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) . [Chuzhoy, Naor] [Chuzhoy, Naor]

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 [Hoyler]Chromatic index: coloring edges [Hoyler] 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