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An NP-completeness Example. The graph CLIQUE problem. Undirected Graph G = (V, E) a clique is a complete subgraph of G Subset V’ of vertices that are all connected to each other CLIQUE = {(G,k) : G is a graph with a clique of size k}

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the graph clique problem
The graph CLIQUE problem
  • Undirected Graph G = (V, E)
    • a cliqueis a complete subgraph of G
      • Subset V’ of vertices that are all connected to each other
  • CLIQUE = {(G,k) : G is a graph with a clique of size k}
  • For a general graph, and large-enough |V|, k, straightforward (naïve) algorithm for CLIQUE takes factorial time
    • OMEGA(k2 * C(|V|, k) )
  • 1. CLIQUE is in NP
    • Exercise: Given encoding of G, and a certificate V’, come up with a poly-time verification algorithm to check that V’ is a clique of size k for G
  • 2. CLIQUE is NP-hard
    • 3-CNF-SAT P CLIQUE
  • 1 + 2 implies CLIQUE is NP-complete
clique is np hard 3 cnf sat p clique
CLIQUE is NP-hard 3-CNF-SAT P CLIQUE
  • From instance I_3_CNF_SAT of 3-CNF-SAT, generate an instance I_CLIQUE of CLIQUE in polynomial time
    • I_CLIQUE returns true exactly whenever I_3_CNF_SAT returns true
      • I_CLIQUE includes a graph G = (V, E) and a vertex subset V’
      • Build the graph G to do the mapping
  • I_CLIQUE
    •  = C1 AND C2 AND … Ck
      • C1 = lit_11 OR lit_12 OR lit_13.
        • Lit_11 = x or NOT x
      • Similarly for C2, C3, … Ck
  • Build Graph G = (V, E) as follows
    • Take the r-th clause in I_3_CNF_SAT
    • For lit_1r OR lit_2r OR lit_3r, place vertices vert_1r, vert_2r, vert_3r in V
    • put an edge from vertices vert_ir to vert_js (i = 1, 2, 3; j = 1, 2, 3) if
      • r != s (I.e., the vertices correspond to different triples in the formula) AND
      • lit_ir != NOT lit_js
        • don’t put an edge from x to (NOT x) even if they are in different triples
    • CAN DO THIS IN POLY TIME!
slide4
 = (x1 OR (NOT x2) OR (NOT x3)) AND ((NOT x1) OR x2 OR x3) AND (x1 OR x2 OR x3) true when x2 =0 and x3 = 1
  • GRAPH (Fig 34.14) HERE
  • Suppose  is TRUE for some values of the variables.
    • Then each Cr is true => at least one of lit_r1, lit_r2, lit_r3 is true.
    • Select one such “true” vertex from each Cr (e.g., if lit_r2 and lit_r3 are both true, select vert_r2 at random from vert_r2 and vert_r3)
      • Call this a subset V’
    • V’ has k elements (one from each Cr) and is a clique
      • Take any two vertices in V’, and there is an edge between them by construction.
        • By the selection method, they are in different triples
        • They are both 1’s by selection method, so they are not negations of each other.
    • We mapped from a satisfying assignment for  to a graph with a clique of size k (vert_12, vert_23, vert_33)
  • Suppose G has a clique V’ of size k (vert_12, vert_23, vert_33)
    • Each vertex in V’ belongs to a different triple and cover all triples between them
      • there cannot be an edge in G if two vertices belong to the same triple.
    • If vert_ri is in V’, assign lit_ri = 1
      • there cannot be an assignment like { xi = 1; (NOT xi) =1 }
        • there cannot be an edge in G if two vertices are negations of each other
    • Each clause has one TRUE literal by the mapping
    • We mapped from a graph with clique of size k to a satisfying assignment for 
slide5
CLIQUE is NP hard for the “special” kind of graphs
    • CLIQUE is “harder” for general kind of graph
    • CLIQUE is NP-hard for general kind of graph.
  • CLIQUE IS NP-COMPLETE.
vertex cover
VERTEX COVER
  • Vertex cover of an undirected graph G = (V, E) is a subset V’ of V such that
    • if (u, v) is in E, then either u or v or both is in V’
    • The vertices in V’ together ‘cover’ all the edges of E
    • |V’| is the size of a vertex cover.
  • VERTEX_COVER = { (G, k) : graph G has a vertex cover of size k}.
  • VERTEX_COVER is NP-complete
  • VERTEX_COVER is in NP
    • exercise given (G, k) and certificate c = V’, find a poly algorithm to verify c
  • VERTEX_COVER is NP-hard
    • CLIQUE p VERTEX_COVER
slide7
Complement of a graph
    • COMP_G = (V, COMP_E)
      • COMP_E = { (u, v) : u, v are in V, u != v, and (u, v) not in E }
      • edges that are not in E
  • FIG 34.15 HERE
  • Reduction algorithm from CLIQUE to VERTEX_COVER
    • from I_CLIQUE = (G, k) Compute COMP_G (poly)
    • Claim: we now have I_VERTEX_COVER = (COMP_G, |V| - k)
      • G has a clique of size k if and only if COMP_G has a vertex cover of size |V| - k
slide8
If part:
  • Suppose G has a clique V’ of size k.
  • Claim: V-V’ is a vertex cover in COMP_G
  • proof: suppose (u, v) is an edge in COMP_E.
    • Then (u, v) is not in E
    • So, either u or v is not in V’ since if both u and v are in V’, there must be an edge between them.
    • That is, either u or v is in V-V’
    • if (u, v) is an edge in COMP_E, then V-V’ covers that edge, hence V-V’ is a vertex cover in COMP_G. and V-V’ has size |V| -k
  • Only-if part:
  • Suppose COMP_G has a vertex cover V’ of size |V| -k
    • claim: V-V’ is a clique in G
    • proof: for all u, v in V, if (u, v) is in COMP_E, then u is in V’ or v is in V’ or both
      • by contrapositive, for all u, v in V, if u is not in V’ and v is not in V’, then (u, v) is in E.
      • I.e., V-V’ is a clique.
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