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Polynomial Time Approximation Schemes and Parameterized Complexity

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  NP

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Polynomial Time Approximation Schemes and Parameterized Complexity

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

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

  3. Some Definitions • FPT: fixed-parameter tractable algorithms: for a given instance x and a parameter k (k is small), solve the problem in time f (k)nc. • PTAS: poly-time approximation schemes 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/ • EPTAS: if the time is f (1/)nc

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

  5. 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

  6. Parameterizing Optimization Problems Q = (IQ, SQ, fQ, optQ): NP optimization problem • If optQ = max: Q = { (x, k) | x IQ and optQ(x)  k} Solving Q: for a yes-instance (x, k), construct y SQ (x) such that fQ (x, y)  k • If optQ = min : Q = { (x, k) | x IQ and optQ(x)  k} Solving Q: for a yes-instance (x, k), construct y SQ (x) such that fQ (x, y)  k

  7. Scalability An optimization problem Q = (IQ, SQ, fQ, optQ) is scalable If there are poly-time computable functions g1 and g2 and a polynomial q: • for any instance x of Q, and integer d > 1, xd = g1(x, d) is an instance of Q such that |optQ(xd) - optQ(x)/d|  q(|x|) 2. for any solution yd to xd, y = g2(xd, yd) is a solution to x such that | fQ(xd, yd) - fQ(x, y)/d|  q(|x|)

  8. Most NP optimization problems are scalable • If fQ(x, y) is bounded by a polynomial of |x|, simply let g1(x, d)=x, g2(xd, yd)= yd • 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 g1(x, d) can be simply “dividing each number in x by d then round it” , and g2(xd, yd) is “the solution of x corresponding to yd”

  9. 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.

  10. 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 • let x1 = g1(x,1); if (x1, 3q(n)/)  Q, then try all instances (x, 1), (x, 2), …, (x, 3q(n)/ + q(n)) to construct an optimal solution for x; STOP. • use binary search on d to find an integer d  1 such that (xd, 3q(n)/)  Q  , but (xd+1, 3 q(n)/)  Q ; • construct an optimal solution yd for xd; • let y0 = g2(xd, yd) and output y0 as a solution for x.

  11. Remarks on the algorithm • (x1, 3q(n)/)  Q, implies optQ(x) < 3q(n)/ + q(n), thus, step 1 construct an optimal solution; • The existence of integer d  1 such that (xd, 3q(n)/)  Q and (xd+1, 3q(n)/)  Q is because (x1, 3q(n)/)  Q and (x2r(n), 3q(n)/)  Q ; • (xd+1, 3 q(n)/)  Q makes optQ(xd) bounded by a polynomial of n and 1/ so the optimal solution yd can be constructed; • (xd, 3q(n)/)  Q provides good lower bound for yd.

  12. Remarks on the theorem • 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); • 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.

  13. Brief Review on PTAS • PTAS has been extremely interesting in theoretical computer science; • Khanna-Motwani’s characterization (1996); • Baker’s algorithms on planar graphs (1994); • Extensions to higher genus graphs (-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)

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

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

  16. 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.

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

  18. 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).

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

  20. FPT versus APX(further discussion) • Fact: all MaxSNP problems are FPT; • Relationship between FPT and APX not clear: longest-path FPT – APX binpacking APX – FPT

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

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

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