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IBM Research ReportA faster Exponential-Time
Algorithm for Max 2-Sat, Max Cut, and Max
k-Cut,Alexander D. Scott, Gregory B. Sorkin,
IBM Research Division

• Presented By Troy McMahon

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Problems Addressed

• Max 2-Sat
• Max Cut
• Max k-Cut

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Background

• The fastest previous algorithm for Max 2-Sat has
  a runtime of O(2m/5), where m is the number of
  clauses
• The fastest previous algorithm for Max Cut has a
  runtime of O(2m/4), where m is the number of
  edges

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Method

• This paper considers a single, more general class
  of problems that includes both Max Cut and 2-Sat.

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Notation

• D-vertex Vertex of degree d
• m and n are the number of vertices and edges in a
  graph.
• r 1,,r is the set of colors a vertex can
  be colored

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Max (r, 2)-CSP

• An instance consists of a graph G(E,V), and a
  set of score functions, and a set of available
  colors r1,,r
• For every edge there exists a dyadic function
  suv(ri,rj) that maps r2 -gt R
• For every vertex there exists a monadic function
  sv(ri) that maps r1 -gt R
• There is single also a niladic score function,
  so() that maps r0 -gt R
• This function has no effect on the solution to
  the problem, and exists only for convenient
  bookkeeping

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Max (r, 2)-CSP

• A candidate solution, fV-gtr is a coloring of
  the vertices (with colorings from r)
• The score of a coloring is given by the function
• The solution to an instance of the Max (r,
  2)-CSP problem is the coloring of the graph that
  maximizes the total score

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Max Cut

• Max Cut is the partition of a graph into two
  classes that maximizes the number of edges cut by
  the petition.

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Max Cut

• Reduction to (r,2)CSP
• Consider each edge to be a function that is 1 if
  the vertices it connects are different colors,
  and 0 otherwise
• Max Cut is equivalent to finding a 2-coloring of
  the vertices that maximizes the sum of all of the
  edge functions
• Max Cut ?(r,2)CSP

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Max 2-Sat

• An instance of Max 2-sat consists of a set of
  2ary Boolean functions
• A solution is a setting of the variables that
  maximizes the number satisfied functions

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Max 2-Sat

• Reduction to (r,2)CSP
• Max 2-sat can be reduced to (r,2)CSP by creating
  a vertex for each variable and an edge fore every
  function
• The score of an edge is 1 whenever the
  corresponding function is satisfied and 0
  otherwise
• Max 2-sat ?(r,2)CSP

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Algorithm

• Reduce vertices of degree 0
• Reduce vertices of degree 1
• Reduce vertices of degree 2
• Branch

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Reducing Vertices of Degree 0

• If a vertex, y has no edges then that vertex has
  no dyadic constraints
• Color y the color that maximizes the function for
  vertex y
• Color(y) maxargCXr(sy(C) )
• An equivalent problem can be formed by increasing
  so by max(sy(r) ), and removing y form the graph
• This can be done in O(r) time

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Reducing Vertices of Degree 1

• Let y be a vertex of degree 1, and x be ys
  neighbor
• Can be reduced to an equivalent problem by
  removing vertex y, and edge xy from the graph,
  and modifying the score functions in the
  following way
• For all CXr, sx(C) sx(C) maxDXrsxy(CD)
  sy(D)
• This can be done in O(r2) time

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Reducing Vertices of Degree 1

• Once an optimal coloring, f is found for the
  reduced graph, and optimal coloring, f for the
  original graph can be found in the following
  manner
• f(y)maxargDXrsxy(f(x)D) sy(D)
• For all other verities v, f(v) f(v)
• This can be done in O(1) time

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Reducing Vertices of Degree 1
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Reducing Vertices of Degree 2

• Let y be a vertex of degree 2, and x and z be ys
  neighbors
• Can be reduced to an equivalent problem by
  removing vertex y, and the edges xy and zy from
  the graph, and modifying the score functions in
  the following way
• For all C,DXr, sxz(CD) sxz(CD)
  maxEXrsxy(CE) syz(ED) sy(E)
• This can be done in O(r3) time

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Reducing Vertices of Degree 2

• Once an optimal coloring, f is found for the
  reduced graph, and optimal coloring, f for the
  original graph can be found in the following
  manner
• f(y)maxargEXrsxy(f(x)E) syz(Ef(z))
  sy(E)
• For all other verities v, f(v) f(v)
• This can be done in O(1) time

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Reducing Vertices of Degree 2
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Branching

• Branching Algorithm
• yhighest degree vertex in G
• VV/y
• EE-neighbors(y)
• For Cevery color in r
• SoSoSy(C)
• For xevery neighbor of y
• Sx(D)Sx(D)Sxy(CD)
• fCrecurse on (V, E, S) f(y)C
• fmax(f, fC)
• Return f
• This algorithm only branches if there are no
  vertices of degree lt3 that it can reduce

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Branching

• Graphical representation of a reduction on
  vertex x.
• In this case the colors in r are red and blue
• Vertex x is removed, and 2 new graphs are created
  (one where the vertex is red and one where it is
  blue)
• The optimal coloring of the original graph is the
  max of the optimal colorings of the two new
  graphs.

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Recursion Depth

• Lemma If a graph has n vertices and m edges,
  than the graph is empty after m/5 branches

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Recursion Depth

• Case 1 If the graph has any vertices of degree
  P5, then the algorithm will branch on one of
  these vertices destroying at least 5 edges

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Recursion Depth

• Graphs containing only vertices of degree 3 and 4

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Recursion Depth

• Case 2 The highest degree vertex has degree 4.
• If the vertex it branches on has at least one
  neighbor of degree 3
• 4 edges will be destroyed directly
• The neighbor of degree 3 will be reduced to
  degree 2
• This vertex will be reduced at the next level
  destroying 1 additional edge
• If the vertex it branches on has at least 4
  neighbor of degree 4
• 4 edges are destroyed
• 4 vertices of degree 3 are created
• The fact that this algorithm terminates with no
  vertices of degree 3 is enough to limit the
  number of such reductions

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Recursion Depth

• Case 3 The graph contains only vertices of
  degree 3
• Branching will eliminate 3 edges directly
• Removing a vertex will reduce the degree of each
  of that vertexs neighbors to 2
• These neighbors will be reduced at the next level
  without branching
• At least 1 edge is destroyed for each of these
  neighbors
• In total at least 6 edges are destroyed

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Recursion Depth

• Every time the algorithm branches, the number of
  edges is reduced by at least 5
• The graph will be empty after m/5 branches
• The maximum number of times this algorithm can
  branch is therefore m/5

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Space Complexity

• This algorithm needs to store the best solution
  found at each level
• The space complexity of this algorithm is
  therefore O(nd), or O(nm/5),

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Time of Algorithm

• This algorithm has a branching factor of r
• The time for each level is bounded by the
  function
• T(l) O(2r3n) rT(l-1)
• This algorithm therefore runs in O(r3nrm/5) time

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Improved Algorithm

• Preference for selecting branching vertices
• Vertices of degree gt6
• Vertices of degree 5 that have at least 1
  neighbor of degree 3 or 4
• Vertices of degree 5 with only degree 5 neighbors
• Vertices of degree 4 that have at least 1
  neighbor of degree 3
• Vertices of degree 4 with only degree 4 neighbors
• Vertices of Degree 3
• All other aspects of this algorithm are the same
  as the first algorithm

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Reduction Forest

• A reduction on a vertex, v, can split component
  of the graph containing G into multiple
  subcomponents
• Let k(v) be the number of sub-components that a
  reduction on v will yield
• This defines a reduction forest
• Nodes in the reduction forest correspond to
  possible vertex reductions
• Each node has a child for every sub-component
  that is created by that nodes corresponding
  reduction
• A node ns children correspond to the first
  reductions in

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Splitting Tree

• Splitting trees can be formed by contracting out
  all of the nodes that correspond to reductions on
  vertices of degreeO2
• The depth of a splitting tree is equal to the
  maximum number of branching reductions that occur
  along the corresponding sequence
• The maximum number of branching reductions is
  bounded by the maximum splitting tree depth

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Depth of Splitting Tree

• Lemma For a graph with m edges, the depth of the
  splitting tree is at most 219m/100

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Depth of Splitting Tree

• Case 1 If the graph has a vertex of degree P6,
  then the algorithm branches on this vertex
  destroying 6 edges
• We only need to consider graphs with vertices of
  degree lt6

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Depth of Splitting Tree

• There are only two types of branching reductions
  that dont destroy at least 6 edges
• Reductions on vertices of degree 5 that have only
  degree 5 neighbors
• Reductions on verities of degree 4 that have only
  degree 4 neighbors
• The improved algorithm works by limiting the
  number of occurrences of these types of
  reductions

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Degree 5 Vertices With 5 Degree 5 Neighbors

• If a component requires a reduction on a degree 5
  vertex with 5 degree 5 neighbors on of 4 things
  must be true
• a. The reduction is the first degree 5 reduction
  in this branch of the splitting tree
• This only occurs once per branch, and contributes
  at most 1 to the height of the tree
• b. The previous reduction produced vertices of
  degree 3 or 4 that were destroyed
• We can pair this reduction with the previous
  reduction

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Degree 5 Vertices With 5 Degree 5 Neighbors

• c. The previous reduction was a reduction on a
  degree 5 vertex that destroyed all 5 of that
  vertexs neighbors
• Again w can pair this reduction with the previous
  
• The previous reduction was on a degree 5 vertex
  and produced vertices of degree 3 or 4 that were
  split off into a different connected component
• Produces a non-empty subcomponent, but does not
  add to the depth of the component of interest
• We can construct a similar argument for vertices
  of degree 4 with 4 neighbors of degree 4

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Depth of Splitting Tree

• Case 2 Graphs containing only vertices of degree
  lt6.

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Depth of Splitting Tree

• The dept of a vertex in the splitting tree is at
  most xm2
• The 2 comes from the first occurrences of degree
  5 and degree 4 vertices
• Solve for x
• x19/100
• The depth of the splitting tree is therefore
  19m/1002
• The recursive depth of this algorithm is
  therefore 19m/1002

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Space Complexity

• This algorithm needs to store the best solution
  found at each level
• The space complexity of this algorithm is
  therefore O(nd), or O((19m/1001)n)

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Time of Algorithm

• This algorithm has a branching factor of r
• The time for each level is bounded by the
  function
• T(l) O(2r3n) rT(l-1)
• This algorithm therefore runs in O(r3nr19m/1002)
  time

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Conclusion

• This paper presents 2 algorithms for solving
  (r,2)-CSP
• The first runs in time O(2r3nrm/5) and space
  O(mn/5)
• The second runs in time O(2r3nr19m/100) and space
  O((19m/1001)n)
• 2-Sat and max cut can be reduced to (r,2)-CSP
• 2-Sat and max cut can therefore be solved in time
  O(2r3nr19m/100) and space O((19m/1001)n)

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References

• IBM Research Report A faster Exponential-Time
  Algorithm for Max 2-Sat, Max Cut, and Max
  k-Cut,Alexander D. Scott, Gregory B. Sorkin,
  IBM Research Division, IBM Research Division,
  Thomas J. Watson Research Center, 2004