A New Algorithm for Optimal 2-Constraint Satisfaction and Its Implications

1
A New Algorithm for Optimal 2-Constraint
Satisfaction and Its Implications

• Ryan Williams
• Computer Science Department, Carnegie Mellon
  University
• Presented by Mu-Fen Hsieh

2
MAX-CUT

• Given a graph G (V,E), positive integer N, is
  there a partition of V into disjoint sets S and
  TV-S such that the number of the edges from E
  that have one endpoint in S and one endpoint in T
  is at least N?
• Proved to be NP-hard by reducing from MAX-2-SAT
  Karp, 1972
• Can be solved in polynomial time if G is planar
  Hadlock, 1975, Orlova et al., 1972
• Can be transformed to an instance of MAX-2-SAT
  in polynomial time

3
MAX 2-Satisfiability

• Given a set U of n variables, a collection C of m
  clauses over U such that each clause
  has c 2, positive integer N C, is there a
  truth assignment for U that simultaneously
  satisfies at least N of the clauses in C? Garey
  et al., 1976
• Solvable in polynomial time if N C Even et
  al., 1976
• Approximated in O((2-e)n), O((2-e)m) Dantsin et
  al.,2001
• Previous best worst-case bound O(2m/5) Hirsch
  et al., 2003
• This paper counts number of truth assignments
  that simultaneously satisfies exactly N of the
  clauses in C in O(m32?n/3) time

4
Encode Max-Cut as Max-2-SAT
T
T
T
F
F
F

• The size of the cut associated to the assignment
  E - number of violated clauses

5
Split and list split the set of n variables into
k partitions of equal size, and list 2n/k
variable assignments for each partition.
MAX-CUT
Polynomial-timetransformation
MAX-2-SAT
Split and list
k-CLIQUE
Matrix Multiplication

• Set k3, we get O(m32?n/3)

6
Fast k-Clique Detecting and Counting

• Locate 3-clique given G (V,E) with n V,
  let A(G) be its adjacency matrix.
• tr(A(G)3) 6 (number of triangles in G)

2 3 3 3 2 3 3 3 2
0 1 1 1 0 1 1 1 0
2 1 1 1 2 1 1 1 2
A(G)3
A(G)
A(G)2
7
Fast k-Clique Detecting and Counting

• Locate k-clique build a graph Gk/3 (Vk/3,Ek/3)
  where Vk/3 (k/3)-cliques in G and Ek/3
  c1,c2c1,c2 Vk/3,c1 c2 is a
  (2k/3)-clique in G
• Each triangle in Gk/3 corresponds to a unique
  k-clique in G?
• Time complexity
• For example, to locate 6-clique

8
Split and List

• Step 1 Delegating responsibility
• Define responsibility map r C ?(k )

x5 x6 0 00 11 01 1
x1 x2 0 00 11 01 1
x3 x4 0 00 11 01 1
9
Split and List (Cont.)

• Step 2 Weighting accordingly
• Consider the partitions as a weighted k-partite
  complete graph. Assign weights to both vertices
  and edges.

x5 x6 0 00 11 01 1
of vertices k2k
x1 x2 0 00 11 01 1
x3 x4 0 00 11 01 1
of cliques 2n
10
Split and List (Cont.)

• Step 3 Enumerate unweighted graphs
  reducing from the weighted graph in step 2. Each
  unweighted graph, constructed in amortized time
  O(1), consists of cliques of weight N.
• Decompose N into different combinations of
  weights ( nonnegative integers)
• Consider variables y1,2, y1,3,,yk-1,k
  where y1,2 0,1,2,,N. Solve the following
  equation y1,2y1,3y1,k y2,1 yk-1,k N
• vi some node in partition iw() weight of
  some node or some edge

w(vj) w(vi,vj)
w(vi,vj)
w(v1) w(v2) w(v1,v2)
11
Time Complexity

• Set k3, we get O(m32?n/3)