Five Representative Problems

• We will look at these quickly
  • They are used for examples in the rest of the book.
  • They sort of serve as a guide for the rest of the book as well.
  • They are interesting problems.

1. Interval Scheduling
   • You have a resource like a room or a tool that is needed for a task to be performed.
   • You have a list tasks, with both the beginning time and ending time (or duration)
     • $T= \{(s_1, f_1), (s_2, f_2), ... (s_n, f_n)\}, s_i \lt f_i$
   • You would like to schedule the maximum number of tasks using the resource.
   • Two tasks $(s_i, f_i)$ and $(s_j, f_j)$ are compatible if they do not overlap.
     • or if $s_i \lt s_j$ then $f_i \le s_j$
   • Formally, select A, the set of all compatible pairs such that $|A|$ is maximal.
   • 
   • To solve this, we use a greedy algorithm design pattern.
     • This is a fairly simple approach.
     • In this case, take the next compatible task to finish first.
     • Generally in greedy, we only consider the next task, no look ahead.
     • Proof that a greedy algorithm is valid is sometimes more challenging.
2. Weighted Interval Scheduling.
   • To the previous problem add a weight or reward for scheduling each task.
   • $T= \{(s_1, f_1, w_1), (s_2, f_2, w_2), ... (s_n, f_n, w_n)\}, s_i \lt f_i, w_i > 0$
   • If for all $i, w_i = c$ for some constant $c$ then this is the previous problem.
   • Greedy will not work.
   • The solution involves dynamic programming.
     • Generally you build a table to compute the solution.
3. Bipartite matching.
   • Given a graph $G=(V,E)$.
   • Can the set of vertexes V be partitioned into two sets $X$ and $Y, X \cap Y = \emptyset, X \cup Y = V, \ni \forall e \in E, e = (v_i, v_j), v_i \in X, v_j \in Y$
   • (From Wikipedia)
   • Bipartite Matching Problem: Given an arbitrary bipartite graph G, find a matching of maximum size.
     • In stable matching, there was a "path" or a possible match between every man and every woman.
     • This is not true in the bipartite matching case.
   • Find the bipartite match in figure 1.5
     • 
   • This might be a set of machines and the jobs matched to those machines.
   • Or courses assigned to a professor.
   • We don't know a straight forward solution.
   • And greedy algorithms don't work.
   • There is an algorithm, with backtracking that solves this problem.
4. Independent Set Problem
   • Given a graph $G=(V,E)$
   • Find the set $I \subset V$ such that for all $u \in I, v \in I$ then $(u,v) \notin E$.
   • The goal is to maximize $|I|$
   • Find $I$ for the graph in figure 1.6
   • 
   • They mention that many of the previous problems are actually special cases of this problem.
     • For interval scheduling
       • Define $G=(V,E)$ where
         • The vertexes are the intervals.
         • There are edges between any two intervals that overlap.
         • Finding the independent set will solve this.
     • He gives an equivalence for the Bipart Matching problem as well.
   • Unfortunately this problem is very difficult to solve.
     • We don't know of an efficient algorithm to solve it.
     • The Brute force algorithm would be $O(n!)$, try all combinations.
     • And we can prove that if we find an easy solution, then many other hard problems become easy too.
     • This is an example of a problem that is NP-complete.
5. Competitive Facility Location
   • Game theory is a branch of mathematics.
   • They propose simple "games" where the players generally make "perfect" decisions.
   • The following is a two player game
     • There is a region divided into $n$ zones labeled 1, 2, ... n.
     • Each zone has a value $b_i$ which a player can claim if they claim the zone.
     • If two zones $i$ and $j$ are connected, (ie an edge $(i,j)$ exists in the graph representing the region) then any player may only claim one of these two zones. (IE no two adjacent zones can be claimed)
     • Players take turns claiming zones.
     • Given a bound $B$ which represents the minimum value player two must achieve to win, is there a winning strategy for $p_2$ regardless of $p_1$ moves?
   • 
   • Try with $B=20$
   • Try with $B=25$
   • It turns out that this is probably harder to do than the previous problem.
     • It fits a class called PSPACE
     • These are algorithms that can be solved in polynomial space.
6. The Halting Problem
   • This is my addition
   • Given an arbitrary computer program and the input, determine if the program will finish running.
   • This problem is undecidable with our current definition of a computer and program.
   • The proof is along the lines

     Assume that there is a function H(P,d) which takes a Program P and an input data set d.
     
     If P halts when give d as input, H returns true, otherwise it returns false.
     
     Consider the following program:
     
     G(G,d) 
          if (H(G,d)) then
              loop forever
          else
              exit
     
     
     If H predicts G will halt, in line 1, it will loop forever in line 2 otherwise it will exit in line 4.
     
     Therefore H does not properly predict G's exit status.
     
     Thus the assumption that H exists is false.