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      Guy Kortsarz
    Rutgers University
          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


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:

   (n/log3n) (Halldorsson, Feige)
    Example: Constrained Satisfaction
   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
       Rare Ratios: Example I
 Until 2000 there was no
  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
   Feige, Halldorsson, Kortsarz, Srinivasan
      The approximation is improved to O (log )
      There is always /ln  solution (complex
      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-
    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 }

   There exists a cluster with  (|B| / log |A| )
   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
     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
                Integrality Gap
        Halperin, Kortsarz, Krauthgamer,

          g1,g3,g2                    g2,g4

g1,g3            g1,g2           g2           g4
   The costs need to decrease by constant factor
   The fractional value is the same at every level
   Thus, if the height is H then the fractional is
   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
   Halperin, Krauthgamer:
     Lower bound: (log k)2-

     (log n / log log n)2

     “Hiding” a trapdor in the integrality gap
       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
   Optimization criteria: Minimize

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

   Due to Vishwanathan
   Idea:

          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

                                  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
   Compose the d-set-cover construction:


   di+1 = exp (di)
   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
   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
   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].

        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]
  Additive Approximation Ratios

 The  cost of the solution returned is
  is called the additive approximation
 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
Poly-log Additive (tight): Radio

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] :

  Radius + O(log n)2
 Lower bound [Elkin, Kortsarz] : For some
  constant c, opt + c  (log n)2 not possible
          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}
   If we choose any subset of size 2j then the set
    of probability (½)j will be informed in log n
   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
   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 don’t understand in approximations

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