Lecture 2: Some Representative Problems
1) Interval Scheduling: Given a set of line segments on the x-axis, choose the largest possible (by count) subset that is non-overlapping. For example, this represents a solution to scheduling a heavily requested conference room, minimizing the number of parties who don’t get the room.
2) Weighted Interval Scheduling: The same as (1), but now each line segment carries a “points” weight. The goal is to maximize the total number of “points.” For example, you could assign more points to groups requesting conference room time who have high-ranking executives in the meeting.
- This is a harder problem than (1). A solution for Weighted Interval Scheduling degrades to Interval Scheduling when the weights are all one, but not vice verse
3) Independent Set: Given a graph G=(V, E), find the largest possible set of independent vertices, that is, the set of vertices not connected by an edge (example).
“Interval scheduling is a special case of Independent Set.”
Given a set S={s1, … , sn} of segments, create a vertex for each interval and connect two vertices if their intervals overlap:
4) Bipartite Matching: First, a definition. “Bipartite graph” means the vertex set is divided into 2 subsets V = X + Y such that X ^ Y = {}, i.e. there are only edges between the members of the two subsets. A perfect bipartite match would use all the vertices. Here, the goal is to find the largest possible matching. It is similar to the stable marriage problem.
“Bipartite matching is a special case of independent set.”
Convert a given bipartite matching problem (BPM) into an independent set (IS) problem by first creating an IS vertex for each edge in the BMP. Connect two vertices in the IS graph if their BPM edges share a vertex. Then select the largest Independent Set.
5) Competitive Facility Location: Given two restaurants, a weighted map of desirable block locations, and zoning laws that forbid adjacent shops, come up with a strategy to maximize your points in placing the restaurant locations turn by turn.
Summary:
- Competitive Facility Location (PSPACE Complete)
- Independent Set (NP Complete)
- Bipartite Matching (Network Flow)
- Weighted Interval Scheduling (Dynamic Programming)
- Interval Scheduling (Greedy)
- Independent Set (NP Complete)
This entry was posted on Wednesday, January 26th, 2005 at 2:09 pm and is tagged with stable marriage problem, representative problems, line segments, line segment, vice verse, room time, bipartite, facility location, vertices, zoning laws, verse 3, vertex, bpm, subsets, interval, bmp, intervals, weights, graph, s1. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback.
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