Maximum Weight Independent Set Problem: Difference between revisions
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The key to finding a [[Algorithms#Dynamic_Programming_Algorithms|dynamic programming algorithm]] is to identify a small set of subproblems whose solution can be computed using the previous subproblems' solutions. In this case, we start with the observation that for the full n vertex path graph, we have two situation: | The key to finding a [[Algorithms#Dynamic_Programming_Algorithms|dynamic programming algorithm]] is to identify a small set of subproblems whose solution can be computed using the previous subproblems' solutions. In this case, we start with the observation that for the full n vertex path graph, we have two situation: | ||
1. v<sub>n-1</sub> belongs to the solution. In this case, v<sub>n-2</sub> does not belong to the solution, by the properties of an independent set, and the maximum weight of the independent set for the graph G<sub>n</sub> is W<sub>n</sub> = w<sub>n-1</sub> + W<sub>n-2</sub>, where W<sub>n-2</sub> is the maximum weight independent set for the path graph | 1. v<sub>n-1</sub> belongs to the solution. In this case, v<sub>n-2</sub> does not belong to the solution, by the properties of an independent set, and the maximum weight of the independent set for the graph G<sub>n</sub> is W<sub>n</sub> = w<sub>n-1</sub> + W<sub>n-2</sub>, where W<sub>n-2</sub> is the maximum weight independent set for the path graph v<sub>0</sub>, .... v<sub>n-3</sub>. | ||
Revision as of 21:45, 27 October 2021
External
- https://www.coursera.org/learn/algorithms-greedy/lecture/t9XAF/wis-in-path-graphs-optimal-substructure
- https://www.coursera.org/learn/algorithms-greedy/lecture/w040v/wis-in-path-graphs-a-linear-time-algorithm
- https://www.coursera.org/learn/algorithms-greedy/lecture/TZgJM/wis-in-path-graphs-a-reconstruction-algorithm
Internal
Overview
This article introduces the maximum weight independent set of a path graph and provides a dynamic programming algorithm to solve it.
The Maximum Weight Independent Set Problem
Given a path graph G=(V, E) where V consists in a set of n vertices v0, v1 ... vn-1 that form a path, each of vertices with its own positive weight wi, compute a maximum weight independent set of the graph. An independent set is a set of vertices in which none is adjacent to the other.
A Dynamic Programming Approach
The key to finding a dynamic programming algorithm is to identify a small set of subproblems whose solution can be computed using the previous subproblems' solutions. In this case, we start with the observation that for the full n vertex path graph, we have two situation:
1. vn-1 belongs to the solution. In this case, vn-2 does not belong to the solution, by the properties of an independent set, and the maximum weight of the independent set for the graph Gn is Wn = wn-1 + Wn-2, where Wn-2 is the maximum weight independent set for the path graph v0, .... vn-3.