Polynomial Time Approximation Schemes and Parameterized Complexity Jianer Chen Texas A&M University Joint work with Xiuzhen Huang, Ge Xia, and I. Kanj

Approximation Algorithms and Parameterized Algorithms Assuming P  NPAssuming P  NP Both approximation algorithms and parameterized algorithms tend to solve intractable problems (in particular, NP- hard problems)Both approximation algorithms and parameterized algorithms tend to solve intractable problems (in particular, NP- hard problems) Approximation algorithms solve NP-hard problems with approximation solutionsApproximation algorithms solve NP-hard problems with approximation solutions Parameterized algorithms solve NP-hard problems with small parameters.Parameterized algorithms solve NP-hard problems with small parameters.

Some Definitions FPT: fixed-parameter tractable algorithms:FPT: fixed-parameter tractable algorithms: for a given instance x and a parameter k (k is small), solve the problem in time f (k)n c. for a given instance x and a parameter k (k is small), solve the problem in time f (k)n c. PTAS: poly-time approximation schemesPTAS: poly-time approximation schemes for a given instance x and   , construct a solution with approximation ratio  in polynomial time. for a given instance x and   , construct a solution with approximation ratio  in polynomial time. FPTAS: if the time is polynomial in both |x| and 1/ FPTAS: if the time is polynomial in both |x| and 1/  EPTAS: if the time is f (1/  )n cEPTAS: if the time is f (1/  )n c

Any Connections? Both FPT and PTAS (in particular FPTAS and EPTAS) problems are “easier” intractable problemsBoth FPT and PTAS (in particular FPTAS and EPTAS) problems are “easier” intractable problems FPTAS  FPT (Cai-Chen, 1997)FPTAS  FPT (Cai-Chen, 1997) EPTAS  FPT (Cesati-Trevisan, 1997)EPTAS  FPT (Cesati-Trevisan, 1997) Max-SNP  FPT (Cai-Chen, 1997)Max-SNP  FPT (Cai-Chen, 1997)

We Discuss More Precise Relationships 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 Problems Q = ( I Q, S Q, f Q, opt Q ): NP optimization problem If opt Q = max :If opt Q = max : Q  = { ( x, k ) | x  I Q and opt Q ( x )  k } Solving Q  : for a yes-instance ( x, k ), construct y  S Q ( x ) such that f Q ( x, y )  k Solving Q  : for a yes-instance ( x, k ), construct y  S Q ( x ) such that f Q ( x, y )  k If opt Q = min :If opt Q = min : Q  = { ( x, k ) | x  I Q and opt Q ( x )  k } Solving Q  : for a yes-instance ( x, k ), construct y  S Q ( x ) such that f Q ( x, y )  k Solving Q  : for a yes-instance ( x, k ), construct y  S Q ( x ) such that f Q ( x, y )  k

Scalability An optimization problem Q = ( I Q, S Q, f Q, opt Q ) is scalable If there are poly-time computable functions g 1 and g 2 and a polynomial q: 1.for any instance x of Q, and integer d > 1, x d = g 1 (x, d) is an instance of Q such that | opt Q (x d ) - opt Q (x)/d|  q(|x|) | opt Q (x d ) - opt Q (x)/d|  q(|x|) 2. for any solution y d to x d, y = g 2 (x d, y d ) is a solution to x such that | f Q (x d, y d ) - f Q (x, y)/d|  q(|x|) | f Q (x d, y d ) - f Q (x, y)/d|  q(|x|)

Most NP optimization problems are scalable If f Q (x, y) is bounded by a polynomial of |x|, simply let g 1 (x, d)=x, g 2 (x d, y d )= y dIf f Q (x, y) is bounded by a polynomial of |x|, simply let g 1 (x, d)=x, g 2 (x d, y d )= y d In general, a “number problem”, such as Knapsack and Makespan, has its solution values bounded by a polynomial of the values of the numbers in its instances. Then g 1 (x, d) can be simply “dividing each number in x by d then round it”, and g 2 (x d, y d ) is “the solution of x corresponding to y d ”In general, a “number problem”, such as Knapsack and Makespan, has its solution values bounded by a polynomial of the values of the numbers in its instances. Then g 1 (x, d) can be simply “dividing each number in x by d then round it”, and g 2 (x d, y d ) is “the solution of x corresponding to y d ”

FPTAS and Efficient-FPT Definition. A parameterized problem is efficient-FPT if its has an algorithm whose running time is bounded by a polynomial of |x| and k on input (x, k). Theorem. Let Q be a scalable NP optimization problem. Then Q has an FPTAS if and only if Q is efficient-FPT.

Proof. FPTAS  efficient-FPT: Cai-Chen 1997 Efficient-FPT  FPTAS: Let Q be a maximization problem, and Q  its parameterized version. For an instance x of Q and   0 1.let x 1 = g 1 (x,1); if (x 1, 3q(n)/  )  Q , then try all instances (x, 1), (x, 2), …, (x, 3q(n)/  + q(n)) to construct an optimal solution for x; STOP. 2.use binary search on d to find an integer d  1 such that (x d, 3q(n)/  )  Q , but (x d+1, 3 q(n)/  )  Q  ; 3.construct an optimal solution y d for x d ; 4.let y 0 = g 2 (x d, y d ) and output y 0 as a solution for x.

Remarks on the algorithm (x 1, 3q(n)/  )  Q , implies opt Q (x) < 3q(n)/  + q(n), thus, step 1 construct an optimal solution;(x 1, 3q(n)/  )  Q , implies opt Q (x) < 3q(n)/  + q(n), thus, step 1 construct an optimal solution; The existence of integer d  1 such that (x d, 3q(n)/  )  Q  and (x d+1, 3q(n)/  )  Q  is because (x 1, 3q(n)/  )  Q  and (x 2 r(n), 3q(n)/  )  Q  ;The existence of integer d  1 such that (x d, 3q(n)/  )  Q  and (x d+1, 3q(n)/  )  Q  is because (x 1, 3q(n)/  )  Q  and (x 2 r(n), 3q(n)/  )  Q  ; (x d+1, 3 q(n)/  )  Q  makes opt Q (x d ) bounded by a polynomial of n and 1/  so the optimal solution y d can be constructed;(x d+1, 3 q(n)/  )  Q  makes opt Q (x d ) bounded by a polynomial of n and 1/  so the optimal solution y d can be constructed; (x d, 3q(n)/  )  Q  provides good lower bound for y d.(x d, 3q(n)/  )  Q  provides good lower bound for y d.

Remarks on the theorem It has been a long time interest to characterize FPTAS;It has been a long time interest to characterize FPTAS; Early research (Ausiello et al 1980, and Paz-Moran 1981) uses p-time computable functions (no clue how to detect the existence of such functions);Early research (Ausiello et al 1980, and Paz-Moran 1981) uses p-time computable functions (no clue how to detect the existence of such functions); Recent research (Woeginger 2001) is based on a dynamic programming scheme;Recent research (Woeginger 2001) is based on a dynamic programming scheme; Ours tells explicitly how an efficient-FPT algorithm is converted to an FPTAS, and seems to be a superclass of Woeginger’s.Ours tells explicitly how an efficient-FPT algorithm is converted to an FPTAS, and seems to be a superclass of Woeginger’s.

Brief Review on PTAS PTAS has been extremely interesting in theoretical computer science;PTAS has been extremely interesting in 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 EPTAS (Downey’s and Fellows’ surveys 2003);Impracticality of general PTAS – introduction of EPTAS (Downey’s and Fellows’ surveys 2003); Khanna-Motwani’s contains non-EPTAS (Cai-Fellows-Juedes-Rosamond 2003)Khanna-Motwani’s contains non-EPTAS (Cai-Fellows-Juedes-Rosamond 2003)

Our Characterization of EPTAS MotivationsMotivations Based on W-hierarchy in parameterized complexity;Based on W-hierarchy in parameterized complexity; Seems to include most EPTAS problems;Seems to include most EPTAS problems; Different from Khanna-Motwani – ours contains only EPTAS;Different from Khanna-Motwani – ours contains only EPTAS; Not a subclass of Khanna-Motwani – ours contains problems not in Khanna-Motwani;Not a subclass of Khanna-Motwani – ours contains problems not in Khanna-Motwani; Contain all FPTAS problems via reductions.Contain all FPTAS problems via reductions.

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

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

Examples in Planar W-hierarchy on planar graphs belongs to Planar Min-W[2] ;Vertex Cover on planar graphs belongs to Planar Min-W[2] ; on planar graphs belongs to Planar Max-W[2] ;Independent Set on planar graphs belongs to Planar Max-W[2] ; of Khanna-Motwani belongs to planar W[2]-SATPlanar MaxSAT of Khanna-Motwani belongs to planar W[2]-SAT

Theorem. All problems in the Planar W-hierarchy have EPTAS Proof. Only need to prove this for Planar Min-W[h], Planar Max-W[h], and Planar W[h]-SAT. Extension of Baker’s constructions or by tree decomposition (Alber-Bodlaender-Fernau- Kloks-Niedermeier, 2002). Extension of Baker’s constructions or by tree decomposition (Alber-Bodlaender-Fernau- Kloks-Niedermeier, 2002).

Corollary. All problems in the Planar W-hierarchy are FPT. Proof. By Cesati-Trevison 1997.

FPT versus APX (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: longest-path  FPT – APX binpacking  APX – FPT

FPT versus APX (further discussion) More problems in FPT – APX:More problems in FPT – APX: Controlled by graph genus: e.g. Independent Set on graphs of genus n c for any 1 < c < 2 (based on Chen-Kanj-Perkovic- Sedgwick-Xia 2003); FPT seems harder than APXFPT seems harder than APX except binpacking  APX – FPT; except binpacking  APX – FPT; Is every W[h]-complete problem non-APX, for h > 0?Is every W[h]-complete problem non-APX, for h > 0?

Concluding Remarks Nice relationships between FPT and approximability;Nice relationships between FPT and approximability; Characterization of efficient PTAS (FPTAS and EPTAS);Characterization of efficient PTAS (FPTAS and EPTAS); Further connections (in particular with APX).Further connections (in particular with APX).