CODEWITHSUNDEEP
algorithmgraph
TheoryQuest1
TheoryQuest1

Reputation: 113

Find Path of a Specific Weight in a Weighted DAG

Given a DAG where are Edges have a Positive Edge Weight. Given a Value N.

Algorithm to calculate a simple (no cycles or node repetitions) Path with the Total weight N?

I am aware of the Algorithm where we have to find a Path of Given Path Length (number of Edges) but somewhat confused about for the Given Path Weight?

Can Dijkstra be modified for this case? Or anything else?

Upvotes: 1

Views: 1226

Answers (1)

j_random_hacker
j_random_hacker

Reputation: 51226

This is NP-complete, so don't expect any reasonably fast (polynomial-time) algorithm. Here's a reduction from the NP-complete Subset Sum problem, where we are given a multiset of n integers X = {x_1, x_2, ..., x_n} and a number k, and asked if there is any submultiset of the n numbers that sum to exactly k:

Create a graph G with n+1 vertices v_1, v_2, ..., v_{n+1}. For each vertex v_i, add edges to every higher-numbered vertex v_j, and give all these edges weight x_i. This graph has O(n^2) edges and can be constructed in O(n^2) time. Clearly it contains no cycles.

Suppose the answer to the Subset Sum problem is YES: That is, there exists a submultiset Y of X such that the numbers in Y total to exactly k. Actually, let Y = {y_1, y_2, ..., y_m} consist of the m <= n indices 1 <= i <= n of the selected elements of X. Then there is a corresponding path in the graph G with exactly the same weight -- namely the path that starts at v_{y_1}, takes the edge to v_{y_2} (which is of weight x_{y_1}), then takes the edge to v_{y_3}, and so on, finally arriving at v_{y_m} and taking a final edge (which is of weight x_{y_m}) to the terminal vertex v_{n+1}.

In the other direction, suppose that there is a simple path in G of total weight exactly k. Since the path is simple, each vertex appears at most once. Thus each edge in the path leaves a unique vertex. For each vertex v_i in the path except the last, add x_i to a set of chosen numbers: these numbers correspond to the edge weights in the path, so clearly they sum to exactly k, implying that the solution to the Subset Sum problem is also YES. (Notice that the position of the final vertex in the path doesn't matter, since we only care about the vertex that it leaves, and all edges leaving a vertex have the same weight.)

A YES answer to either problem implies a YES answer to the other problem, so a NO answer to either problem implies a NO answer to the other problem. Thus the answer to any Subset Sum problem instance can be found by first constructing the specified instance of your problem in polynomial time, and then using any algorithm for your problem to solve that instance -- so if an algorithm exists that can solve any instance of your problem in polynomial time, the NP-hard Subset Sum problem can also be solved in polynomial time.

Upvotes: 0

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