CS4102 Algorithms Fall 2018 Warm up: Grab cookies! Start with 2, leftovers will be at regrade office hours Vegan & Gluten-free are available Courtesy of Nate and Robbie
Today’s Keywords • Reductions • Bipartite Matching • Vertex Cover • Independent Set • NP-Completeness 2
CLRS Readings • Chapter 34 3
Homeworks • HW9 due Friday 12/7 at 11pm – Written (use LaTeX) – Graphs 4
𝑙 Independent Set Is there a set of non-adjacent nodes of size 𝑙 ? 5
𝑙 Vertex Cover Is there a set of nodes of size 𝑙 which covers every edge? 6
Reduction Idea 𝑇 is an independent set of 𝐻 iff 𝑊 − 𝑇 is a vertex cover of 𝐻 Independent Set Vertex Cover 7
Reduction Idea 𝑇 is an independent set of 𝐻 iff 𝑊 − 𝑇 is a vertex cover of 𝐻 Vertex Cover Independent Set 8
MacGyver’s Reduction Problem we do know how to solve Problem we don’t know how to solve Opening a door Lighting a fire Aim duct at door, 𝐶 𝐵 insert keg How? Solution for 𝑩 Solution for 𝑪 Keg cannon Alcohol, wood, battering ram matches Put fire under the Keg Reduction 9
MaxVertCov 𝑊 -Time Reducable to MinIndSet MinVertCov O(V) Time MaxIndSet Do nothing 𝐶 𝐵 Using any Algorithm for MinVertCov Take complement of solution Solution for MinVertCov Solution for MaxIndSet 𝑍 𝑌 Reduction 10
Corollary MinVertCov O(V) Time MaxIndSet Do nothing 𝐶 𝐵 If Solving 𝑩 was Using any Algorithm for MinIndSet always slow Then this shows solving 𝑪 is also slow Take complement of solution Solution for MinVertCov Solution for MaxIndSet 𝑍 𝑌 Reduction 11
MaxIndSet 𝑊 -Time Reducable to MinVertCov MaxIndSet MinVertCov O(V) Time Do nothing 𝐶 𝐵 Using any Algorithm for MaxIndSet Solution for MinVertCov Solution for MaxIndSet Take complement of solution 𝑍 𝑌 Reduction 12
Corollary MaxIndSet MinVertCov O(V) Time Do nothing 𝐶 𝐵 If Solving 𝑩 was Using any Algorithm for MaxVertCov always slow Then this shows solving 𝑪 is also slow Solution for MinVertCov Solution for MaxIndSet Take complement of solution 𝑍 𝑌 Reduction 13
Conclusion • MaxIndSet and MinVertCov are either both fast, or both slow – Spoiler alert: We don’t know which! • (But we think they’re both slow) – Both problems are NP-Complete 14
Problem Types • Decision Problems: If we can solve this – Is there a solution? • Output is True/False – Is there a vertex cover of size 𝑙 ? • Search Problems: Then we can solve this – Find a solution • Output is complex – Give a vertex cover of size 𝑙 • Verification Problems: – Given a potential solution, is it valid? • Output is True/False – Is this a vertex cover of size 𝑙 ? 15
P vs NP • P – Deterministic Polynomial Time – Problems solvable in polynomial time • 𝑃(𝑜 𝑞 ) for some number 𝑞 • NP NP – Non-Deterministic Polynomial Time P – Problems verifiable in polynomial time • 𝑃(𝑜 𝑞 ) for some number 𝑞 • Open Problem: Does P=NP? – Certainly 𝑄 ⊆ 𝑂𝑄 16
𝑙 -Independent Set is NP • To show: Given a potential solution, can we verify it in 𝑃(𝑜 𝑞 ) ? [𝑜 = 𝑊 + 𝐹] How can we verify it? 1. Check that it’s of size 𝑙 𝑃(𝑊) 2. Check that it’s an independent set 𝑃(𝑊 2 ) 17
𝑙 -Vertex Cover is NP • To show: Given a potential solution, can we verify it in 𝑃(𝑜 𝑞 ) ? [𝑜 = 𝑊 + 𝐹] How can we verify it? 1. Check that it’s of size 𝑙 𝑃(𝑊) 2. Check that it’s a Vertex Cover 𝑃(𝐹) 18
NP-Hard • How can we try to figure out if P=NP? NP-H • Identify problems at least as “hard” as NP At least as “hard” as NP – If any of these “hard” problems can be solved in polynomial time, then all NP problems can be solved in polynomial time. NP • Definition: NP-Hard: – 𝐶 is NP-Hard if ∀𝐵 ∈ 𝑂𝑄, 𝐵 ≤ 𝑞 𝐶 P – 𝐵 ≤ 𝑞 𝐶 means 𝐵 reduces to 𝐶 in polynomial time 19
NP-Hardness Reduction 𝑃(𝑜 𝑞 ) Problem to show is NP-Hard Any NP-Hard Problem 𝐶 𝐵 If This could be done in Polynomial time Then this could be done in polynomial time Solution for 𝑩 Solution for 𝑪 𝑍 𝑌 Reduction 20
NP-Complete • “Together they stand, together they fall” NP-H • Problems solvable in polynomial time iff ALL NP problems are • NP-Complete = NP ∩ NP-Hard NP-C • How to show a problem is NP-Complete? NP – Show it belongs to NP P • Give a polynomial time verifier – Show it is NP-Hard • Give a reduction from another NP-H problem We now just need a FIRST NP-Hard problem 21
NP-Completeness 𝑃(𝑜 𝑞 ) Any other NP-Complete Problem Any NP-Complete Problem 𝐶 𝐵 If This could be done in Polynomial time Then this could be done in polynomial time Solution for 𝑩 Solution for 𝑪 𝑍 𝑌 Reduction 22
NP-Completeness 𝑃(𝑜 𝑞 ) Any other NP-Complete Problem Any NP-Complete Problem 𝐶 𝐵 If this cannot be done in polynomial time Then this cannot be done in polynomial time Solution for 𝑩 Solution for 𝑪 𝑍 𝑌 Reduction 23
3-SAT • Shown to be NP-Hard by Cook and Levin (independently) • Given a 3-CNF formula (logical AND of clauses, each an OR of 3 variables), Is there an assignment of true/false to each variable to make the formula true? 𝑦 ∨ 𝑧 ∨ 𝑨 ∧ 𝑦 ∨ 𝑧 ∨ 𝑧 ∧ 𝑣 ∨ 𝑧 ∨ 𝑨 ∧ 𝑨 ∨ 𝑦 ∨ 𝑣 ∧ (𝑦 ∨ 𝑧 ∨ 𝑨 ) 𝒚 = 𝒖𝒔𝒗𝒇 Clause 𝒛 = 𝒈𝒃𝒎𝒕𝒇 Variables 𝒜 = 𝒈𝒃𝒎𝒕𝒇 𝒗 = 𝒖𝒔𝒗𝒇 24
𝑙 -Independent Set is NP-Complete 1. Show that it belongs to NP – Give a polynomial time verifier (slide 21) 2. Show it is NP-Hard – Give a reduction from a known NP-Hard problem Show 3𝑇𝐵𝑈 ≤ 𝑞 𝑙𝐽𝑜𝑒𝑇𝑓𝑢 – 25
3𝑇𝐵𝑈 ≤ 𝑞 𝑙𝐽𝑜𝑒𝑇𝑓𝑢 𝑃(𝑜 𝑞 ) 𝑙 -Ind Set 3SAT 𝐶 𝐵 Solution for 3SAT Solution for 𝑙 -Ind Set 𝑍 𝑌 Reduction 26
Instance of 3SAT to Instance of 𝑙 IndSet 𝑦 ∨ 𝑧 ∨ 𝑨 ∧ 𝑦 ∨ 𝑧 ∨ 𝑧 ∧ 𝑣 ∨ 𝑧 ∨ 𝑨 ∧ 𝑨 ∨ 𝑦 ∨ 𝑣 ∧ (𝑦 ∨ 𝑧 ∨ 𝑨 ) 𝑨 𝑦 𝑣 𝑣 𝑦 𝑣 𝑦 𝑧 𝑧 𝑨 𝑨 𝑧 𝑧 𝑨 𝑧 For each clause, produce a triangle graph with its three variables as nodes Connect each node to all of its opposites Let 𝑙 = number of clauses There is a 𝑙 -IndSet in this graph, iff there is a satisfying assignment 27
𝑙 IndSet ⇒ Satisfying Assignment 𝑦 ∨ 𝑧 ∨ 𝑨 ∧ 𝑦 ∨ 𝑧 ∨ 𝑧 ∧ 𝑣 ∨ 𝑧 ∨ 𝑨 ∧ 𝑨 ∨ 𝑦 ∨ 𝑣 ∧ (𝑦 ∨ 𝑧 ∨ 𝑨 ) 𝑨 𝑦 𝑣 𝑣 𝑦 𝑣 𝑦 𝑧 𝑧 𝑨 𝑧 𝑨 𝑧 𝑨 𝑧 𝒚 = 𝒖𝒔𝒗𝒇 𝒛 = 𝒈𝒃𝒎𝒕𝒇 𝒜 = 𝒈𝒃𝒎𝒕𝒇 One node per triangle is in the Independent set: 𝒗 = 𝒖𝒔𝒗𝒇 because we can have exactly 𝑙 total in the set, and 2 in a triangle would be adjacent If 𝑦 is selected in some triangle, 𝑦 is not selected in any triangle: Because every 𝑦 is adjacent to every 𝑦 Set the variable which each included node represents to “true” 28
Satisfying Assignment ⇒ 𝑙 IndSet 𝑦 ∨ 𝑧 ∨ 𝑨 ∧ 𝑦 ∨ 𝑧 ∨ 𝑧 ∧ 𝑣 ∨ 𝑧 ∨ 𝑨 ∧ 𝑨 ∨ 𝑦 ∨ 𝑣 ∧ (𝑦 ∨ 𝑧 ∨ 𝑨 ) 𝑨 𝑦 𝑣 𝑣 𝑦 𝑣 𝑦 𝑧 𝑧 𝑨 𝑧 𝑨 𝑧 𝑨 𝑧 𝒚 = 𝒖𝒔𝒗𝒇 𝒛 = 𝒈𝒃𝒎𝒕𝒇 𝒜 = 𝒈𝒃𝒎𝒕𝒇 Use one true variable from the assignment for each triangle 𝒗 = 𝒖𝒔𝒗𝒇 The independent set has 𝑙 nodes, because there are 𝑙 clauses If any variable 𝑦 is true then 𝑦 cannot be true 29
3𝑇𝐵𝑈 ≤ 𝑞 𝑙𝐽𝑜𝑒𝑇𝑓𝑢 𝑃(𝑜 𝑞 ) 𝑙 -Ind Set 3SAT 𝐶 Make triangles, connect 𝐵 opposites, 𝑙 = num clauses If This could be done in Polynomial time Then This could be done in polynomial time Solution for 3SAT Solution for 𝑙 -Ind Set Assign true to variables from selected nodes 𝑍 𝑌 Reduction 30
𝑙 -Vertex Cover is NP-Complete 1. Show that it belongs to NP – Give a polynomial time verifier (slide 22) 2. Show it is NP-Hard – Give a reduction from a known NP-Hard problem We showed 𝑙𝐽𝑜𝑒𝑇𝑓𝑢 ≤ 𝑞 𝑙𝑊𝑓𝑠𝑢𝐷𝑝𝑤 – • (Last Class) 31
𝑙𝐽𝑜𝑒𝑇𝑓𝑢 ≤ 𝑞 𝑙𝑊𝑓𝑠𝑢𝐷𝑝𝑤 𝑃(𝑜 𝑞 ) 𝑙𝐽𝑜𝑒𝑇𝑓𝑢 𝑙𝑊𝑓𝑠𝑢𝐷𝑝𝑤 𝐶 𝐵 𝑙 = 𝑊 − 𝑙 If This could be done in Polynomial time Then This could be done in polynomial time Solution for 𝑙𝐽𝑜𝑒𝑇𝑓𝑢 Solution for 𝑙𝑊𝑓𝑠𝑢𝐷𝑝𝑤 Take Complement of nodes 𝑍 𝑌 Reduction 32
𝑙 -Clique Problem • Clique: A complete subgraph 4-Clique • 𝑙 -Clique Problem: – Given a graph 𝐻 and a number 𝑙 , is there a clique of size 𝑙 ? 3-Clique 33
𝑙 -Clique is NP-Complete 1. Show that it belongs to NP – Give a polynomial time verifier 2. Show it is NP-Hard – Give a reduction from a known NP-Hard problem We will show 3𝑇𝐵𝑈 ≤ 𝑞 𝑙𝐷𝑚𝑗𝑟𝑣𝑓 – 34
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