NP Complete: The Exciting Conclusion Review For Final CS 332: Algorithms NP Complete: The Exciting Conclusion Review For Final David Luebke 1 4/21/2017

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 2 4/21/2017

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 3 4/21/2017

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 4 4/21/2017

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 5 4/21/2017

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 6 4/21/2017

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 7 4/21/2017

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 8 4/21/2017

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 9 4/21/2017

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 10 4/21/2017

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 11 4/21/2017

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 12 4/21/2017

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 13 4/21/2017

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 14 4/21/2017

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 15 4/21/2017

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 16 4/21/2017

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 17 4/21/2017

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 18 4/21/2017

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 19 4/21/2017

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 20 4/21/2017

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 4/21/2017

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 22 4/21/2017

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 23 4/21/2017

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 24 4/21/2017

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 25 4/21/2017

Graph Representation Adjacency list Adjacency matrix Tradeoffs: What makes a graph dense? What makes a graph sparse? What about planar graphs? David Luebke 26 4/21/2017

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 27 4/21/2017

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 28 4/21/2017

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 29 4/21/2017

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 30 4/21/2017

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 31 4/21/2017

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 32 4/21/2017

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 33 4/21/2017

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 34 4/21/2017

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 35 4/21/2017

The End David Luebke 36 4/21/2017