Polynomial Time Polynomial Time Approximation Schemes Approximation Schemes and Parameterized and Parameterized ComplexityComplexity

Jianer ChenJianer ChenTexas A&M UniversityTexas A&M University

Joint work with Xiuzhen Huang, Ge Xia, and I. Kanj

Approximation Algorithms Approximation Algorithms and Parameterized and Parameterized

AlgorithmsAlgorithms• Assuming P Assuming P NP NP• Both approximation algorithms and Both approximation algorithms and

parameterized algorithms tend to solve parameterized algorithms tend to solve intractable problems (in particular, NP-intractable problems (in particular, NP-hard problems)hard problems)

• Approximation algorithms solve NP-Approximation algorithms solve NP-hard problems with approximation hard problems with approximation solutionssolutions

• Parameterized algorithms solve NP-Parameterized algorithms solve NP-hard problems with small parameters. hard problems with small parameters.

Some DefinitionsSome Definitions• FPTFPT: fixed-parameter tractable algorithms: : fixed-parameter tractable algorithms: for a given instance for a given instance xx and a parameter and a parameter kk ( (kk is small), is small),

solve the problem in time solve the problem in time f f ((kk))nncc..

• PTASPTAS: poly-time approximation schemes : poly-time approximation schemes for a given instance for a given instance xx and and , construct a solution with , construct a solution with

approximation ratio approximation ratio in polynomial time. in polynomial time.

• FPTASFPTAS: : if the time is polynomial in both |x| and 1/if the time is polynomial in both |x| and 1/

• EPTASEPTAS: : if the time is if the time is f f (1/(1/))nncc

Any Connections?Any Connections?• Both FPT and PTAS (in particular Both FPT and PTAS (in particular

FPTAS and EPTAS) problems are FPTAS and EPTAS) problems are “easier” intractable problems“easier” intractable problems

• FPTAS FPTAS FPT (Cai-Chen, 1997) FPT (Cai-Chen, 1997)• EPTAS EPTAS FPT (Cesati-Trevisan, FPT (Cesati-Trevisan,

1997)1997)• Max-SNP Max-SNP FPT (Cai-Chen, 1997) FPT (Cai-Chen, 1997)

We Discuss We Discuss More PreciseMore Precise Relationships in This TalkRelationships in This Talk

• Under a very general condition, we present a precise characterization of FPTAS in terms of parameterized complexity

• Based on the W-hierarchy in parameterized complexity, we introduce a syntactic EPTAS class that seems to characterize most EPTAS problems

Parameterizing Optimization Parameterizing Optimization ProblemsProblems

Q = (Q = (IIQQ, , SSQQ, , ffQQ, , optoptQQ): NP optimization problem): NP optimization problem• If If optoptQQ = = maxmax::

QQ = { ( = { (x, kx, k) | ) | xx IIQQ and and optoptQQ((xx) ) kk}} Solving QSolving Q: for a yes-instance (: for a yes-instance (x, kx, k), ),

construct construct yy SSQ Q ((xx) such that ) such that ffQ Q ((x, yx, y) ) kk• If If optoptQQ = = minmin : :

QQ = { ( = { (x, kx, k) | ) | xx IIQQ and and optoptQQ((xx) ) kk}} Solving QSolving Q: for a yes-instance (: for a yes-instance (x, kx, k), ),

construct construct yy SSQ Q ((xx) such that ) such that ffQ Q ((x, yx, y) ) kk

ScalabilityScalabilityAn optimization problem Q = (An optimization problem Q = (IIQQ, , SSQQ, , ffQQ, , optoptQQ) is ) is

scalablescalable If there are poly-time computable If there are poly-time computable functions functions gg11 and and gg22 and a polynomial and a polynomial qq: :

1.1. for any instance x of Q, and integer d > 1, for any instance x of Q, and integer d > 1, xxdd = = gg11(x, d) is an instance of Q such that(x, d) is an instance of Q such that

||optoptQQ(x(xdd) - ) - optoptQQ(x)/d| (x)/d| q(|x|) q(|x|) 2. for any solution y2. for any solution ydd to x to xdd, y = , y = gg22(x(xdd, y, ydd) is a ) is a

solution to x such that solution to x such that | | ffQQ(x(xdd, y, ydd) - ) - ffQQ(x, y)/d| (x, y)/d| q(|x|) q(|x|)

Most NP optimization Most NP optimization problems are scalableproblems are scalable• If If ffQQ(x, y) is bounded by a polynomial of |x|, (x, y) is bounded by a polynomial of |x|,

simply let simply let gg11(x, d)=x, (x, d)=x, gg22(x(xdd, y, ydd)= y)= ydd • In general, a “number problem”, such as In general, a “number problem”, such as

KnapsackKnapsack and and MakespanMakespan, has its solution , has its solution values bounded by a polynomial of the values bounded by a polynomial of the values of the numbers in its instances. values of the numbers in its instances. ThenThen g g11(x, d) can be simply “dividing each (x, d) can be simply “dividing each number in x by d then round it” , and number in x by d then round it” , and gg22(x(xdd, , yydd) is “the solution of x corresponding to y) is “the solution of x corresponding to ydd””

FPTAS and Efficient-FPTFPTAS and Efficient-FPTDefinition.Definition. A parameterized problem is A parameterized problem is

efficient-FPTefficient-FPT if its has an algorithm if its has an algorithm whose running time is bounded by a whose running time is bounded by a polynomial of |x| and k on input (x, k).polynomial of |x| and k on input (x, k).

Theorem.Theorem. Let Q be a scalable NP optimization Let Q be a scalable NP optimization problem. Then Q has an FPTAS if and problem. Then Q has an FPTAS if and only if Q is efficient-FPT.only if Q is efficient-FPT.

Proof.Proof.FPTAS FPTAS efficient-FPT efficient-FPT: Cai-Chen 1997: Cai-Chen 1997Efficient-FPT Efficient-FPT FPTAS: FPTAS:Let Q be a maximization problem, and QLet Q be a maximization problem, and Q its its

parameterized version. For an instance x of Q and parameterized version. For an instance x of Q and 0 0

1.1. let xlet x11 = g = g11(x,1); if (x(x,1); if (x11, 3q(n)/, 3q(n)/) ) Q Q, then try all , then try all instances (x, 1), (x, 2), …, (x, 3q(n)/instances (x, 1), (x, 2), …, (x, 3q(n)/ + q(n)) to + q(n)) to construct an optimal solution for x; STOP.construct an optimal solution for x; STOP.

2.2. use binary search on d to find an integer d use binary search on d to find an integer d 1 such 1 such that (xthat (xdd, 3q(n)/, 3q(n)/) ) Q Q , but (x , but (xd+1d+1, 3 q(n)/, 3 q(n)/) ) Q Q ;;

3.3. construct an optimal solution yconstruct an optimal solution ydd for x for xdd; ; 4.4. let ylet y00 = g = g22(x(xdd, y, ydd) and output y) and output y00 as a solution for x. as a solution for x.

Remarks on the algorithmRemarks on the algorithm• (x(x11, 3q(n)/, 3q(n)/) ) Q Q, implies , implies optoptQQ(x) < 3q(n)/(x) < 3q(n)/ + +

q(n), thus, step 1 construct an optimal solution;q(n), thus, step 1 construct an optimal solution;• The existence of integer d The existence of integer d 1 such that (x 1 such that (xdd, ,

3q(n)/3q(n)/) ) Q Q and (x and (xd+1d+1, 3q(n)/, 3q(n)/) ) Q Q is because is because (x (x11, 3q(n)/, 3q(n)/) ) Q Q and (x and (x22r(n)r(n), 3q(n)/, 3q(n)/) ) Q Q ;;

• (x(xd+1d+1, 3 q(n)/, 3 q(n)/) ) Q Q makes makes optoptQQ(x(xdd) bounded by a ) bounded by a polynomial of n and 1/polynomial of n and 1/ so the optimal solution y so the optimal solution yd d can be constructed;can be constructed;

• (x(xdd, 3q(n)/, 3q(n)/) ) Q Q provides good lower bound for provides good lower bound for yydd. .

Remarks on the theoremRemarks on the theorem• It has been a long time interest to characterize It has been a long time interest to characterize

FPTAS;FPTAS;• Early research (Ausiello et al 1980, and Paz-Early research (Ausiello et al 1980, and Paz-

Moran 1981) uses p-time computable functions Moran 1981) uses p-time computable functions (no clue how to detect the existence of such (no clue how to detect the existence of such functions);functions);

• Recent research (Woeginger 2001) is based on Recent research (Woeginger 2001) is based on a dynamic programming scheme;a dynamic programming scheme;

• Ours tells explicitly how an efficient-FPT Ours tells explicitly how an efficient-FPT algorithm is converted to an FPTAS, and seems algorithm is converted to an FPTAS, and seems to be a superclass of Woeginger’s.to be a superclass of Woeginger’s.

Brief Review on PTASBrief Review on PTAS• PTAS has been extremely interesting in PTAS has been extremely interesting in

theoretical computer science;theoretical computer science;• Khanna-Motwani’s characterization (1996);Khanna-Motwani’s characterization (1996);• Baker’s algorithms on planar graphs (1994);Baker’s algorithms on planar graphs (1994);• Extensions to higher genus graphs (-2003);Extensions to higher genus graphs (-2003);• Impracticality of general PTAS – introduction of Impracticality of general PTAS – introduction of

EPTAS (Downey’s and Fellows’ surveys 2003);EPTAS (Downey’s and Fellows’ surveys 2003);• Khanna-Motwani’s contains non-EPTAS Khanna-Motwani’s contains non-EPTAS

(Cai-Fellows-Juedes-Rosamond 2003)(Cai-Fellows-Juedes-Rosamond 2003)

Our Characterization of Our Characterization of EPTASEPTAS• MotivationsMotivations• Based on W-hierarchy in parameterized Based on W-hierarchy in parameterized

complexity;complexity;• Seems to include most EPTAS problems;Seems to include most EPTAS problems;• Different from Different from Khanna-Motwani – ours Khanna-Motwani – ours

contains only EPTAS;contains only EPTAS;• Not a subclass of Khanna-Motwani – ours Not a subclass of Khanna-Motwani – ours

contains problems not in Khanna-Motwani;contains problems not in Khanna-Motwani;• Contain all FPTAS problems via reductions.Contain all FPTAS problems via reductions.

DefinitionsDefinitions

• Planar Min-W[h]:Planar Min-W[h]: given a planar monotone given a planar monotone -circuit -circuit of depth h, construct a satisfying of depth h, construct a satisfying assignment of min weight. (planar assignment of min weight. (planar circuits: become planar after circuits: become planar after removing the output gate);removing the output gate);

• Similarly define Similarly define Planar Max-W[h]Planar Max-W[h] and and planar W[h]-SAT.planar W[h]-SAT.

Planar W-hierarchyPlanar W-hierarchy

Optimization problems that are Optimization problems that are FPTAS-reducible to one of FPTAS-reducible to one of Planar Min-W[h]Planar Min-W[h], , Planar Max-Planar Max-W[h]W[h], and , and Planar W[h]-SATPlanar W[h]-SAT..

Examples in Planar W-Examples in Planar W-hierarchyhierarchy• Vertex Cover on planar graphs on planar graphs

belongs to belongs to Planar Min-W[2]Planar Min-W[2];;• Independent Set on planar graphs on planar graphs

belongs to belongs to Planar Max-W[2]Planar Max-W[2];;• Planar MaxSAT of Khanna- of Khanna-

Motwani belongs to Motwani belongs to planar W[2]-planar W[2]-SATSAT

Theorem. Theorem. All problems in the All problems in the Planar W-hierarchy have Planar W-hierarchy have EPTASEPTAS Proof.Proof.Only need to prove this for Only need to prove this for Planar Min-Planar Min-

W[h]W[h], , Planar Max-W[h]Planar Max-W[h], and , and Planar W[h]-SATPlanar W[h]-SAT..

Extension of Baker’s constructions or by Extension of Baker’s constructions or by tree decomposition (Alber-Bodlaender-tree decomposition (Alber-Bodlaender-Fernau-Kloks-Niedermeier, 2002).Fernau-Kloks-Niedermeier, 2002).

Corollary. Corollary. All problems in the All problems in the Planar W-hierarchy are FPT.Planar W-hierarchy are FPT.

Proof. By Cesati-Trevison 1997.Proof. By Cesati-Trevison 1997.

FPT versus APXFPT versus APX(further discussion)(further discussion)

• Fact: all MaxSNP problems are FPT;Fact: all MaxSNP problems are FPT;• Relationship between FPT and APXRelationship between FPT and APX

not clear:not clear:longest-pathlongest-path FPT – APX FPT – APXbinpackingbinpacking APX – FPT APX – FPT

FPT versus APXFPT versus APX(further discussion)(further discussion)• More problems in FPT – APX: More problems in FPT – APX:

Controlled by graph genus: Controlled by graph genus: e.g. e.g. Independent SetIndependent Set on graphs of genus on graphs of genus nnc c for any 1 < c < 2 (based on Chen-Kanj-for any 1 < c < 2 (based on Chen-Kanj-Perkovic-Sedgwick-Xia 2003);Perkovic-Sedgwick-Xia 2003);

• FPT seems harder than APX FPT seems harder than APX except except binpackingbinpacking APX – FPT; APX – FPT;• Is every W[h]-complete problem non-APX, Is every W[h]-complete problem non-APX,

for h > 0?for h > 0?

Concluding RemarksConcluding Remarks• Nice relationships between FPT and Nice relationships between FPT and

approximability;approximability;• Characterization of efficient PTAS Characterization of efficient PTAS

(FPTAS and EPTAS);(FPTAS and EPTAS);• Further connections (in particular Further connections (in particular

with APX).with APX).