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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 24/3/2014

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 34/3/2014

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 44/3/2014

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: PNP, but no one knows whether P = NP

David Luebke 54/3/2014

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 64/3/2014

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 PNP p Q
- What does it mean if Q is NP-Complete?
- Q is NP-Hard and Q NP

David Luebke 74/3/2014

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 Ris NP-Complete?
- A: Show R NP, and reduce a known NP-Complete problem Q to R

David Luebke 84/3/2014

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 vin 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 94/3/2014

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 104/3/2014

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 114/3/2014

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 124/3/2014

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 134/3/2014

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 144/3/2014

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 154/3/2014

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 164/3/2014

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 174/3/2014

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 184/3/2014

Clique Vertex Cover

- First, show vertex cover in NP (How?)
- Next, reduce k-clique to vertex cover
- The complementGC 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 194/3/2014

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 204/3/2014

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 214/3/2014

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 224/3/2014

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 234/3/2014

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 244/3/2014

Final Exam: Study Tips

- Study tips:
- 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 254/3/2014

Graph Representation

- Adjacency list
- Adjacency matrix
- Tradeoffs:
- What makes a graph dense?
- What makes a graph sparse?
- What about planar graphs?

David Luebke 264/3/2014

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 274/3/2014

Topological Sort, MST

- Topological sort
- 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 284/3/2014

MST Algorithms

- Prim’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 294/3/2014

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?
- What is the running time?
- When does Dijkstra’s algorithm not apply?

David Luebke 304/3/2014

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 314/3/2014

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 324/3/2014

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 334/3/2014

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 344/3/2014

Greedy Algorithms

- Indicators:
- 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 354/3/2014

The End

David Luebke 364/3/2014

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