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

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

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 P1 P2 Mystery NP-Complete Problem Reference halting problem and undecidability Q P3 P4

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)

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)

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

Example: Clique CLIQUE = { <G,k> | 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?

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

Example: Independent Set INDEPENDENT SET = { <G,k> | 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?

Independent Set to CLIQUE This is the dual problem!

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

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

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

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

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

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

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

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

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

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

Put every satisfying literal in the cover

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

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

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

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

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

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

Example:

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

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

is satisfied with since the respective literals satisfy the clauses

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

Dominating Set (Definition) Problem: Dominating-set = {<G, K> | 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

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

Dominating Set (Proof Sketch) Steps: Show that Dominating-set ∈ NP. 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 Show the correspondence of “yes” instances between the reduction

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

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

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 T G G’ <G,k> ∈ L Vertex-cover Dominating-set <G’, k> ∈ L’

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 1 2 3 4 {2, 4}, {3, 4}, {1, 2, 3} are vertex covers

Dominating Set – Graph Transformation Example w v z u vz wu vw zu vu G' v w z u G

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

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)

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)

Dominating Set - (3) Correspondence vw v w v w vz wu vu z u z u zu Vertex-cover in G Dominating-set in G'