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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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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

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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

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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)

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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)

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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

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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?

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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

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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?

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Independent Set to CLIQUE This is the dual problem!

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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:

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

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

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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)

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

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

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

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

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

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

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Put every satisfying literal in the cover

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

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

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

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

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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

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

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Example:

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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

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

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since the respective literals satisfy the clauses is satisfied with

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

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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

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

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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

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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

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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

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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

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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 12 3 4

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

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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

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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)

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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)

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

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