# rare

Document Sample

```					RARE APPROXIMATION
RATIOS

Guy Kortsarz
Rutgers University
Camden
Approximation Ratios
   NP-Hard problems
   Coping with the difficulty: approximation
   Minimization or maximization.
   Approximation ratio (for minimization):

 ALG ( I ) 
           
max                  
I Inputs  opt ( I ) 
           
A Generic Problem: Set-Cover

SETS
ELEMENTS

A
B
Frequent Approximation Ratios

 Constants. Example:
 Max-3-SAT: Tight 8/7 ratio

 Logarithmic for minimization problems:

 Set-cover

 PTAS (1 + ) for all  > 0

 Example: Euclidean TSP
Frequent Ratios continued
 Polynomial Ratios:
 sqrt (n), n {1 - }

 Example:

 Clique: n {1 - } lower bound

 Upper bound:

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
"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
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
The Domatic Number Problem

          (V, E)
Given: G
 Find: V=V1  V2  ….  Vk

so that Vi dominating set (in G).
 Goal: Maximize k

 Example: A maximal independent set

and its complement is dominating. k ≥ 2
A Simple Algorithm
   Create           bins

3 ln n
   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:

 3 ln n    1
1        3
      n
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                3 ln n
   Remark:  + 1 is a trivial upper bound
   This implies O(ln n) ratio
Large Minimum Degree
opt = 2
More Lower and Upper Bounds
 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  > 0
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
Perhaps log n for Maximization:
Unique Set Cover
Special Case: Every Element in B
has Degree d
    Choose every a  A with probability 1/d
d 1
 1             1 1
Pr(Unique Neighbour for b)  d 1             
 d             d e

    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
General Case:
   Cluster the degrees into powers of 2:
i 1
D i  {b  B | 2  deg(b)  2 }
i

   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
Lower Bounds

    Demaine, Feige, Hajiaghayi, Salvatipour:
 Hard to find complete bipartite graphs,
Implies log n best possible
n
 NP has no 2      algorithm implies (log n)
hard to approximate
 Hard to refute random 3-sat instances,
implies ( log n ) 1/3 hard
Polylogarithmic for Minimization
   Group Steiner problem on trees:

g1     g2       g3              g5
g4
Integrality Gap
Halperin, Kortsarz, Krauthgamer,
Srinivasan,Wang
g1,g2
g3,g4

g1,g3,g2                    g2,g4

g1,g3            g1,g2           g2           g4
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
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
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]
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

max{dist (u,U )}
uU
A log* n Approximation

   Due to Vishwanathan
   Idea:

k
Lower Bound: log* n
   Due to: Chuzhoy, Guha, Halperin, Khanna,
Kortsarz, Krauthgamer, J. Naor
   Based on hardness for d-set-cover
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
Hardness: Based on d-Set Cover
Hardness: d – 1 - 
Dinur, Guruswami, Khot, Regev:
Gap Reduction for d – Set - Cover

Yes instance    d-set-cover    3/d  |A| enough to cover

I
Any (1-2/d)|A| subset covers at
d-set-cover   most (1-f(d)) fraction of B.
No instance
f(d)=(1/2) {poly d}
A Hardness Result for Directed
k-Center
   Compose the d-set-cover construction:

d2
d1

   di+1 = exp (di)
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
Complete partitions of graphs
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  log n sets
   Prove that with high probability its complete
Complete Partitions Continued
   For non-regular graphs complex algorithm and
proof.
   However      log n possible
   Lower bound ( log n )
   Uses the domatic number lower bound
 Complex analysis

 Gives ( log n ) lower bound for
achromatic number
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) . [Andrews, Zhang]
    Maximum cycle packing.
  log n upper bound [M. Krivelevich, Z. Nutov, M.
Salavatipour, R. Yuster].
   log n lower bound. Salavatipour (private communication)
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.
    log n upper bound [Agrawal etal].

   Under the UNIQUE GAME CONJECTURE
no constant ratio [Khot]
More Between log n and O(1)

   Sparsest cut:
     log n upper bound [Arora, Rao and Vazirani]
   Under UGC no c  loglog n ratio, constant c
[Chawla etal]
    Point set width.
    log n   upper bound [Varadarajan etal]
   (log n) lower bound [Varadarajan etal]

 The  cost of the solution returned is
opt+
  is called the additive approximation
ratio
 Much less common (or studied(?)) than

multiplicative ratios
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 d+2.
Additive ratio 2 [Goemans, FOCS 2006]
The Ultimate Approximation
    Some problems admit +1 approximation
    Known examples:
 Coloring a planar graph

 Chromatic index: coloring edges [Vizing]

 Find spanning tree with minimum
maximum degree [Furer Ragavachari]
    Some less known +1 approximation:
Achromatic Number
Achromatic Number of Trees
   The problem is hard on trees
 opt 
     
       n 1
 2 
     
   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

R1                 R4
R2    R3
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] :

 Lower bound [Elkin, Kortsarz] : For some
constant c, opt + c  (log n)2 not possible
unless
NP  DTIME (n {poly-log n})
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}
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 > j
   However, if k < j, size 2j causes collisions for
k, hence is of little help
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(?))