Linear Time Approximation Schemes for the GaleBerlekamp Game and Related Minimization Problems

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Linear Time Approximation Schemes for
theGale-Berlekamp Game and Related Minimization
Problems

• Marek Karpinski (Bonn)
• Warren Schudy (Brown)
• STOC 2009

Please see http//www.cs.brown.edu/ws/papers/gb.p
df for the most current version of the paper.
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Gale-Berlekamp Game (1960s)
n/2

• Minimize number of lit light bulbs
• NP hard Roth Viswanathan 08
• PTAS runtime nO(1/e²) Bazgan, Fernandez de la
  Vega, Karpinski 03
• We give PTAS linear runtime O(n2)2O(1/e²)

Animating
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Dense MIN-UNCUT
Uncut (monochromatic) edge

• Approximate 2-coloring
• General case
• O(v log n) approx is best known
• no PTAS unless PNP
• Everywhere- dense case, i.e. every vertex has
  degree O(n)
• Previous best PTAS nO(1/e²) Arora, Karger,
  Karpinski 95
• We give PTAS with linear runtime O(n2)2O(1/e²)
• If three colors no PTAS unless PNP
• Average degree O(n) is insufficient for PTAS
  unless PNP

Added complete bipartite graph
Animating
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Generalization Fragile dense MIN-k-CSP

• n variables taking values from constant-sized
  domain
• GB-Game switches
• MIN UNCUT vertices
• Soft constraints, which each depend on k
  variables
• GB Game lightbulbs
• MIN UNCUT edges
• These constraints are fragile, i.e. changing
  value of a variable makes all satisfied
  constraints it participates in unsatisfied. (For
  all assignments.)
• Dense, i.e. each variable appears in O(nk-1)
  constraints

GB Game
Dense MIN UNCUT
First conceptual contribution unifying these
PTASs (and others) using new fragile framework
We give first PTAS for all fragile dense
MIN-k-CSPs, which has linear runtime
O(nk)2O(1/e²)
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Another fragile problem Multiway cut
Vertices are variables Edges are soft
constraints These constraints are fragile, i.e.
changing value of a variable makes all satisfied
constraints it participates in unsatisfied

• General case has O(1) approx. but no PTAS
• Dense case
• Previous best PTAS nO(1/e²) Arora, Karger,
  Karpinski 95
• We give PTAS with runtime O(n2)2O(1/e²)
  (linear-time)

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Summary of results
Runtimes for 1e approximation on everywhere-
dense instances
Essentially optimal

• Reference key
• AKK 95Arora, Karger, Karpinski 95
• BFK 03Bazgan, Fernandez de la Vega,
  Karpinski 03
• GG 06Giotis Guruswami 06

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Additive error algorithms

• Whenever OPT f(e)nk we have f(e)enk
  O(eOPT), so existing algorithms achieving
  additive error f(e)enk suffice for a PTAS.
  Arora, Karger, Karpinski 95, Fernandez de la
  Vega 96, Goldreich, Goldwasser Ron 98, Frieze
  Kannan 99, Alon, Fernandez de la Vega, Kannan,
  Karpinski 02, Mathieu Schudy 08
• Typical runtime O(nk)2O(1/e²)
• Rest of talk focuses on
• OPT small and
• MIN-UNCUT

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Previous algorithm (1/3)
analysis version
Assumes OPT e ?0 n2 where ?0 is a constant

• Let S be random sample of V of size O(1/e²)log n
  
• For each coloring x0 of S
• partial coloring x2 ? if margin of v w.r.t. x0
  is large then color v greedily w.r.t. x0, else
  label v ambiguous
• Extend x2 to a complete coloring x3 greedily
• Return the best coloring x3 found

Let x0 x restricted to S
Runtime 2S 2O(1/e²)log n nO(1/e²)
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Previous Algorithm (2/3)
Blue 1 to 0 margin is too small
Blue 2 to 0
A
B
A
B
A
B
Blue 1 to 0 margin is too small
Blue 1 to 0 margin is too small
D
E
D
E
D
E
C
C
C
OPT
F
F
F
Blue 2 to 1 margin is too small
Sample x0 of OPT
partial coloring x2 ?if margin of v w.r.t. x0 is
largethen color v greedily w.r.t. x0 else label
v ambiguous
Blue 2 to 0

• Define the margin of vertex v w.r.t. coloring x
  to be(number of green neighbors of v in x) -
  (number of red neighbors of v in x).
• Key facts (recall dense assumption)
• Partial coloring x2 agrees with the optimal
  coloring x
• There are few ambiguous vertices

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Previous algorithm (3/3)
A
B
A
B
D
E
D
E
C
C
F
F
x2
x3 extends x2 greedily
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Previous algorithm
Our
Intermediate
Assume OPT e ?0 n2
?1 n2
?2

• Let S be random sample of V of size O(1/e²)log n
• For each coloring x0 of S
• partial coloring x2 ? if margin of v w.r.t. x1
  is large then color v greedily w.r.t. x1 else
  label v ambiguous
• Extend x2 to a complete coloring x3 greedily
• Return the best coloring x3 found

Second conceptual contribution two greedy
phases before assigning ambiguity allows constant
sample size

• x1 ? greedy w.r.t. x0

Third conceptual contribution use additive error
algorithm to color ambiguous vertices.
using an algorithm with additive error at most
Err?3 e n ( ambiguous)
O(n2)2O(1/e4)
O(n2)2O(1/e²)
Runtime nO(1/e²)
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More Algorithm (1/2)
C is blue so I like being red
E is red so I like being blue
My reasoning exactly
Me too
A
A
A
C
C
B
D
B
D
C
D
E
B
E
E
OPT
F
F
F
C is Blue so I like being red
Sample x0 of OPT
x1 is greedy w.r.t. (with respect to) x0
E is red so Ill go blue
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More Algorithm (2/2)
Blue 2 to 1 margin is too small
Ambiguous run additive error algorithm to color
Red 2 to 1 margin is too small
Blue 4 to 0
A
A
Red 2 to 1 margin is too small
C
C
B
D
B
D
E
Blue 3 to 0
E
Red 2 to 0
F
F
x1
x2 is greedy w.r.t. x1
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Plan of analysis

• Main Lemma ( Lemma 16)
• Coloring x2 agrees with the optimal coloring x
• The additive error Err?3 e n ( ambiguous) is
  at most e OPT

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Proof (1/3) Bounding OPT

• Assume all degrees are at least d n
• Vertex v is balanced if its margin w.r.t. x is
  at most d n / 3.
• Lemma 12 (balanced vert.) 6 OPT / (d n)
• Proof
• If v is balanced then v is incident in x to at
  least d n / 3 uncut edges
• OPT ½?v (uncut edges incident to v)
  ½?v balanced (uncut edges incident to v)
  ½ (balanced vert.) (dn / 3)

Optimum assignment x
F
D
C
B
A
E
G
Balanced 13
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Proof (2/3) relating x1 to OPT coloring

• Lemma 14 with probability at least 90 at most d
  n / 24 vertices are colored different colors in
  x1 and x
• Proof
• Corollary with probability at least 90 all
  vertices have margin w.r.t. x within d n / 12 of
  margin w.r.t. x1

Case 1 balanced vertices By Lemma 1 (balanced)
6 OPT / (d n) 6 (k1 n2) / (d n) d n / 48.
Case 2 unbalanced vertices Chernoff and Markov
bounds imply that the number unbalanced vertices
is at most d n / 48.
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Proof (3/3) Proof of main lemma

• Proof that x2 agrees with the optimal coloring x
• Assume v is colored by x2
• Then v has a big margin w.r.to x1
• Then by Corollary v is colored by x in the same
  way as by x2
• Proof that the additive errorErr?3 e n (
  ambiguous) is at most e OPT
• Assume v is not colored by x2 (ambiguous)
• Then v has a small margin w.r.to x1
• Then by Corollary v has small margin w.r.to x
  (balanced)
• So ( ambiguous) ( balanced)
• Bound ( ambiguous) by ( balanced) in Err, and
  use Lemma 12 to get Err e OPT.

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Correlation Clustering with d clusters

• Previous best PTAS runtime nO(1/e²) Giotis
  Guruswami 06
• We give PTAS with runtime n22O(1/e²) (linear
  time)
• Cor. Clust. constraints not fragile for dgt2, but
  it satisfies a generalization we call rigidity

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Correlation Clustering and Rigidity

• Definition of rigid CSP in any assignment, a
  vertex in a large cluster is either incident to
  many incorrect edges or would be incident to many
  if moved to any other cluster.
• Fragility implies rigidity
• Key additional algorithmic technique (also used
  in GG 06) after identifying some clear-cut
  variables fix them and recurse on the remaining
  variables

v
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Directions

• More applications of the fragility and rigidity
  methods for other minimization problems. Might
  require generalizing the notion of rigidity to
  k-CSP problems.
• Improving runtimes for Correlation Clustering,
  replacing "" with "" in O(n2)2O(1/e²)
• Designing linear time (1 e)-approximation
  algorithms for the k-Clustering (MIN-SUM) problem.

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Bonus slides
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MIN-3-UNCUT
Uncut (monochromatic) edge

• MIN-3-UNCUT constraints are not fragile
• Dense MIN-3-UNCUT is at least as hard as general
  MIN-2-UNCUT so no PTAS unless PNP

General MIN-2-UNCUT instance
Dense MIN-3-UNCUT instance
10n2 vert.
Reduction
10n2 vert.
n vertices
10n2 vert.
n vertices
Complete tripartite graph