The M-Hierarchy, and XP-Optimality

We relate NP to parameterized complexity via the ETH, SETH, and prove results on XP-optimality. Namely, we give methods to show that, under reasonable hypotheses, current algorithms for various computational tasks are optimal up to an O-factor.

[1]  Ge Xia,et al.  On the computational hardness based on linear FPT-reductions , 2006, J. Comb. Optim..

[2]  Dimitrios M. Thilikos,et al.  Catalan structures and dynamic programming in H-minor-free graphs , 2008, SODA '08.

[3]  Dimitrios M. Thilikos,et al.  Confronting intractability via parameters , 2011, Comput. Sci. Rev..

[4]  Patrick C. Fischer,et al.  Refining Nondeterminism in Relativized Polynomial-Time Bounded Computations , 1980, SIAM J. Comput..

[5]  Michael R. Fellows,et al.  Parameterized complexity: A framework for systematically confronting computational intractability , 1997, Contemporary Trends in Discrete Mathematics.

[6]  Alan Bundy,et al.  Constructing Induction Rules for Deductive Synthesis Proofs , 2006, CLASE.

[7]  Michael R. Fellows,et al.  On the complexity of some colorful problems parameterized by treewidth , 2011, Inf. Comput..

[8]  J. Geske.  On the structure of intractable sets (binary relation) , 1987 .

[9]  Erik D. Demaine,et al.  Exponential Speedup of Fixed-Parameter Algorithms for Classes of Graphs Excluding Single-Crossing Graphs as Minors , 2005, Algorithmica.

[10]  Robin Thomas,et al.  Quickly Excluding a Planar Graph , 1994, J. Comb. Theory, Ser. B.

[11]  Ljubomir Perkovic,et al.  Improved Parameterized Algorithms for Planar Dominating Set , 2002, MFCS.

[12]  Erik D. Demaine,et al.  Bidimensionality: new connections between FPT algorithms and PTASs , 2005, SODA '05.

[13]  Liming Cai,et al.  On the Amount of Nondeterminism and the Power of Verifying , 1997, SIAM J. Comput..

[14]  Rolf Niedermeier,et al.  Fixed-Parameter Algorithms for CLOSEST STRING and Related Problems , 2003, Algorithmica.

[15]  Jörg Flum,et al.  Bounded fixed-parameter tractability and log2n nondeterministic bits , 2004, J. Comput. Syst. Sci..

[16]  Dimitrios M. Thilikos,et al.  Subexponential parameterized algorithms , 2008, Comput. Sci. Rev..

[17]  Hisao Tamaki,et al.  Optimal branch-decomposition of planar graphs in O(n3) Time , 2005, TALG.

[18]  Paul Young,et al.  Collapsing degrees in subexponential time , 1994, Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory.

[19]  Liming Cai,et al.  On the Amount of Nondeterminism and the Power of Verifying (Extended Abstract) , 1993, MFCS.

[20]  Michael R. Fellows,et al.  Fixed Parameter Tractability and Completeness , 1992, Complexity Theory: Current Research.

[21]  Henning Fernau,et al.  Graph Separator Algorithms: A Refined Analysis , 2002, WG.

[22]  Dániel Marx,et al.  Lower bounds based on the Exponential Time Hypothesis , 2011, Bull. EATCS.

[23]  Noga Alon,et al.  Fast Fast , 2009, ICALP.

[24]  Ge Xia,et al.  Tight lower bounds for certain parameterized NP-hard problems , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..

[25]  Michael R. Fellows,et al.  Cutting Up is Hard to Do: the Parameterized Complexity of k-Cut and Related Problems , 2003, CATS.

[26]  Fedor V. Fomin,et al.  Beyond bidimensionality: Parameterized subexponential algorithms on directed graphs , 2010, Inf. Comput..

[27]  Frederic Dorn,et al.  Designing Subexponential Algorithms: Problems, Techniques & Structures , 2007 .

[28]  Rodney G. Downey,et al.  The Birth and Early Years of Parameterized Complexity , 2012, The Multivariate Algorithmic Revolution and Beyond.

[29]  Erik D. Demaine,et al.  Fixed-parameter algorithms for (k, r)-center in planar graphs and map graphs , 2005, TALG.

[30]  Mihai Patrascu,et al.  On the possibility of faster SAT algorithms , 2010, SODA '10.

[31]  Russell Impagliazzo,et al.  Complexity of k-SAT , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[32]  Erik D. Demaine,et al.  The Bidimensional Theory of Bounded-Genus Graphs , 2006, SIAM J. Discret. Math..

[33]  Albert R. Meyer,et al.  Logic at Botik '89, Symposium on Logical Foundations of Computer Science, Pereslav-Zalessky, USSR, July 3-8, 1989, Proceedings , 1989, Logic at Botik.

[34]  Siamak Tazari,et al.  Faster Approximation Schemes and Parameterized Algorithms on H-Minor-Free and Odd-Minor-Free Graphs , 2010, MFCS.

[35]  Ge Xia,et al.  Genus Characterizes the Complexity of Graph Problems: Some Tight Results , 2003, ICALP.

[36]  Henning Fernau,et al.  A Geometric Approach to Parameterized Algorithms for Domination Problems on Planar Graphs , 2004, MFCS.

[37]  Don Coppersmith,et al.  Rectangular Matrix Multiplication Revisited , 1997, J. Complex..

[38]  Liming Cai,et al.  On Fixed-Parameter Tractability and Approximability of NP Optimization Problems , 1997, J. Comput. Syst. Sci..

[39]  Dániel Marx,et al.  Known algorithms on graphs of bounded treewidth are probably optimal , 2010, SODA '11.

[40]  Dimitrios M. Thilikos,et al.  Subexponential parameterized algorithms for degree-constrained subgraph problems on planar graphs , 2010, J. Discrete Algorithms.

[41]  Raymond E. Miller,et al.  Complexity of Computer Computations , 1972 .

[42]  Petr A. Golovach,et al.  Contraction Bidimensionality: the Accurate Picture , 2009, Parameterized complexity and approximation algorithms.

[43]  Robin Thomas,et al.  Call routing and the ratcatcher , 1994, Comb..

[44]  Ton Kloks,et al.  New Algorithms for k-Face Cover, k-Feedback Vertex Set, and k -Disjoint Cycles on Plane and Planar Graphs , 2002, WG.

[45]  Milan Ruzic,et al.  Uniform deterministic dictionaries , 2008, TALG.

[46]  Dániel Marx,et al.  Slightly superexponential parameterized problems , 2011, SODA '11.

[47]  Dimitrios M. Thilikos,et al.  Planar Feedback Vertex Set and Face Cover: Combinatorial Bounds and Subexponential Algorithms , 2010, Algorithmica.

[48]  Erik D. Demaine,et al.  Linearity of grid minors in treewidth with applications through bidimensionality , 2008, Comb..

[49]  Judy Goldsmith,et al.  Nondeterminism Within P , 1993, SIAM J. Comput..

[50]  R. Solovay,et al.  Relativizations of the $\mathcal{P} = ?\mathcal{NP}$ Question , 1975 .

[51]  Liming Cai,et al.  On the existence of subexponential parameterized algorithms , 2003, J. Comput. Syst. Sci..

[52]  Saharon Shelah,et al.  Nearly Linear Time , 1989, Logic at Botik.

[53]  Rolf Niedermeier,et al.  Fixed Parameter Algorithms for DOMINATING SET and Related Problems on Planar Graphs , 2002, Algorithmica.

[54]  David S. Johnson,et al.  The Planar Hamiltonian Circuit Problem is NP-Complete , 1976, SIAM J. Comput..

[55]  Ge Xia,et al.  Strong computational lower bounds via parameterized complexity , 2006, J. Comput. Syst. Sci..

[56]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[57]  Rolf Niedermeier,et al.  Fixed Parameter Algorithms for PLANAR DOMINATING SET and Related Problems , 2000, SWAT.

[58]  Fedor V. Fomin,et al.  Subexponential parameterized algorithm for minimum fill-in , 2011, SODA.

[59]  Erik D. Demaine,et al.  Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs , 2005, JACM.

[60]  Michal Pilipczuk,et al.  Solving Connectivity Problems Parameterized by Treewidth in Single Exponential Time , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[61]  Michael R. Fellows,et al.  Fixed-parameter intractability , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.

[62]  Dimitrios M. Thilikos,et al.  Dominating sets in planar graphs: branch-width and exponential speed-up , 2003, SODA '03.

[63]  Russell Impagliazzo,et al.  Which Problems Have Strongly Exponential Complexity? , 2001, J. Comput. Syst. Sci..

[64]  Mikkel Thorup,et al.  Generalized Dominators for Structured Programs , 1996, Algorithmica.

[65]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[66]  David A. Schmidt,et al.  Proceedings of the Third International Symposium on Static Analysis , 1996 .

[67]  Erik D. Demaine,et al.  Bidimensional Parameters and Local Treewidth , 2004, SIAM J. Discret. Math..

[68]  Yoshio Okamoto,et al.  On Problems as Hard as CNF-SAT , 2012, Computational Complexity Conference.

[69]  Michael R. Fellows,et al.  On the complexity of fixed parameter problems , 1989, 30th Annual Symposium on Foundations of Computer Science.

[70]  Liming Cai,et al.  Subexponential Parameterized Algorithms Collapse the W-Hierarchy , 2001, ICALP.

[71]  Hisao Tamaki,et al.  Improved Bounds on the Planar Branchwidth with Respect to the Largest Grid Minor Size , 2010, ISAAC.

[72]  Dimitrios M. Thilikos,et al.  Bidimensionality and kernels , 2010, SODA '10.

[73]  Dimitrios M. Thilikos.  Fast Sub-exponential Algorithms and Compactness in Planar Graphs , 2011, ESA.

[74]  Kenneth W. Regan.  Finitary substructure languages with application to the theory of NP-completeness , 1989, [1989] Proceedings. Structure in Complexity Theory Fourth Annual Conference.

[75]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[76]  Fedor V. Fomin,et al.  Efficient Exact Algorithms on Planar Graphs: Exploiting Sphere Cut Decompositions , 2010, Algorithmica.

[77]  Michael R. Fellows,et al.  Fixed-Parameter Tractability and Completeness IV: On Completeness for W[P] and PSPACE Analogues , 1995, Ann. Pure Appl. Log..

[78]  Fedor V. Fomin,et al.  Bidimensionality and EPTAS , 2010, SODA '11.

[79]  Andrew Chi-Chih Yao,et al.  Separating the Polynomial-Time Hierarchy by Oracles (Preliminary Version) , 1985, FOCS.

[80]  Yijia Chen,et al.  An Isomorphism Between Subexponential and Parameterized Complexity Theory , 2007, SIAM J. Comput..