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NP-Completeness: Reductions CSc 4520/6520 Fall 2013 Slides adapted from David Evans (Virgina) and Costas Busch (RPI)

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Presentation on theme: "NP-Completeness: Reductions CSc 4520/6520 Fall 2013 Slides adapted from David Evans (Virgina) and Costas Busch (RPI)"— Presentation transcript:

1 NP-Completeness: Reductions CSc 4520/6520 Fall 2013 Slides adapted from David Evans (Virgina) and Costas Busch (RPI)

2 Definition of NP-Complete Q is an NP-Complete problem if: 1) Q is in NP 2) every other NP problem polynomial time is reducible to Q

3 Getting Started How do you show that EVERY NP problem reduces to Q? One way would be to already have an NP- Complete problem and just reduce from that P1P1 P2P2 P3P3 P4P4 Mystery NP-Complete Problem Q

4 3-SAT 3-SAT = { f | f is in Conjunctive Normal Form, each clause has exactly 3 literals and f is satisfiable } 3-SAT is NP-Complete (2-SAT is in P)

5 NP-Complete To prove a problem is NP-Complete show a polynomial time reduction from 3-SAT Other NP-Complete Problems: PARTITION SUBSET-SUM CLIQUE HAMILTONIAN PATH (TSP) GRAPH COLORING MINESWEEPER (and many more)

6 NP-Completeness Proof Method To show that Q is NP-Complete: 1) Show that Q is in NP 2) Pick an instance, R, of your favorite NP- Complete problem (ex: Φ in 3-SAT) 3) Show a polynomial algorithm to transform R into an instance of Q

7 Example: Clique CLIQUE = { | G is a graph with a clique of size k } A clique is a subset of vertices that are all connected Why is CLIQUE in NP?

8 Reduce 3-SAT to Clique Pick an instance of 3-SAT, Φ, with k clauses Make a vertex for each literal Connect each vertex to the literals in other clauses that are not the negation Any k-clique in this graph corresponds to a satisfying assignment

9

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11 Example: Independent Set INDEPENDENT SET = { | where G has an independent set of size k } An independent set is a set of vertices that have no edges How can we reduce this to clique?

12 Independent Set to CLIQUE This is the dual problem!

13 Vertex Cover Vertex cover of a graph is a subset of nodes such that every edge in the graph touches one node in S = red nodes Example:

14 |S|=4Example: Size of vertex-cover is the number of nodes in the cover

15 graph contains a vertex cover of size } VERTEX-COVER = { : Corresponding language: Example:

16 Theorem: 1. VERTEX-COVER is in NP 2. We will reduce in polynomial time 3CNF-SAT to VERTEX-COVER Can be easily proven VERTEX-COVER is NP-complete Proof: (NP-complete)

17 Let be a 3CNF formula with variables and clauses Example: Clause 1 Clause 2Clause 3

18 Formula can be converted to a graph such that: is satisfied if and only if Contains a vertex cover of size

19 Clause 1 Clause 2Clause 3 Clause 1 Clause 2Clause 3 Variable Gadgets Clause Gadgets nodes

20 Clause 1 Clause 2Clause 3 Clause 1 Clause 2Clause 3

21 If is satisfied, then contains a vertex cover of size First direction in proof:

22 Satisfying assignment Example: We will show that contains a vertex cover of size

23 Put every satisfying literal in the cover

24 Select one satisfying literal in each clause gadget and include the remaining literals in the cover

25 This is a vertex cover since every edge is adjacent to a chosen node

26 Explanation for general case: Edges in variable gadgets are incident to at least one node in cover

27 Edges in clause gadgets are incident to at least one node in cover, since two nodes are chosen in a clause gadget

28 Every edge connecting variable gadgets and clause gadgets is one of three types: Type 1 Type 2 Type 3 All adjacent to nodes in cover

29 If graph contains a vertex-cover of size then formula is satisfiable Second direction of proof:

30 Example:

31 exactly one literal in each variable gadget is chosen exactly two nodes in each clause gadget is chosen To include “internal’’ edges to gadgets, and satisfy chosen out of

32 For the variable assignment choose the literals in the cover from variable gadgets

33 since the respective literals satisfy the clauses is satisfied with

34 Dominating Set Problem: Dominating-set = { | A dominating set of size K for G exists} Goal: Show that Dominating-set is NP-Complete

35 Dominating Set (Definition) Problem: Dominating-set = { | A dominating set of size (at most) K for G exists} Let G=(V,E) be an undirected graph A dominating set D is a set of vertices that covers all vertices i.e., every vertex of G is either in D or is adjacent to at least one vertex from D

36 Dominating Set (Example) Size-2 example : {Yellow vertices} e

37 Dominating Set (Proof Sketch) Steps: 1)Show that Dominating-set ∈ NP. 2)Show that Dominating-set is not easier than a NPC problem We choose this NPC problem to be Vertex cover Reduction from Vertex-cover to Dominating-set 3)Show the correspondence of “yes” instances between the reduction

38 Dominating Set - (1) NP It is trivial to see that Dominating-set ∈ NP Given a vertex set D of size K, we check whether (V-D) are adjacent to D i.e., for each vertex, v, in (V-D), whether v is adjacent to some vertex u in D

39 Dominating Set - (2) Reduction Reduction - Graph transformation Construct a new graph G' by adding new vertices and edges to the graph G as follows: G G’ ∈ L ∈ L ’ T Vertex-cover Dominating-set

40 Dominating Set - (2) Reduction Reduction - Graph transformation (Con’t) For each edge (v, w) of G, add a vertex vw and the edges (v, vw) and (w, vw) to G' Furthermore, remove all vertices with no incident edges; such vertices would always have to go in a dominating set but are not needed in a vertex cover of G G G’ ∈ L ∈ L ’ T Vertex-cover Dominating-set

41 Vertex cover A vertex cover, C, is a set of vertices that covers all edges i.e., each edge is at least adjacent to some node in C {2, 4}, {3, 4}, {1, 2, 3} are vertex covers

42 Dominating Set – Graph Transformation Example vw zu G wv zu vzwu vw zu vu G'

43 Dominating Set - (3) Correspondence A dominating set of size K in G’  A vertex cover of size K in G  Let D be a dominating set of size K in G’ Case 1): D contains only vertices from G Then, all new vertices have an edge to a vertex in D D covers all edges D is a valid vertex cover of G

44 Dominating Set - (3) Correspondence A dominating set of size K in G’  A vertex cover of size K in G  Let D be a dominating set of size K in G’ Case 2): D contains some new vertices (vertex in the form of uv) (We show how to construct a vertex cover using only old vertices, otherwise we cannot obtain a vertex cover for G) For each new vertex uv, replace it by u (or v) If u ∈ D, this node is not needed Then the edge u-v in G will be covered After new edges are removed, it is a valid vertex cover of G (of size at most K)

45 Dominating Set - (3) Correspondence A dominating set of size K in G’  A vertex cover of size K in G  Let C be a vertex cover of size K in G For an old vertex, v ∈ G’ : By the definition of VC, all edges incident to v are covered v is also covered For a new vertex, uv ∈ G’ : Edge u-v must be covered, either u or v ∈ C This node will cover uv in G’ Thus, C is a valid dominating for G’ (of size at most K)

46 Dominating Set - (3) Correspondence vw zu wv zu vzwu vw zu vu Vertex-cover in G Dominating-set in G'


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