to use the Hungarian algorithm with complexity $O(N^3)$ , forming a cost matrix with all the squared distances between $X$ elements and $Y$ elements. Another option is to solve this approximately in $O(N log N)$ using a K-D tree. But is there any method that is faster than cubic (perhaps quadratic) but better than the K-D tree approach?

The fact that these are one-dimensional real numbers suggests we could somehow exploit the fact that we can sort the numbers. For example, we could sort each set of numbers and pair the elements with the same rank order. If this makes things any easier, I don't need to work with squared distances necessarily; sum of absolute distances would work too.

• co.combinatorics

2 Answers 2

Suppose the $X,Y$ are sorted. Then $X_1$ (the smallest $X$ -value) must always be matched with $Y_1$ (the smallest $Y$ -value). (Why? If you consider some other pairing, say one that includes $(X_1,Y_j)$ and $(X_i,Y_1)$ , then changing those two pairs to $(X_1,Y_1)$ and $(X_i,Y_j)$ yields a lower-cost pairing. I'll let you prove this. The case analysis is not too difficult.)

Continuing onward, we see that you can sort the $X$ and $Y$ and then match $(X_i,Y_i)$ for each $i$ , to obtain the optimal solution.

Thus, the problem can be solved in $O(N \log N)$ time.

It turns out the sorting idea is at the core of the sliced Wasserstein distance for approximating the true Wasserstein distance (which requires either solving a linear assignment problem or running the Sinkhorn algorithm) with 1d projections. See the following paper for more details:

Bonneel, Nicolas, et al. "Sliced and radon wasserstein barycenters of measures." Journal of Mathematical Imaging and Vision 51 (2015): 22-45.

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• Prof. Alexander Postnikov

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2 (page 7) p. 7 Four types of problem

• Published: April 2016
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‘Four types of problem’ explains that combinatorics is concerned with four types of problem: existence problems (does x exist?); construction problems (if x exists, how can we construct it?); enumeration problems (how many x are there?); and optimization problems (which x is best?). Existence problems discussed include tilings, placing dominoes on a chess board, the knight’s tour problem, the Königsberg bridges problem, the Gas–Water–Electricity problem, and the map-colour problem. Construction problems include solving mazes, and the two types of enumeration problems considered are counting problems and listing problems. Examples of an optimization problem include the minimum connector problem and the travelling salesman problem. The efficiency of algorithms is also explained.

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Problem-Solving Strategies for solving Combinatorics Assignments

Embarking on the journey of conquering combinatorics assignments requires more than just mathematical prowess. Combinatorics, a captivating branch of mathematics dealing with the art of counting and arrangement, demands a strategic approach and a solid foundation in various concepts. As you embark on your journey to tackle assignments in combinatorics, it's crucial to have a solid understanding of fundamental topics and effective problem-solving strategies. In this blog, we will delve into the critical topics that lay the groundwork for successful combinatorics assignments and explore effective strategies to help with your Combinatorics assignment in this intricate field.

Understanding the Basics of Combinatorics

Before delving into more complex concepts, it's essential to grasp the foundational ideas in combinatorics. These concepts serve as building blocks for solving more intricate problems later on. Here are a few key topics you should familiarize yourself with:

Permutations and Combinations

Permutations involve arranging objects in a specific order, while combinations focus on selecting objects without considering the order. Understanding the formulas for permutations and combinations is crucial.

The Multiplication Principle

The multiplication principle, also known as the counting principle, helps you count the total number of outcomes for multiple tasks performed sequentially. It's a fundamental concept that underlies many combinatorial problems.

The Addition Principle

The addition principle allows you to count the total number of outcomes for tasks that can be performed in different ways or cases. It's particularly useful for solving problems involving choices or alternatives.

Advanced Topics for Combinatorics Assignments

Once you have a solid grasp of the basics, you can move on to more advanced topics that will enhance your ability to tackle complex combinatorics assignments.

Binomial Theorem and Pascal's Triangle

Understanding the binomial theorem and Pascal's Triangle can be immensely beneficial. These concepts are often used to expand binomial expressions and calculate combinations efficiently.

Inclusion-Exclusion Principle

The inclusion-exclusion principle helps solve problems involving multiple sets or conditions. It's a powerful tool for counting elements that satisfy certain criteria while considering overlaps.

Generating Functions

Generating functions provide a formal way to represent sequences of numbers, making it easier to solve problems involving recurrence relations and counting.

Pigeonhole Principle

The pigeonhole principle might seem intuitive, but it has deep implications in combinatorics. It helps you recognize when there must be repetitions or patterns in a given set of objects.

Strategies for Solving Combinatorics Assignments

Equipped with a solid understanding of the essential and advanced topics in combinatorics, let's dive into strategies for approaching and solving assignments effectively.

Understand the Problem Statement

Understanding the problem statement is the crucial first step in navigating the world of combinatorics assignments. It involves dissecting the given scenario, deciphering the information provided, and grasping the specific task you need to accomplish. This initial comprehension sets the stage for effective problem-solving.

To truly understand the problem, read it multiple times, identifying key details, constraints, and requirements. Rephrasing the problem in your own words can often clarify its essence. Break down any complex sentences or scenarios into simpler components. This process helps you avoid misinterpretations and ensures you are addressing the problem accurately.

A solid grasp of the problem statement guides your approach. It helps you determine whether permutations, combinations, or other principles are applicable. Without this foundational understanding, attempting to solve the problem can lead to confusion and errors.

Remember, a well-defined problem is halfway solved. Take your time, analyze the information provided, and ensure you're clear on what's being asked. This practice not only enhances your problem-solving skills in combinatorics but also cultivates a mindset that's attentive, analytical, and detail-oriented – qualities that extend beyond mathematics into various facets of problem-solving in the real world.

Visualize the Problem

In the realm of combinatorics, where abstract concepts meet real-world scenarios, visualization becomes an invaluable tool. Many combinatorial problems involve arranging objects, making choices, or following specific sequences. Visualizing the problem allows you to create a mental or graphical representation of the situation, which can provide insights and clarity.

When faced with a combinatorial problem, consider creating diagrams, tables, or even simple models to represent the elements involved. For example, if you're dealing with a permutation problem where you need to arrange people in a queue, you can sketch out a line with placeholders for each person. Visual aids like this help you see patterns, repetitions, or symmetries that might not be immediately apparent from the problem statement alone.

Visualizing the problem doesn't just make it more tangible; it also guides your thought process. As you work on assignments, remember that a clear mental image or a visual representation on paper can lead you to potential solutions more efficiently. Combining mathematical analysis with visual insight allows you to approach problems from multiple angles, increasing your chances of finding an elegant and accurate solution.

Break Down the Problem

Combinatorics problems, especially complex ones, can seem overwhelming at first glance. Breaking down the problem into smaller, manageable components is a powerful strategy to tackle the challenge systematically. This approach not only makes the problem more approachable but also helps you identify patterns and connections that might not be obvious initially.

Start by analyzing the problem statement and identifying its key elements. What are the given values, constraints, and requirements? Then, consider if you can divide the problem into sub-problems or cases. Each sub-problem can be addressed independently, simplifying the overall solution process.

Additionally, breaking down the problem often involves recognizing similarities between different parts of the problem. If you encounter repetitive tasks or scenarios, you can solve one instance and then generalize the solution for others.

By deconstructing the problem, you transform a complex challenge into a series of manageable steps. This approach not only reduces the risk of overlooking crucial details but also boosts your confidence as you steadily work through each component. As you gain experience, you'll become more skilled at identifying how to break down problems effectively, enhancing your problem-solving prowess in combinatorics.

Apply Relevant Concepts

When faced with a combinatorics problem, applying the appropriate concepts is akin to selecting the right tools for a task. Each problem has unique characteristics, and understanding which principles, theorems, or formulas are relevant is crucial. This step requires a deep understanding of the concepts you've studied, so take the time to review and internalize them.

For instance, if the problem involves arranging objects in a specific order, permutations are likely the key. On the other hand, if the order doesn't matter, combinations come into play. Similarly, problems that involve sequential tasks might call for the multiplication principle. By carefully analyzing the problem statement and identifying its underlying structure, you can confidently choose the relevant combinatorial approach.

Double-Check Your Solution

While the sense of accomplishment from finding a solution is undeniable, it's equally important to exercise caution and rigor in your approach. The intricate nature of combinatorial problems leaves room for errors, oversights, or miscalculations. This is where the art of double-checking steps in.

Revisiting your solution with a critical eye helps ensure its accuracy. Verify whether you've accurately applied the chosen concept, whether calculations are error-free, and if your solution aligns with the problem's requirements. Moreover, assessing your solution from different angles can reveal potential shortcuts or alternative methods, leading to a more elegant solution.

The process of double-checking fosters a sense of confidence in your solution and helps you catch any slips that might have otherwise gone unnoticed. Remember, precision is paramount in combinatorics, where small mistakes can lead to vastly different outcomes. By making double-checking an integral part of your problem-solving routine, you're bound to refine your skills and produce consistently reliable solutions.

Practice, Practice, Practice

Practice lies at the heart of mastering combinatorics assignments. As with any skill, consistent practice is essential for honing your problem-solving abilities in this field. Combinatorics problems come in various forms, each presenting unique challenges. By working through a diverse range of problems, you develop a deeper understanding of the underlying concepts and strategies required.

Repetition allows you to internalize the methods used to approach different scenarios. It enhances your ability to recognize patterns, apply relevant principles, and make connections between seemingly unrelated problems. Through practice, you not only improve your technical skills but also gain the confidence to tackle even the most complex combinatorial puzzles.

Remember that it's not just about the quantity of problems you solve, but the quality of your engagement with each one. Analyze your solutions critically, identifying areas for improvement and alternative approaches. With every problem you conquer, you're building a mental toolkit that equips you to face new challenges with resilience and creativity.

Seek Help When Needed

In the realm of combinatorics, seeking help is not a sign of weakness, but a testament to your commitment to understanding and solving problems effectively. When you encounter a particularly challenging or perplexing problem, don't hesitate to reach out for assistance.

Consult your textbooks, online resources, or course materials to gain alternative perspectives on the problem. Sometimes, a fresh explanation can provide the clarity you need to unlock the solution. Engage with peers, teachers, or online communities to discuss the problem. Explaining your thought process to others often leads to insights you might have missed on your own.

Collaboration is a powerful tool in mastering combinatorics. A fellow student might approach a problem from a different angle, offering you a novel way to solve it. By seeking help, you're tapping into a wealth of collective knowledge, fostering a collaborative learning environment, and expanding your problem-solving toolkit.

Remember, the willingness to ask for help demonstrates your dedication to understanding and growing within the field of combinatorics. It's a trait that sets you on a path to becoming a more proficient and confident problem solver.

In conclusion, embarking on the realm of combinatorics assignments is a journey that combines mathematical acumen with strategic thinking. By establishing a strong foundation in fundamental concepts such as permutations, combinations, and the core principles of counting, you lay the groundwork for confidently approaching problems. As you delve into more advanced topics like the inclusion-exclusion principle, generating functions, and Pascal's Triangle, you unlock the tools to tackle even the most intricate combinatorial challenges. Remember, success in combinatorics assignments is not solely about equations and formulas; it's about cultivating a problem-solving mindset. Break down complex problems, visualize scenarios, and apply relevant principles. Mistakes and hurdles are part of the process – learn from them, iterate, and refine your approach.

Combinatorics assignments offer not only mathematical satisfaction but also the opportunity to hone critical thinking and analytical skills that transcend mathematics. So, whether you're arranging objects or counting possibilities, the journey through combinatorics equips you with skills that extend far beyond the classroom, enriching your problem-solving toolkit for a lifetime. Embrace the challenges, persist in your practice, and master the art of counting and arrangement with confidence.

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NP completeness of a variant of assignment problem

I have the following variant of assignment problem:

Suppose we have $m$ machines and $n$ jobs. Each machine is capable of doing a subset of jobs and each machine $i$ has capacity $C_i$ . Each job $j$ has a load of $D_j$ and the goal is to find if there exists a satisfying assignment where each job is assigned to one machine and no machine's load exceeds its capacity.

Hence the input is: $m$ positive numbers $C_1,...,C_m$ , $n$ positive numbers $D_1,...,D_n$ and the knowledge of whether machine $i$ is capable of job $j$ for all $i,j$ .

This variant is a bit different from original assignment problem as now each machine is only capable of a certain subset of jobs. I personally believe it is NP complete, however, I am not able to prove it...

• combinatorics
• optimization
• computational-complexity
• np-complete

• $\begingroup$ When you say "each machine is capable of doing a subset of jobs," do you mean that each machine is tasked with some number of the $n$ jobs, or that each machine has its own subset of the jobs which it is able to perform? $\endgroup$ –  Stephen Donovan Commented Jul 23, 2021 at 5:48
• $\begingroup$ This looks like it can be solved with Min-Cost MaxFlow: en.wikipedia.org/wiki/Minimum-cost_flow_problem $\endgroup$ –  Sudix Commented Jul 23, 2021 at 15:51
• 1 $\begingroup$ @Sudix minimum-cost flow might violate the "each job is assigned to one machine" rule. $\endgroup$ –  RobPratt Commented Jul 23, 2021 at 17:29

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