RARE APPROXIMATION RATIOS

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RARE APPROXIMATION RATIOS

• Guy Kortsarz
• Rutgers University
• Camden

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Approximation Ratios

• NP-Hard problems
• Coping with the difficulty approximation
• Minimization or maximization.
• Approximation ratio (for minimization)

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A Generic Problem Set-Cover
SETS
ELEMENTS
A
B
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Frequent Approximation Ratios

• Constants. Example
• Max-3-SAT Tight 8/7 ratio
• Logarithmic for minimization problems
• Set-cover
• PTAS (1 ?) for all ? gt 0
• Example Euclidean TSP

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Frequent Ratios continued

• Polynomial Ratios
• sqrt (n), n 1 - ?
• Example
• Clique n 1 - ? lower bound
• Upper bound
• ?(n/log3n) (Halldorsson, Feige)

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Example Constrained Satisfaction Problems

• Given a collection of Boolean formulas, satisfy
  all constrains. Maximize true variables.
• Possible ratios
• 1) Solvable in polynomial time
• 2) n?
• 3) Constant
• 4) Unbounded
• Due to Khanna, Sudan, Williamson

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"Natural" Problems

• It is possible to artificially design problems to
  get any desired ratio
• See for example the NP-complete column of D.
  Johnson The many limits of approximation
• If in set-cover we take the objective function to
  be sqrt(S) then the ratio is sqrt(ln n)
• I discuss rare ratios that appeared as a natural
  consequence of the problem/techniques
• This sheds light on special problems/techniques

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Rare Ratios Example I

• Until 2000 there was no
• MAXIMIZATION PROBLEM
• with log n threshold
• Example Domatic Number
• Input G (V, E)
• Dominating set U U ? N(U) V

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The Domatic Number Problem

• Given G (V, E)
• Find VV1 ? V2 ? . ? Vk
• so that Vi dominating set (in G).
• Goal Maximize k
• Example A maximal independent set
• and its complement is dominating. k 2

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A Simple Algorithm

• Create bins
• Throw every vertex into a bin at random
• The expected number of neighbors of every v in
  bin i is 3 ln n
• The probability that bin i has no neighbor of v

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Domatic Number Continued

• The number of bad events is n2 or less.
• Each one has probability 1/n3 to hold
• By the union bound size partition
  exists
• Remark ? 1 is a trivial upper bound
• This implies O(ln n) ratio

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Large Minimum Degree opt 2
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More Lower and Upper Bounds

• Feige, Halldorsson, Kortsarz, Srinivasan
• The approximation is improved to O (log ?) (LLL)
• There is always ?/ln ? solution (complex proof)
• Can not be approximated within (1 - ?) ? ln n
  for any constant ? gt 0

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Remarks on the Lower Bound

• Lower Bound Method 1R2P
• Generalizes (or improves) the paper of Feige
  from 1996, (1 - ?) ? ln n , lower bound for
  set-cover
• Recycling solutions One Set Cover implies many
  set-cover exist
• Uses Zero-Knowledge techniques

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Perhaps log n for Maximization Unique Set Cover
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Special Case Every Element in B has Degree d

• Choose every a ? A with probability 1/d
• Hence, expected number of uniquely covered
  elements of B, a constant fraction
• Hence, there always is a subset A? A that
  uniquely covers a fraction

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General Case

• Cluster the degrees into powers of 2
• There exists a cluster with ? (B / log A )
  vertices
• Corollary There always exists A? A that
  uniquely covers a 1 / log n fraction of B

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Lower Bounds

• Demaine, Feige, Hajiaghayi, Salvatipour
• Hard to find complete bipartite graphs, Implies
  log n best possible
• NP has no algorithm implies (log n)? hard
  to approximate
• Hard to refute random 3-sat instances, implies (
  log n ) 1/3 hard

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Polylogarithmic for Minimization

• Group Steiner problem on trees

g1
g2
g3
g5
g4
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Integrality Gap

• Halperin, Kortsarz, Krauthgamer, Srinivasan,Wang

g1,g2 g3,g4
g1,g3,g2
g2,g4
g1,g3
g4
g1,g2
g2
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Analysis

• The costs need to decrease by constant factor
  HST
• The fractional value is the same at every level
• Thus, if the height is H then the fractional is
  O(H)
• The integral H2 ? log k (k is groups)
• (log k)2 gap
• The same paper HKKSW gives O ( (log k)2 ) upper
  bound

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More Upper Bounds

• Garg, Ravi, Konjevod
• O( (log n)2) using Linear Programming
• Randomized rounding plus Jansen inequalities
• Halperin, Krauthgamer
• Lower bound (log k)2-?
• (log n / log log n)2
• Hiding a trapdor in the integrality gap
  construction

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Directed Steiner and Below

• Directed Steiner O( (log n)3) quasi-polynomial
  time and n ? for every ? polynomial time
  Charikar etal
• Special case Group Steiner on general graphs
• O( (log n)3) polynomial (reduction to trees
  using Bartal Trees)
• In quasi-polynomial tine O( (log n)2) for general
  graphs Chekuri, Pal
• Group Steiner trees log2 n / log log n,
  quasi-polynomial time Chekuri, Even, Kortsarz

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The Asymmetric k-Center Problem

• Given Directed graph G(V, E) and length l(e) on
  edges and a number k
• Required choose a subset U, U k of the
  vertices
• Optimization criteria Minimize

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A log n Approximation

• Due to Vishwanathan
• Idea

k
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Lower Bound log n

• Due to Chuzhoy, Guha, Halperin, Khanna,
  Kortsarz, Krauthgamer, J. Naor
• Based on hardness for d-set-cover

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Simple Algorithm for d-Set-Cover

• Choose all the neighbors of some b? B and add
  them to the solution
• The algorithm adds d elements to the solution
• The optimum is reduced by 1
• An inductive proof gives d ratio

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Hardness Based on d-Set Cover Hardness d 1 - ?

• Dinur, Guruswami, Khot, Regev
• Gap Reduction for d Set - Cover

3/d ? A enough to cover
d-set-cover
Yes instance
I
Any (1-2/d)A subset covers at most (1-f(d))
fraction of B. f(d)(1/2) poly d
d-set-cover
No instance
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A Hardness Result for Directed k-Center

• Compose the d-set-cover construction
• di1 exp (di)

d2
d1
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Analysis

• Choose k (V1/d1) - 1
• For a YES instance get dist 1
• For a NO instance
• We may assume all centers are at V1
• But the number of uncovered vertices remains
  larger than 0
• Approaches 0 at log (previous) speed
• Gives log n gap

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Complete partitions of graphs
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Approximation for d - Regular Graphs

• sqrt(m/2) is an upper bound
• Partition to sqrt(m/2) classes at random
• There is an expected O(1) edges per sets
• Merge randomly to groups of 3 ?
  sets
• Prove that with high probability its complete

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Complete Partitions Continued

• For non-regular graphs complex algorithm and
  proof.
• However possible
• Lower bound
• Uses the domatic number lower bound
• Complex analysis
• Gives lower bound for
  achromatic number

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More Between log n and O(1)

• Minimum congestion routing
• Given a collection of pairs (undirected graph)
  choose a path for each pair. Minimize the
  congestion
• Upper bound O(log n / loglog n) . Raghavan ,
  Thompson
• Lower bound ?(log log n) . Chuzhoy, Naor
• Maximum cycle packing.
• upper bound M. Krivelevich, Z.
  Nutov, M.
• 
  Salavatipour, R. Yuster.
• lower bound. Salavatipour (private
  communication)

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More Between log n and O(1)

• Directed congestion minimization
• O(log n / loglog n) upper bound Raghavan and
  Thompson
• (log n) 1-? lower bound.
• Andrews and Zhang
• Min 2CNF deletion.
• upper bound Agrawal etal.
• Under the UNIQUE GAME CONJECTURE no constant
  ratio Khot

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More Between log n and O(1)

• Sparsest cut
• upper bound Arora, Rao and
  Vazirani
• Under UGC no c ? loglog n ratio, constant c
• Chawla etal
• Point set width.
• upper bound Varadarajan etal
• (log n)? lower bound Varadarajan etal

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Additive Approximation Ratios

• The cost of the solution returned is
• opt?
• ? is called the additive approximation ratio
• Much less common (or studied(?)) than
• multiplicative ratios

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New Result

• Let G (V,E,c) be a graph that admits a spanning
  tree of cost at most c and maximum degree at
  most d
• Then, there exists a polynomial time algorithm
  that finds a spanning tree of cost at most c and
  maximum degree d2. Additive ratio 2 Goemans,
  FOCS 2006

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The Ultimate Approximation

• Some problems admit 1 approximation
• Known examples
• Coloring a planar graph
• Chromatic index coloring edges Hoyler
• Find spanning tree with minimum maximum degree
  Furer Ragavachari
• Some less known 1 approximation

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Achromatic Number
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Achromatic Number of Trees

• The problem is hard on trees
• Thus opt is bounded by roughly sqrt n
• This bound is achievable within 1 (in polynomial
  time)
• Similarly Minimum Harmonious coloring of trees
  1 approximation

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Poly-log Additive (tight) Radio Broadcast
R1
R2
R3
R4
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Upper and Lower Bounds

• Since one can cover 1/log n uniquely, in
• O( (log n)2) rounds the other side of a
  Bipartite graph can be informed
• Thus, in a BFS fashion Radius ? (log n)2
• Best known Kowalski, Pelc
• Radius O(log n)2
• Lower bound Elkin, Kortsarz For some constant
  c, opt c ? (log n)2 not possible unless
• NP ? DTIME (n poly-log n)

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A graph with radius 1, opt ? (log n)2

• A construction by Alon, Bar-Noy, Lineal, Peleg

P(1/2)0.4?log n
P(1/2) 0.6?log n
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Analysis

• If we choose any subset of size 2j then the set
  of probability (½)j will be informed in log n
  rounds
• Since there are 0.2? ln n sets, it will take
  O( (log n)2)
• The difficulty A size 2j does not affect the
  sets of p (½)k, k gt j
• However, if k lt j, size 2j causes collisions for
  k, hence is of little help

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Conclusion

• No real conclusion
• The NPC problem seems to admit little order if at
  all regarding approximation
• The problems are unstable
• There does not seem to be a deep reason these
  ratios are rare (because of techniques(?))
• Very good advances.
• Still much we dont understand in approximations