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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):. A Generic Problem: Set-Cover. SETS. ELEMENTS. A. B. - PowerPoint PPT Presentation
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RARE APPROXIMATION RARE APPROXIMATION RATIOSRATIOS
Guy KortsarzGuy Kortsarz
Rutgers UniversityRutgers University
CamdenCamden
Approximation RatiosApproximation Ratios
NP-Hard problems NP-Hard problems Coping with the difficulty: approximationCoping with the difficulty: approximation Minimization or maximization.Minimization or maximization. Approximation ratio (for minimization):Approximation ratio (for minimization):
)(
)(max
Inputs Iopt
IALGI
A Generic Problem: Set-CoverA Generic Problem: Set-Cover
A
SETSELEMENTS
B
Frequent Approximation RatiosFrequent Approximation Ratios
Constants. Example: Constants. Example: Max-3-SAT: Tight 8/7 ratioMax-3-SAT: Tight 8/7 ratio
Logarithmic for minimization problems:Logarithmic for minimization problems: Set-coverSet-cover
PTAS (1 + PTAS (1 + ) for all ) for all > 0 > 0 Example: Euclidean TSPExample: Euclidean TSP
Frequent Ratios continuedFrequent Ratios continued
Polynomial Ratios: Polynomial Ratios: sqrt (sqrt (nn), ), nn {1 - {1 - }}
Example: Example: Clique: Clique: nn {1 - {1 - }} lower bound lower bound
Upper bound: Upper bound: ((nn/log/log33nn) (Halldorsson, Feige)) (Halldorsson, Feige)
Example: Constrained Satisfaction Example: Constrained Satisfaction ProblemsProblems
Given a collection of Boolean formulas, satisfy all Given a collection of Boolean formulas, satisfy all constrains. Maximize # true variables. constrains. Maximize # true variables.
Possible ratios:Possible ratios:
1) Solvable in polynomial time 1) Solvable in polynomial time
2) 2) nn
3) Constant3) Constant
4) Unbounded4) Unbounded Due to Khanna, Sudan, WilliamsonDue to Khanna, Sudan, Williamson
""NaturalNatural"" Problems Problems
It is possible to artificially design problems to It is possible to artificially design problems to get any desired ratioget any desired ratio
See for example the NP-complete column of D. See for example the NP-complete column of D. Johnson: The many limits of approximationJohnson: The many limits of approximation
If in set-cover we take the objective function to If in set-cover we take the objective function to be sqrt(|S|) then the ratio is sqrt(ln be sqrt(|S|) then the ratio is sqrt(ln nn))
I discuss rare ratios that appeared as a natural I discuss rare ratios that appeared as a natural consequence of the problem/techniquesconsequence of the problem/techniques
This sheds light on special problems/techniquesThis sheds light on special problems/techniques
Rare Ratios: Example IRare Ratios: Example I
Until 2000 there was no Until 2000 there was no
MAXIMIZATION PROBLEM MAXIMIZATION PROBLEM
with with log log nn threshold threshold Example: Domatic NumberExample: Domatic Number
Input: Input: G G ((VV, , EE)) Dominating set Dominating set UU: : U U NN((UU) = ) = VV
The Domatic Number ProblemThe Domatic Number Problem
Given: Given: G G ((VV, , EE))
Find: Find: VV==VV11 VV22 …. …. VVkk
so that so that VVii dominating set (in dominating set (in GG).). Goal: Maximize Goal: Maximize kk Example: A maximal independent set Example: A maximal independent set
and its complement is dominating. and its complement is dominating. kk ≥ ≥ 22
A Simple AlgorithmA Simple Algorithm
Create binsCreate bins
Throw every vertex into a bin at randomThrow every vertex into a bin at random The expected number of neighbors of every The expected number of neighbors of every vv in bin in bin ii
is is 3 ln 3 ln nn The probability that bin i has no neighbor of The probability that bin i has no neighbor of vv::
nln3
3
1ln31
n
n
Domatic Number ContinuedDomatic Number Continued
The number of bad events is The number of bad events is nn22 or less. or less. Each one has probability Each one has probability 1/1/nn3 3 to hold to hold
By the union bound size partition By the union bound size partition existsexists
Remark: Remark: + 1 + 1 is a trivial upper bound is a trivial upper bound This implies This implies OO(ln (ln nn)) ratio ratio
nln3
Large Minimum Degree
opt = 2
More Lower and Upper BoundsMore Lower and Upper Bounds Feige, Halldorsson, Kortsarz, Srinivasan Feige, Halldorsson, Kortsarz, Srinivasan
The approximation is improved to The approximation is improved to O O (log (log )) (LLL)(LLL)
There is always There is always /ln /ln solution (complex proof) solution (complex proof) Can not be approximated within Can not be approximated within (1 - (1 - ) ) ln ln nn
for any constant for any constant > 0 > 0
Remarks on the Lower BoundRemarks on the Lower Bound
Lower Bound Method: Lower Bound Method: 1R2P1R2P Generalizes (or improves) the paper of Feige Generalizes (or improves) the paper of Feige
from 1996, from 1996, (1 - (1 - ) ) ln ln nn , lower bound for set-, lower bound for set-covercover
Recycling solutions: One Set Cover implies Recycling solutions: One Set Cover implies many set-cover existmany set-cover exist
Uses Zero-Knowledge techniquesUses Zero-Knowledge techniques
Perhaps Perhaps loglog n n for Maximization: for Maximization: Unique Set CoverUnique Set Cover
Special Case: Every Element in Special Case: Every Element in BB has Degree has Degree dd
Choose every Choose every aa AA with probability 1/ with probability 1/dd
Hence, expected number of uniquely covered Hence, expected number of uniquely covered elements of elements of BB, a constant fraction, a constant fraction
Hence, there always is a subset Hence, there always is a subset AA’’ AA that uniquely that uniquely covers a fraction covers a fraction
eddd
d111
1)bforNeighbourUniquePr(1
General Case:General Case:
Cluster the degrees into powers of 2:Cluster the degrees into powers of 2:
There exists a cluster with There exists a cluster with (|(|BB| / log || / log |A| A| )) verticesvertices
Corollary: There always exists Corollary: There always exists AA’’ AA that that uniquely covers a uniquely covers a 1 / log 1 / log nn fraction of fraction of BB
}2)deg(2|{ 1 iii bBbD
Lower BoundsLower Bounds
Demaine, Feige, Hajiaghayi, Salvatipour:Demaine, Feige, Hajiaghayi, Salvatipour: Hard to find complete bipartite graphs, Hard to find complete bipartite graphs,
Implies Implies log log nn best possible best possible NP has no algorithm implies NP has no algorithm implies (log (log nn))
hard to approximatehard to approximate Hard to refute random 3-sat instances, Hard to refute random 3-sat instances,
implies implies ( log ( log nn ) ) 1/31/3 hardhard
n2
Polylogarithmic for Polylogarithmic for MinimizationMinimization
Group Steiner problem on trees:Group Steiner problem on trees:
g1 g2 g3 g4 g5
Integrality GapIntegrality Gap
Halperin, Kortsarz, Krauthgamer, Halperin, Kortsarz, Krauthgamer, Srinivasan,WangSrinivasan,Wang
g1,g2g3,g4
g1,g3,g2 g2,g4
g1,g3 g1,g2 g2 g4
Analysis:Analysis: The costs need to decrease by constant factor The costs need to decrease by constant factor
[HST][HST] The fractional value is the same at every levelThe fractional value is the same at every level Thus, if the height is Thus, if the height is HH then the fractional is then the fractional is
OO((HH)) The integral The integral HH22 log log kk ( (kk is # groups) is # groups) (log (log kk))22 gap gap The same paper [HKKSW] gives The same paper [HKKSW] gives OO ( (log ( (log kk))22 ) )
upper boundupper bound
More Upper BoundsMore Upper Bounds Garg, Ravi, KonjevodGarg, Ravi, Konjevod : :
OO( (log ( (log nn))22)) using Linear Programming using Linear Programming Randomized rounding plus Jansen Randomized rounding plus Jansen
inequalitiesinequalities Halperin, Krauthgamer: Halperin, Krauthgamer:
Lower bound: Lower bound: (log (log kk))2-2- (log (log nn / log log / log log nn))22
“ “Hiding” a trapdor in the integrality gap Hiding” a trapdor in the integrality gap constructionconstruction
Directed Steiner and BelowDirected Steiner and Below Directed Steiner:Directed Steiner: OO( (log ( (log nn))33)) quasi-polynomial time quasi-polynomial time
and and n n for every for every polynomial time [Charikar etal] polynomial time [Charikar etal] Special case:Special case: Group Steiner on general graphs: Group Steiner on general graphs: OO( (log ( (log nn))33)) polynomial (reduction to trees using Bartal polynomial (reduction to trees using Bartal
Trees)Trees) In quasi-polynomial tine In quasi-polynomial tine OO( (log ( (log nn))22)) for general graphs for general graphs
[Chekuri, Pal][Chekuri, Pal] Group Steiner trees: Group Steiner trees: loglog22 nn / log log / log log nn,, quasi- quasi-
polynomial time [Chekuri, Even, Kortsarz]polynomial time [Chekuri, Even, Kortsarz]
The Asymmetric The Asymmetric kk-Center Problem-Center Problem
Given: Directed graph Given: Directed graph GG((VV, , EE)) and length and length ll((ee)) on edges and a number on edges and a number kk
Required: choose a subset Required: choose a subset UU, |, |UU| = | = kk of the of the verticesvertices
Optimization criteria: Minimize Optimization criteria: Minimize
)},({max UudistUu
A log* A log* nn Approximation Approximation
Due to VishwanathanDue to Vishwanathan Idea:Idea:
k
Lower Bound: log* Lower Bound: log* nn Due to: Chuzhoy, Guha, Halperin, Khanna, Due to: Chuzhoy, Guha, Halperin, Khanna,
Kortsarz, Krauthgamer, J. NaorKortsarz, Krauthgamer, J. Naor Based on hardness for d-set-coverBased on hardness for d-set-cover
Simple Algorithm for Simple Algorithm for dd-Set-Cover-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 Hardness: Based on dd-Set Cover -Set Cover Hardness: Hardness: d d – 1 - – 1 -
Dinur, Guruswami, Khot, Regev: Dinur, Guruswami, Khot, Regev:
Gap Reduction for Gap Reduction for d d – Set - Cover– Set - Cover
I
d-set-cover
d-set-cover
No instance
Yes instance 3/d |A| enough to cover
Any (1-2/d)|A| subset covers at most (1-f(d)) fraction of B.
f(d)=(1/2) {poly d}
A Hardness Result for Directed A Hardness Result for Directed kk-Center-Center
Compose the d-set-cover construction:Compose the d-set-cover construction:
ddii+1 +1 = exp (= exp (ddii))
d1d2
AnalysisAnalysis Choose Choose k k = (= (VV11//dd11)) - 1- 1 For a YES instance get dist =1For a YES instance get dist =1 For a NO instance:For a NO instance:
We may assume all centers are at We may assume all centers are at VV11
But the number of uncovered vertices But the number of uncovered vertices remains larger than 0remains larger than 0
Approaches 0 at Approaches 0 at log (previous)log (previous) speed speed Gives Gives log* log* nn gap gap
Complete partitions of graphsComplete partitions of graphs
Approximation for Approximation for d d - Regular - Regular GraphsGraphs
sqrt(sqrt(mm/2)/2) is an upper bound is an upper bound Partition to Partition to sqrt(sqrt(mm/2)/2) classes at random classes at random There is an expected There is an expected OO(1)(1) edges per sets edges per sets
Merge randomly to groups of Merge randomly to groups of 33 sets sets Prove that with high probability its completeProve that with high probability its complete
nlog
Complete Partitions ContinuedComplete Partitions Continued
For non-regular graphs complex algorithm and For non-regular graphs complex algorithm and proof. proof.
However possibleHowever possible Lower bound Lower bound Uses the domatic number lower boundUses the domatic number lower bound
Complex analysisComplex analysis Gives lower bound for Gives lower bound for
achromatic numberachromatic number
nlog
)log( n
)log( n
More Between log More Between log nn and and OO(1)(1) Minimum congestion routing: Minimum congestion routing:
Given a collection of pairs (undirected graph) choose a Given a collection of pairs (undirected graph) choose a path for each pair. Minimize the congestion:path for each pair. Minimize the congestion:
Upper bound: Upper bound: O(log n / loglog n) . [Raghavan , Thompson] . [Raghavan , Thompson] Lower bound: Lower bound: (log log n) . [Andrews, Zhang] [Andrews, Zhang]
Maximum cycle packing. Maximum cycle packing. upper bound [M. Krivelevich, Z. Nutov, M.upper bound [M. Krivelevich, Z. Nutov, M.
Salavatipour, R. YusterSalavatipour, R. Yuster]]. . lower bound. Salavatipour (private lower bound. Salavatipour (private
communication)communication)
nlog
nlog
More Between log More Between log nn and and OO(1)(1)
Directed congestion minimization: Directed congestion minimization: O(log n / loglog n) upper bound upper bound
[Raghavan and Thompson] [Raghavan and Thompson] (log n) 1- lower bound. bound. [Andrews and Zhang][Andrews and Zhang]
Min 2CNF deletion. Min 2CNF deletion. upper bound [Agrawal etal].upper bound [Agrawal etal]. Under the UNIQUE GAME CONJECTURE Under the UNIQUE GAME CONJECTURE
no constant ratio [Khot]no constant ratio [Khot]
nlog
More Between log More Between log nn and and OO(1)(1)
Sparsest cut:Sparsest cut: upper bound [Arora, Rao and Vazirani]upper bound [Arora, Rao and Vazirani] Under UGC no Under UGC no c loglog n ratio, constant ratio, constant c
[Chawla etal][Chawla etal] Point set width.Point set width.
upper bound [Varadarajan etal]upper bound [Varadarajan etal] (log n) lower bound [Varadarajan etal]lower bound [Varadarajan etal]
nlog
nlog
Additive Approximation RatiosAdditive Approximation Ratios
The cost of the solution returned is The cost of the solution returned is
opt+opt+ is called the additive approximation is called the additive approximation
ratioratio Much less common (or studied(?)) than Much less common (or studied(?)) than
multiplicative ratiosmultiplicative ratios
New ResultNew Result
Let Let G G ((VV,,EE,,cc) be a graph that admits a ) be a graph that admits a spanning tree of cost at most spanning tree of cost at most cc* and * and maximum degree at most maximum degree at most dd
Then, there exists a polynomial time Then, there exists a polynomial time algorithm that finds a spanning tree of cost algorithm that finds a spanning tree of cost at most at most cc* and maximum degree * and maximum degree dd+2. +2. Additive ratio 2 [Goemans, FOCS 2006]Additive ratio 2 [Goemans, FOCS 2006]
The Ultimate ApproximationThe Ultimate Approximation
Some problems admit Some problems admit ++1 approximation1 approximation Known examples:Known examples:
Coloring a planar graphColoring a planar graph Chromatic index: coloring edges [Vizing]Chromatic index: coloring edges [Vizing] Find spanning tree with minimum Find spanning tree with minimum
maximum degree [Furer Ragavachari]maximum degree [Furer Ragavachari] Some less known +1 approximation:Some less known +1 approximation:
Achromatic NumberAchromatic Number
Achromatic Number of TreesAchromatic Number of Trees The problem is hard on treesThe problem is hard on trees
Thus opt is bounded by roughly Thus opt is bounded by roughly sqrt sqrt nn This bound is achievable within +1 (in This bound is achievable within +1 (in
polynomial time) polynomial time) Similarly: Minimum Harmonious coloring of Similarly: Minimum Harmonious coloring of
trees: +1 approximationtrees: +1 approximation
1n2
opt
Poly-log Additive (tight): Radio Poly-log Additive (tight): Radio BroadcastBroadcast
R1R2 R3
R4
Upper and Lower BoundsUpper and Lower Bounds Since one can cover Since one can cover 1/log 1/log nn uniquely, in uniquely, in OO( (log ( (log nn))22)) rounds the other side of a Bipartite rounds the other side of a Bipartite
graph can be informedgraph can be informed Thus, in a BFS fashion: Thus, in a BFS fashion: RadiusRadius (log (log nn))22
Best known [Kowalski, Pelc] : Best known [Kowalski, Pelc] : RadiusRadius + + OO(log (log nn))22
Lower bound [Elkin, Kortsarz] : For some Lower bound [Elkin, Kortsarz] : For some constant constant cc, , opt + c opt + c (log (log nn))2 2 not possible not possible unless unless
NP NP DTIME DTIME ((nn {poly-log {poly-log nn}}))
A graph with radius = 1, A graph with radius = 1, opt = opt = (log (log nn))22
A construction by Alon, Bar-Noy, Lineal, PelegA construction by Alon, Bar-Noy, Lineal, Peleg
P=(1/2){0.4log n} P=(1/2) {0.6log n}
AnalysisAnalysis
If we choose any subset of size If we choose any subset of size 22jj then the set then the set of probability of probability (½)(½)jj will be informed in will be informed in log log nn roundsrounds
Since there are Since there are 0.20.2 ln ln nn sets, it will take sets, it will take OO( (log ( (log nn))22))
The difficulty: A size The difficulty: A size 22jj does not affect the sets does not affect the sets of of p p = (½)= (½)kk, , k k > > jj
However, if However, if kk < < jj,, size size 22jj causes collisions for causes collisions for kk, hence is of little help, hence is of little help
ConclusionConclusion
No real conclusionNo real conclusion The NPC problem seems to admit little order if at all The NPC problem seems to admit little order if at all
regarding approximationregarding approximation The problems are ``unstable”The problems are ``unstable” There does not seem to be a ``deep” reason these There does not seem to be a ``deep” reason these
ratios are rare (because of techniques(?))ratios are rare (because of techniques(?)) Very good advances. Very good advances. Still much we don’t understand in approximationsStill much we don’t understand in approximations