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CS 332: Algorithms. NP Complete: The Exciting Conclusion Review For Final. 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

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cs 332 algorithms

CS 332: Algorithms

NP Complete: The Exciting Conclusion

Review For Final

David Luebke 14/3/2014

administrivia
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
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
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 np5
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 54/3/2014

review np complete
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 complete7
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 74/3/2014

review proving problems np complete
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
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 cycle10
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
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 tsp12
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
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
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 clique15
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 clique16
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 clique17
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
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 cover19
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 cover20
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 cover21
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
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
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
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
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
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
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, 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
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
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
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
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
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
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
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
The End

David Luebke 364/3/2014