1. NP Complete: The Exciting Conclusion Review For Final
CS 332: Algorithms
NP Complete: The Exciting Conclusion
Review For Final
David Luebke /21/2017

2. Administrivia Homework 5 due now
All previous homeworks available after class
Undergrad TAs still needed (before finals)
Final exam
Wednesday, December 13
9 AM - noon
You are allowed two 8.5“ x 11“ cheat sheets
Both sides okay
Mechanical reproduction okay (sans microfiche)
David Luebke /21/2017

3. Homework 5 Optimal substructure:
Given an optimal subset A of items, if remove item j, remaining subset A’ = A-{j} is optimal solution to knapsack problem (S’ = S-{j}, W’ = W - wj)
Key insight is figuring out a formula for c[i,w], value of soln for items 1..i and max weight w:
Time: O(nW)
David Luebke /21/2017

4. Review: P and NP What do we mean when we say a problem is in P?
What do we mean when we say a problem is in NP?
What is the relation between P and NP?
David Luebke /21/2017

5. Review: P and NP What do we mean when we say a problem is in P?
A: A solution can be found in polynomial time
What do we mean when we say a problem is in NP?
A: A solution can be verified in polynomial time
What is the relation between P and NP?
A: P  NP, but no one knows whether P = NP
David Luebke /21/2017

6. Review: NP-Complete
What, intuitively, does it mean if we can reduce problem P to problem Q?
How do we reduce P to Q?
What does it mean if Q is NP-Hard?
What does it mean if Q is NP-Complete?
David Luebke /21/2017

7. Review: NP-Complete
What, intuitively, does it mean if we can reduce problem P to problem Q?
P is “no harder than” Q
How do we reduce P to Q?
Transform instances of P to instances of Q in polynomial time s.t. Q: “yes” iff P: “yes”
What does it mean if Q is NP-Hard?
Every problem PNP p Q
What does it mean if Q is NP-Complete?
Q is NP-Hard and Q  NP
David Luebke /21/2017

8. Review: Proving Problems NP-Complete
What was the first problem shown to be NP-Complete?
A: Boolean satisfiability (SAT), by Cook
How do we usually prove that a problem R is NP-Complete?
A: Show R NP, and reduce a known NP-Complete problem Q to R
David Luebke /21/2017

9. Review: Directed  Undirected Ham. Cycle
Given: directed hamiltonian cycle is NP-Complete (draw the example)
Transform graph G = (V, E) into G’ = (V’, E’):
Every vertex v in V transforms into 3 vertices v1, v2, v3 in V’ with edges (v1,v2) and (v2,v3) in E’
Every directed edge (v, w) in E transforms into the undirected edge (v3, w1) in E’ (draw it)
David Luebke /21/2017

10. Review: Directed  Undirected Ham. Cycle
Prove the transformation correct:
If G has directed hamiltonian cycle, G’ will have undirected cycle (straightforward)
If G’ has an undirected hamiltonian cycle, G will have a directed hamiltonian cycle
The three vertices that correspond to a vertex v in G must be traversed in order v1, v2, v3 or v3, v2, v1, since v2 cannot be reached from any other vertex in G’
Since 1’s are connected to 3’s, the order is the same for all triples. Assume w.l.o.g. order is v1, v2, v3.
Then G has a corresponding directed hamiltonian cycle
David Luebke /21/2017

11. Review: Hamiltonian Cycle  TSP
The well-known traveling salesman problem:
Complete graph with cost c(i,j) from city i to city j
 a simple cycle over cities with cost < k ?
How can we prove the TSP is NP-Complete?
A: Prove TSP  NP; reduce the undirected hamiltonian cycle problem to TSP
TSP  NP: straightforward
Reduction: need to show that if we can solve TSP we can solve ham. cycle problem
David Luebke /21/2017

12. Review: Hamiltonian Cycle  TSP
To transform ham. cycle problem on graph G = (V,E) to TSP, create graph G’ = (V,E’):
G’ is a complete graph
Edges in E’ also in E have weight 0
All other edges in E’ have weight 1
TSP: is there a TSP on G’ with weight 0?
If G has a hamiltonian cycle, G’ has a cycle w/ weight 0
If G’ has cycle w/ weight 0, every edge of that cycle has weight 0 and is thus in G. Thus G has a ham. cycle
David Luebke /21/2017

13. Review: Conjunctive Normal Form
3-CNF is a useful NP-Complete problem:
Literal: an occurrence of a Boolean or its negation
A Boolean formula is in conjunctive normal form, or CNF, if it is an AND of clauses, each of which is an OR of literals
Ex: (x1  x2)  (x1  x3  x4)  (x5)
3-CNF: each clause has exactly 3 distinct literals
Ex: (x1  x2  x3)  (x1  x3  x4)  (x5  x3  x4)
Notice: true if at least one literal in each clause is true
David Luebke /21/2017

14. 3-CNF  Clique What is a clique of a graph G?
A: a subset of vertices fully connected to each other, i.e. a complete subgraph of G
The clique problem: how large is the maximum-size clique in a graph?
Can we turn this into a decision problem?
A: Yes, we call this the k-clique problem
Is the k-clique problem within NP?
David Luebke /21/2017

15. 3-CNF  Clique What should the reduction do?
A: Transform a 3-CNF formula to a graph, for which a k-clique will exist (for some k) iff the 3-CNF formula is satisfiable
David Luebke /21/2017

16. 3-CNF  Clique The reduction:
Let B = C1  C2  …  Ck be a 3-CNF formula with k clauses, each of which has 3 distinct literals
For each clause put a triple of vertices in the graph, one for each literal
Put an edge between two vertices if they are in different triples and their literals are consistent, meaning not each other’s negation
Run an example: B = (x  y  z)  (x  y  z )  (x  y  z )
David Luebke /21/2017

17. 3-CNF  Clique Prove the reduction works:
If B has a satisfying assignment, then each clause has at least one literal (vertex) that evaluates to 1
Picking one such “true” literal from each clause gives a set V’ of k vertices. V’ is a clique (Why?)
If G has a clique V’ of size k, it must contain one vertex in each clique (Why?)
We can assign 1 to each literal corresponding with a vertex in V’, without fear of contradiction
David Luebke /21/2017

18. Clique  Vertex Cover
A vertex cover for a graph G is a set of vertices incident to every edge in G
The vertex cover problem: what is the minimum size vertex cover in G?
Restated as a decision problem: does a vertex cover of size k exist in G?
Thm 36.12: vertex cover is NP-Complete
David Luebke /21/2017

19. Clique  Vertex Cover First, show vertex cover in NP (How?)
Next, reduce k-clique to vertex cover
The complement GC of a graph G contains exactly those edges not in G
Compute GC in polynomial time
G has a clique of size k iff GC has a vertex cover of size |V| - k
David Luebke /21/2017

20. Clique  Vertex Cover
Claim: If G has a clique of size k, GC has a vertex cover of size |V| - k
Let V’ be the k-clique
Then V - V’ is a vertex cover in GC
Let (u,v) be any edge in GC
Then u and v cannot both be in V’ (Why?)
Thus at least one of u or v is in V-V’ (why?), so edge (u, v) is covered by V-V’
Since true for any edge in GC, V-V’ is a vertex cover
David Luebke /21/2017

21. Clique  Vertex Cover
Claim: If GC has a vertex cover V’  V, with |V’| = |V| - k, then G has a clique of size k
For all u,v  V, if (u,v)  GC then u  V’ or v  V’ or both (Why?)
Contrapositive: if u  V’ and v  V’, then (u,v)  E
In other words, all vertices in V-V’ are connected by an edge, thus V-V’ is a clique
Since |V| - |V’| = k, the size of the clique is k
David Luebke /21/2017

22. General Comments
Literally hundreds of problems have been shown to be NP-Complete
Some reductions are profound, some are comparatively easy, many are easy once the key insight is given
You can expect a simple NP-Completeness proof on the final
David Luebke /21/2017

23. Other NP-Complete Problems
Subset-sum: Given a set of integers, does there exist a subset that adds up to some target T?
0-1 knapsack: you know this one
Hamiltonian path: Obvious
Graph coloring: can a given graph be colored with k colors such that no adjacent vertices are the same color?
Etc…
David Luebke /21/2017

24. Final Exam Coverage: 60% stuff since midterm, 40% stuff before midterm
Goal: doable in 2 hours
This review just covers material since the midterm review
David Luebke /21/2017

25. Final Exam: Study Tips Study tips: Re-make your midterm cheat sheet
Study each lecture since the midterm
Study the homework and homework solutions
Study the midterm
Re-make your midterm cheat sheet
I recommend handwriting or typing it
Think about what you should have had on it the first time…cheat sheet is about identifying important concepts
David Luebke /21/2017

26. Graph Representation Adjacency list Adjacency matrix Tradeoffs:
What makes a graph dense?
What makes a graph sparse?
What about planar graphs?
David Luebke /21/2017

27. Basic Graph Algorithms
Breadth-first search
What can we use BFS to calculate?
A: shortest-path distance to source vertex
Depth-first search
Tree edges, back edges, cross and forward edges
What can we use DFS for?
A: finding cycles, topological sort
David Luebke /21/2017

28. Topological Sort, MST Topological sort Minimum spanning tree
Examples: getting dressed, project dependency
What kind of graph do we do topological sort on?
Minimum spanning tree
Optimal substructure
Min edge theorem (enables greedy approach)
David Luebke /21/2017

29. MST Algorithms Prim’s algorithm Kruskal’s algorithm
What is the bottleneck in Prim’s algorithm?
A: priority queue operations
Kruskal’s algorithm
What is the bottleneck in Kruskal’s algorithm?
Answer: depends on disjoint-set implementation
As covered in class, disjoint-set union operations
As described in book, sorting the edges
David Luebke /21/2017

30. Single-Source Shortest Path
Optimal substructure
Key idea: relaxation of edges
What does the Bellman-Ford algorithm do?
What is the running time?
What does Dijkstra’s algorithm do?
When does Dijkstra’s algorithm not apply?
David Luebke /21/2017

31. Disjoint-Set Union
We talked about representing sets as linked lists, every element stores pointer to list head
What is the cost of merging sets A and B?
A: O(max(|A|, |B|))
What is the maximum cost of merging n 1-element sets into a single n-element set?
A: O(n2)
How did we improve this? By how much?
A: always copy smaller into larger: O(n lg n)
David Luebke /21/2017

32. Amortized Analysis
Idea: worst-case cost of an operation may overestimate its cost over course of algorithm
Goal: get a tighter amortized bound on its cost
Aggregate method: total cost of operation over course of algorithm divided by # operations
Example: disjoint-set union
Accounting method: “charge” a cost to each operation, accumulate unused cost in bank, never go negative
Example: dynamically-doubling arrays
David Luebke /21/2017

33. Dynamic Programming
Indications: optimal substructure, repeated subproblems
What is the difference between memoization and dynamic programming?
A: same basic idea, but:
Memoization: recursive algorithm, looking up subproblem solutions after computing once
Dynamic programming: build table of subproblem solutions bottom-up
David Luebke /21/2017

34. LCS Via Dynamic Programming
Longest common subsequence (LCS) problem:
Given two sequences x[1..m] and y[1..n], find the longest subsequence which occurs in both
Brute-force algorithm: 2m subsequences of x to check against n elements of y: O(n 2m)
Define c[i,j] = length of LCS of x[1..i], y[1..j]
Theorem:
David Luebke /21/2017

35. Greedy Algorithms Indicators: Example problems: Optimal substructure
Greedy choice property: a locally optimal choice leads to a globally optimal solution
Example problems:
Activity selection: Set of activities, with start and end times. Maximize compatible set of activities.
Fractional knapsack: sort items by $/lb, then take items in sorted order
MST
David Luebke /21/2017

36. The End
David Luebke /21/2017