Prize Collecting Cuts

1
Prize Collecting Cuts

• Daniel Golovin
• Carnegie Mellon University
• Lamps of ALADDIN 2005
• Joint work with Mohit Singh Viswanath Nagarajan

2
The Problem

• Input Undirected graph G (V,E), root vertex r,
  and integer K, 0 lt K lt V
• Goal find a set of vertices S, not containing
  the root, minimizing cap(?S), subject to the
  constraint S K, (?S edges out of S)

root
S
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Motivation

• Protect at least K nodes (servers, cities, etc)
  from an infected node in a network.

root
S
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Motivation, continued

• Separate at least K enemy units from your base.

S
root
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Related Work

• A polylogarithmic approximation of the minimum
  bisection, Feige Krauthgamer, SIAM Journal on
  Computing 2002
• On cutting a few vertices from a graph, Feige,
  Krauthgamer, Nissim, Discrete Applied
  Mathematics 2003
• Global min-cuts in RNC, and other ramifications
  of a simple min-cut algorithm, Karger, SODA 1993

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Related Work

• Hayrapetyan, Kempe, Pál and Svitkina claim a
  (2,2) bicriteria approx for the problem of
  minimizing the number of vertices on the root
  side of the cut, subject to cap(?S) B, though
  we have not seen the manuscript.
• We can get a (2,2) approx via Lagrangian
  relaxation and Markovs inequality

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What was known

• Feige Krauthgamer consider the problem of
  removing exactly K vertices from a graph, obtain
  an O(log3/2(n)) approx for all values of K.
• F.K.N. consider this problem for small K, obtain
  a (1 eK/log(n)) approx, for any fixed e gt 0.
  They use ideas from Kargers min-cut algorithm.

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Results

• For K O(n), we obtain an (const, const)
  bi-criteria approx
• For small K, we match the F.K.N. result (i.e. an
  (1 eK/log(n)) approx)

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PTAS for K O(log(n))

• First run the FKN PTAS for all K in K,8K. At
  all times, keep the best solution cut around.
• While G still has edges, contract an edge
  uniformly at random, compute the minimum cost
  root-cluster cut for the new cluster, and
  continue.
• Output the best solution cut seen.

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Contraction

• Contract (u,v) Keep parallel edges

v
u
u,v
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PTAS for K O(log(n))

• First run the FKN PTAS for all K in K,8K. At
  all times, keep the best solution cut around.
• While G still has edges, contract an edge
  uniformly at random, compute the minimum cost
  root-cluster cut for the new cluster, and
  continue.
• Output the best solution cut seen.

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Analysis

• Suppose OPT has cost B, and cuts away S.
• If FKN returns a solution of cost (1e)B, we are
  done. Otherwise, S gt 8K, and for every subset
  R of size between K and 8K, cap(?R) gt B.

R1
R3
root
S
R2
R4
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Analysis

• Suppose OPT has cost B, and cuts away S.
• If FKN returns a solution of cost (1e)B, we are
  done. Otherwise, S gt 8K, and for every subset
  R of size between K and 8K, cap(?R) gt B.

Lots of inter-cluster edges
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Analysis, cont.

• If we generate a cluster of size at least K in S,
  its min-cut from the root has cost at most B, and
  we will return it (or some better solution).
• Safe to assume each cluster in S has at most K
  vertices

?S is a min root to C cut
root
C
S
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Analysis, cont.

• Each cluster in S has at most K vertices
• Partition the clusters of S into groups such that
  each group has between K and 2K vertices.

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Analysis, cont.

• There are at least S/2K groups in S, each has
  at least B edges leaving it.
• Each edge is counted at most twice, so there are
  at least (SB)/(4K) edges incident on vertices
  of S.
• At most B of these edges leave S. If we contract
  such an edge, we abort the run.

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Analysis PrAbort
Probability of Aborting exactly B red (bad)
edges, at least SB/4K red black edges
Pre red, given e is not blue
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Analysis PrAbort

• At each step, Prabort 4K/S, so we succeed
  with probability at least 1-4K/S

Pre black, given e is not blue
R1
R3
root
R2
S
R4
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Analysis PrSuccess

• We may run only S-1 contractions of edges in
  (?S)U(SxS) (i.e. red black edges) before either
  aborting or contracting S into a single node
• The probability of generating a cluster of size
  at least K in S before aborting is

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Analysis PrSuccess
If x 2, (1-1/x)x 1/4 (via Bernoullis
ineq.) Since S gt 8K (1-4K/S)S/4K 1/4
Raise both sides to the 4K power (1-4K/S)S
(¼)4k 4-O(log(n)) n-O(1)
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Analysis, cont.

• So either the FKN preprocessing gives us an
  (1e)B solution, or with high probability in
  polynomial many independent runs we obtain the
  optimal solution.

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Bi-criteria approx for K O(n)

• For large K, prize collecting cut starts to look
  like sparsest cut with demands D(root,v) 1 for
  all vertices v, and the constraint that at least
  K vertices are cut away.
• Note we can solve sparsest cut exactly on inputs
  with a single source of demands.

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Bi-criteria approx for K O(n)

• Idea Iteratively run sparsest cut with these
  demands, chopping off more and more of the graph,
  until at least K/2 vertices have been removed.

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Analysis

• At each step, we know the sparsest cut has
  sparsity at most 2B/K. Thus the cost per vertex
  separated is at most 2B/K.

OPT cut
The shaded region has at least K/2 vertices, and
can be separated from the root at cost at most B.
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Analysis

• Cost per vertex separated is at most 2B/K.
• If the output separates L vertices from the root,
  its cost is at most L(2B/K).
• Since L n and K O(n), L(2B/K) O(B).

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Ongoing Work

• The middle ground log(n) ltlt K ltlt n
• Strictly enforcing the budget constraint and
  approximating the prize collected

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Thank You

• Questions?