Polynomial-time reduction is a powerful concept in algorithm design that allows you to compare the difficulty of different problems and use existing solutions to solve new ones. In this article, you will learn what polynomial-time reduction is, how to apply it to some common problems, and why it is useful for finding efficient algorithms.
Polynomial-time reduction is a way of transforming one problem into another problem in such a way that the solution of the second problem can be used to solve the first problem. The transformation must be done in polynomial time, which means that the time required to perform the transformation is bounded by some polynomial function of the input size. For example, if you can transform problem A into problem B in O(n^2) time, where n is the size of the input for problem A, then you have a polynomial-time reduction from A to B.
To apply polynomial-time reduction, you need to find a problem that is already known to have an efficient algorithm and show how to transform your problem into that problem. For example, suppose you want to solve the problem of finding the longest common subsequence (LCS) of two strings. You can reduce this problem to the problem of finding the edit distance between two strings, which is the minimum number of insertions, deletions, or substitutions needed to transform one string into another. The edit distance problem has a well-known dynamic programming algorithm that runs in O(mn) time, where m and n are the lengths of the strings. To reduce the LCS problem to the edit distance problem, you can assign a cost of 1 to each insertion or deletion, and a cost of -1 to each substitution. Then, the edit distance between two strings is equal to the sum of their lengths minus twice the length of their LCS. Therefore, you can use the edit distance algorithm to find the LCS by reversing the cost function and dividing the result by two.
In finance, polynomial-time reduction can be used to optimize portfolio risk. Imagine reducing the complex problem of portfolio optimization to a simpler problem, like finding the shortest path in a network. By representing assets as nodes and correlations as edges, existing shortest-path algorithms can be applied, allowing efficient computation of the optimal asset allocation.
Polynomial-time reduction is useful for two main reasons. First, it helps you find efficient algorithms for new problems by reusing existing algorithms for known problems. This can save you a lot of time and effort in designing and implementing algorithms from scratch. Second, it helps you classify problems according to their difficulty and complexity. If you can reduce problem A to problem B in polynomial time, then you can say that problem A is no harder than problem B, or that problem A is in the same complexity class as problem B. For example, if you can reduce the problem of finding a Hamiltonian cycle in a graph to the problem of finding a satisfying assignment for a Boolean formula, then you can say that both problems are NP-complete, which means that they are among the hardest problems in the class NP. This implies that there is no known polynomial-time algorithm for either problem, unless P = NP, which is a major open question in computer science.
Polynomial-time reduction is a widely used technique in algorithm design and analysis, and there are many examples of problems that can be reduced to each other. For instance, the problem of finding a maximum matching in a bipartite graph can be reduced to the problem of finding a maximum flow in a network. A bipartite graph is a graph whose vertices can be divided into two disjoint sets such that no edge connects vertices from the same set. To reduce the matching problem to the flow problem, you can add a source and a sink to the graph, connect the source to all vertices in one set, connect the sink to all vertices in the other set, and assign a capacity of 1 to each edge. The value of the maximum flow is equal to the size of the maximum matching. Additionally, the problem of finding a minimum spanning tree in a graph can be reduced to the problem of finding a minimum cut in a network. To reduce the spanning tree problem to the cut problem, you can assign a large capacity to each edge in the graph, add a source and a sink to the graph, connect the source to one vertex, connect the sink to another vertex, and assign a small capacity to the edges that connect the source and the sink. The capacity of the minimum cut is equal to the weight of the minimum spanning tree plus the small capacity. Lastly, the problem of finding a shortest path between two vertices in a graph can be reduced to the problem of finding a negative cycle in a graph. To reduce the path problem to the cycle problem, you can add a new vertex to the graph, connect it to the source vertex with an edge of length 0, and connect it to all other vertices with edges of length infinity. The length of the shortest path from the source to the destination is equal to the negative of the length of the shortest negative cycle that contains the new vertex.