A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions.

[1]  Anders Yeo,et al.  Kernel bounds for disjoint cycles and disjoint paths , 2009, Theor. Comput. Sci..

[2]  Vangelis Th. Paschos,et al.  Parameterized Algorithms for the Max K-set Cover and Related Satisfiability Problems Parameterized Exact and Approximation Algorithms for Maximum K-set Cover and Related Satisfiability Problems , 2022 .

[3]  Jan Vondrák,et al.  Multiway cut, pairwise realizable distributions, and descending thresholds , 2014, STOC.

[4]  Shi Li,et al.  Constant Approximation for Capacitated k-Median with (1 + ε)-Capacity Violation , 2016, ArXiv.

[5]  Erik Jan van Leeuwen,et al.  Approximation and Parameterized Algorithms for Geometric Independent Set with Shrinking , 2016, MFCS.

[6]  David B. Shmoys,et al.  A unified approach to approximation algorithms for bottleneck problems , 1986, JACM.

[7]  Miroslav Chlebík,et al.  The Steiner tree problem on graphs: Inapproximability results , 2008, Theor. Comput. Sci..

[8]  Amit Kumar,et al.  A simple linear time (1 + /spl epsiv/)-approximation algorithm for k-means clustering in any dimensions , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[9]  Leslie E. Trotter,et al.  Vertex packings: Structural properties and algorithms , 1975, Math. Program..

[10]  Bruce A. Reed,et al.  A Simple Algorithm for the Graph Minor Decomposition - Logic meets Structural Graph Theory , 2013, SODA.

[11]  Dror Rawitz,et al.  Local ratio: A unified framework for approximation algorithms. In Memoriam: Shimon Even 1935-2004 , 2004, CSUR.

[12]  Carsten Lund,et al.  Efficient probabilistically checkable proofs and applications to approximations , 1993, STOC.

[13]  Pasin Manurangsi,et al.  Losing Treewidth by Separating Subsets , 2019, SODA.

[14]  Sanjeev Arora,et al.  Probabilistic checking of proofs: a new characterization of NP , 1998, JACM.

[15]  Mohammad R. Salavatipour,et al.  On Sum Coloring of Graphs , 2003, Discret. Appl. Math..

[16]  Saket Saurabh,et al.  Subquadratic Kernels for Implicit 3-Hitting Set and 3-Set Packing Problems , 2018, SODA.

[17]  Johan Håstad,et al.  Some optimal inapproximability results , 2001, JACM.

[18]  Marek Cygan,et al.  Improved Approximation for 3-Dimensional Matching via Bounded Pathwidth Local Search , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[19]  S. E. Dreyfus,et al.  The steiner problem in graphs , 1971, Networks.

[20]  Bart M. P. Jansen,et al.  Polynomial Kernels for Hitting Forbidden Minors under Structural Parameterizations , 2018, ESA.

[21]  Alexander Vardy,et al.  The Parametrized Complexity of Some Fundamental Problems in Coding Theory , 1999, SIAM J. Comput..

[22]  László Lovász,et al.  Interactive proofs and the hardness of approximating cliques , 1996, JACM.

[23]  Saket Saurabh,et al.  Polylogarithmic Approximation Algorithms for Weighted-ℱ-deletion Problems , 2020, ACM Trans. Algorithms.

[24]  David P. Woodruff,et al.  Strong Coresets for k-Median and Subspace Approximation: Goodbye Dimension , 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[25]  Saket Saurabh,et al.  Capacitated Domination and Covering: A Parameterized Perspective , 2008, IWPEC.

[26]  Chaitanya Swamy,et al.  Improved Approximation Algorithms for Matroid and Knapsack Median Problems and Applications , 2013, APPROX-RANDOM.

[27]  Vijay V. Vazirani,et al.  Finding k Cuts within Twice the Optimal , 1995, SIAM J. Comput..

[28]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs , 1990, Inf. Comput..

[29]  Pasin Manurangsi,et al.  ETH-Hardness of Approximating 2-CSPs and Directed Steiner Network , 2018, ITCS.

[30]  Dana Moshkovitz,et al.  The Projection Games Conjecture and the NP-Hardness of ln n-Approximating Set-Cover , 2012, Theory Comput..

[31]  Sanjeev Khanna,et al.  Polynomial flow-cut gaps and hardness of directed cut problems , 2007, STOC '07.

[32]  Vivek Madan,et al.  Approximating Multicut and the Demand Graph , 2017, SODA.

[33]  Henning Fernau,et al.  A novel parameterised approximation algorithm for minimum vertex cover , 2013, Theor. Comput. Sci..

[34]  Xinhui Wang,et al.  Dynamic Programming for Minimum Steiner Trees , 2007, Theory of Computing Systems.

[35]  Yixin Cao,et al.  Minimum Fill-In: Inapproximability and Almost Tight Lower Bounds , 2016, SODA.

[36]  Fabrizio Grandoni,et al.  Parameterized Approximation Schemes for Independent Set of Rectangles and Geometric Knapsack , 2019, ESA.

[37]  Pasin Manurangsi,et al.  On the Parameterized Complexity of Approximating Dominating Set , 2019, J. ACM.

[38]  Michal Pilipczuk,et al.  Minimum bisection is fixed parameter tractable , 2013, STOC.

[39]  Euiwoong Lee,et al.  Improved and simplified inapproximability for k-means , 2015, Inf. Process. Lett..

[40]  Charles J. Colbourn,et al.  Unit disk graphs , 1991, Discret. Math..

[41]  David M. Mount,et al.  A local search approximation algorithm for k-means clustering , 2002, SCG '02.

[42]  Ken-ichi Kawarabayashi,et al.  The Minimum k-way Cut of Bounded Size is Fixed-Parameter Tractable , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[43]  Irit Dinur,et al.  Exponentially Small Soundness for the Direct Product Z-test , 2017, Electron. Colloquium Comput. Complex..

[44]  Mohammad R. Salavatipour,et al.  Local Search Yields a PTAS for k-Means in Doubling Metrics , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[45]  Michael Lampis A kernel of order 2 k-c log k for vertex cover , 2011, Inf. Process. Lett..

[46]  Vangelis Th. Paschos,et al.  Structural Parameters, Tight Bounds, and Approximation for (k, r)-Center , 2017, ISAAC.

[47]  Jörg Flum,et al.  Parameterized Complexity Theory , 2006, Texts in Theoretical Computer Science. An EATCS Series.

[48]  Pasin Manurangsi,et al.  On the parameterized complexity of approximating dominating set , 2017, Electron. Colloquium Comput. Complex..

[49]  Vincent Cohen-Addad,et al.  A Fast Approximation Scheme for Low-Dimensional k-Means , 2017, SODA.

[50]  Juraj Hromkovic,et al.  The Parameterized Approximability of TSP with Deadlines , 2007, Theory of Computing Systems.

[51]  Dániel Marx,et al.  Parameterized graph separation problems , 2004, Theor. Comput. Sci..

[52]  David R. Karger,et al.  A new approach to the minimum cut problem , 1996, JACM.

[53]  Saket Saurabh,et al.  Approximate Counting of k-Paths: Deterministic and in Polynomial Space , 2019, ICALP.

[54]  Siu On Chan,et al.  Approximation resistance from pairwise independent subgroups , 2013, STOC '13.

[55]  Satish Rao,et al.  Approximation schemes for Euclidean k-medians and related problems , 1998, STOC '98.

[56]  Saket Saurabh,et al.  Kernelization Lower Bounds Through Colors and IDs , 2014, ACM Trans. Algorithms.

[57]  Henning Fernau,et al.  Combining Two Worlds: Parameterised Approximation for Vertex Cover , 2010, ISAAC.

[58]  Saket Saurabh,et al.  A Parameterized Approximation Scheme for Min $k$-Cut , 2020, 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS).

[59]  Andreas Wiese Fixed-Parameter Approximation Schemes for Weighted Flowtime , 2018, APPROX-RANDOM.

[60]  David Saulpic,et al.  Near-Linear Time Approximation Schemes for Clustering in Doubling Metrics , 2018, J. ACM.

[61]  Michael Dinitz,et al.  The Densest k-Subhypergraph Problem , 2016, APPROX-RANDOM.

[62]  Venkatesan Guruswami,et al.  Inapproximability of H-Transversal/Packing , 2015, SIAM J. Discret. Math..

[63]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems , 1998, JACM.

[64]  Tony Huynh,et al.  A tight Erdős-Pósa function for planar minors , 2018, SODA.

[65]  Michal Pilipczuk,et al.  Lower bounds for approximation schemes for Closest String , 2015, SWAT.

[66]  Yijia Chen,et al.  The Constant Inapproximability of the Parameterized Dominating Set Problem , 2015, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[67]  Jan Vondrák,et al.  Maximizing a Monotone Submodular Function Subject to a Matroid Constraint , 2011, SIAM J. Comput..

[68]  Rajiv Gandhi,et al.  Bicovering: Covering edges with two small subsets of vertices , 2016, Electron. Colloquium Comput. Complex..

[69]  Teofilo F. GONZALEZ,et al.  Clustering to Minimize the Maximum Intercluster Distance , 1985, Theor. Comput. Sci..

[70]  David Zuckerman Simulating BPP using a general weak random source , 2005, Algorithmica.

[71]  Marek Karpinski,et al.  Approximation schemes for clustering problems , 2003, STOC '03.

[72]  Udi Rotics,et al.  On the Relationship between Clique-Width and Treewidth , 2001, WG.

[73]  Dániel Marx Chordal Deletion is Fixed-Parameter Tractable , 2008, Algorithmica.

[74]  Stefan Kratsch,et al.  Polynomial Kernelizations for MIN F+Π1 and MAX NP , 2009, Algorithmica.

[75]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[76]  Euiwoong Lee,et al.  Partitioning a graph into small pieces with applications to path transversal , 2016, Mathematical Programming.

[77]  Vangelis Th. Paschos,et al.  Efficient Approximation of Combinatorial Problems by Moderately Exponential Algorithms , 2009, WADS.

[78]  Ge Xia,et al.  Linear FPT reductions and computational lower bounds , 2004, STOC '04.

[79]  Dániel Marx,et al.  Efficient Approximation Schemes for Geometric Problems? , 2005, ESA.

[80]  Prasad Raghavendra,et al.  A Birthday Repetition Theorem and Complexity of Approximating Dense CSPs , 2016, ICALP.

[81]  Michal Wlodarczyk Parameterized Inapproximability for Steiner Orientation by Gap Amplification , 2020, ICALP.

[82]  J. Håstad Clique is hard to approximate withinn1−ε , 1999 .

[83]  Rolf Niedermeier,et al.  Parameterized Complexity of Vertex Cover Variants , 2007, Theory of Computing Systems.

[84]  Frank Thomson Leighton,et al.  Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms , 1999, JACM.

[85]  Jason Li,et al.  On the Fixed-Parameter Tractability of Capacitated Clustering , 2022, ICALP.

[86]  Michael R. Fellows,et al.  Fundamentals of Parameterized Complexity , 2013 .

[87]  Piotr Berman,et al.  On the Complexity of Approximating the Independent Set Problem , 1989, Inf. Comput..

[88]  Ken-ichi Kawarabayashi,et al.  A nearly 5/3-approximation FPT Algorithm for Min-k-Cut , 2020, SODA.

[89]  M MountDavid,et al.  A local search approximation algorithm for k-means clustering , 2004 .

[90]  Vangelis Th. Paschos,et al.  New Results on Polynomial Inapproximabilityand Fixed Parameter Approximability of Edge Dominating Set , 2014, Theory of Computing Systems.

[91]  Prasad Raghavendra,et al.  Graph expansion and the unique games conjecture , 2010, STOC '10.

[92]  Evripidis Bampis,et al.  Parameterized Power Vertex Cover , 2016, WG.

[93]  Liming Cai,et al.  Advice Classes of Parameterized Tractability , 1997, Ann. Pure Appl. Log..

[94]  Harry B. Hunt,et al.  NC-Approximation Schemes for NP- and PSPACE-Hard Problems for Geometric Graphs , 1998, J. Algorithms.

[95]  Hans L. Bodlaender A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC '93.

[96]  Andreas Wiese,et al.  Approximation Schemes for Maximum Weight Independent Set of Rectangles , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[97]  Pasin Manurangsi,et al.  Inapproximability of Maximum Biclique Problems, Minimum k-Cut and Densest At-Least-k-Subgraph from the Small Set Expansion Hypothesis , 2017, Algorithms.

[98]  Saket Saurabh,et al.  Polylogarithmic Approximation Algorithms for Weighted-ℱ-deletion Problems , 2017, APPROX-RANDOM.

[99]  Petr A. Golovach,et al.  Spanning Circuits in Regular Matroids , 2019, ACM Trans. Algorithms.

[100]  Amit Kumar,et al.  Linear Time Algorithms for Clustering Problems in Any Dimensions , 2005, ICALP.

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

[102]  Nikhil Bansal,et al.  LP-Based Robust Algorithms for Noisy Minor-Free and Bounded Treewidth Graphs , 2017, SODA.

[103]  V. N. Muralidhara,et al.  Minimizing Total Flow-Time: The Unrelated Case , 2008, ISAAC.

[104]  Saket Saurabh,et al.  Packing Cycles Faster Than Erdos-Posa , 2017, ICALP.

[105]  Philip N. Klein,et al.  Local Search Yields Approximation Schemes for k-Means and k-Median in Euclidean and Minor-Free Metrics , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[106]  Xi Wu,et al.  A Completeness Theory for Polynomial (Turing) Kernelization , 2013, Algorithmica.

[107]  Jochen Könemann,et al.  A (1+ε)-Embedding of Low Highway Dimension Graphs into Bounded Treewidth Graphs , 2018, SIAM J. Comput..

[108]  Satish Rao,et al.  A Nearly Linear-Time Approximation Scheme for the Euclidean k-Median Problem , 2007, SIAM J. Comput..

[109]  David Zuckerman,et al.  Electronic Colloquium on Computational Complexity, Report No. 100 (2005) Linear Degree Extractors and the Inapproximability of MAX CLIQUE and CHROMATIC NUMBER , 2005 .

[110]  Kunal Talwar,et al.  Bypassing the embedding: algorithms for low dimensional metrics , 2004, STOC '04.

[111]  Hans L. Bodlaender,et al.  Treewidth: Structure and Algorithms , 2007, SIROCCO.

[112]  Dániel Marx Completely Inapproximable Monotone and Antimonotone Parameterized Problems , 2010, 2010 IEEE 25th Annual Conference on Computational Complexity.

[113]  Samir Khuller,et al.  Greedy strikes back: improved facility location algorithms , 1998, SODA '98.

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

[115]  J. Matou On Approximate Geometric K-clustering , 1999 .

[116]  Gerhard J. Woeginger,et al.  Approximability and nonapproximability results for minimizing total flow time on a single machine , 1996, STOC '96.

[117]  Lee-Ad Gottlieb,et al.  A Light Metric Spanner , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[118]  Michal Pilipczuk,et al.  Designing FPT Algorithms for Cut Problems Using Randomized Contractions , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[119]  Bingkai Lin,et al.  A Simple Gap-producing Reduction for the Parameterized Set Cover Problem , 2019, ICALP.

[120]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[121]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[122]  Jianer Chen,et al.  Directed Feedback Vertex Set Problem is FPT , 2007, Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs.

[123]  L. Pósa,et al.  On Independent Circuits Contained in a Graph , 1965, Canadian Journal of Mathematics.

[124]  Irit Dinur,et al.  Mildly exponential reduction from gap 3SAT to polynomial-gap label-cover , 2016, Electron. Colloquium Comput. Complex..

[125]  Michal Pilipczuk,et al.  Everything you always wanted to know about the parameterized complexity of Subgraph Isomorphism (but were afraid to ask) , 2013, STACS.

[126]  Subhash Khot,et al.  Ruling out PTAS for graph min-bisection, densest subgraph and bipartite clique , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[127]  Venkatesh Raman,et al.  Revisiting Connected Vertex Cover: FPT Algorithms and Lossy Kernels , 2017, Theory of Computing Systems.

[128]  Yixin Cao,et al.  Interval Deletion Is Fixed-Parameter Tractable , 2012, SODA.

[129]  Bart M. P. Jansen,et al.  A Turing Kernelization Dichotomy for Structural Parameterizations of F-Minor-Free Deletion , 2019, WG.

[130]  Blair D. Sullivan,et al.  Structural Rounding: Approximation Algorithms for Graphs Near an Algorithmically Tractable Class , 2018, ESA.

[131]  Aviad Rubinstein,et al.  SETH vs Approximation , 2019, SIGA.

[132]  Pasin Manurangsi,et al.  A Note on Max k-Vertex Cover: Faster FPT-AS, Smaller Approximate Kernel and Improved Approximation , 2018, SOSA.

[133]  Jaroslaw Byrka,et al.  Constant factor FPT approximation for capacitated k-median , 2018, ESA.

[134]  Eran Halperin,et al.  Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs , 2000, SODA '00.

[135]  Bingkai Lin,et al.  The Parameterized Complexity of the k-Biclique Problem , 2018, J. ACM.

[136]  Robert Krauthgamer,et al.  Polylogarithmic inapproximability , 2003, STOC '03.

[137]  Vangelis Th. Paschos,et al.  Improved (In-)Approximability Bounds for d-Scattered Set , 2019, WAOA.

[138]  Fedor V. Fomin,et al.  Subquadratic Kernels for Implicit 3-Hitting Set and 3-Set Packing Problems , 2019, SODA.

[139]  Vangelis Th. Paschos,et al.  Structurally Parameterized $d$-Scattered Set , 2017, WG.

[140]  Michael Lampis,et al.  Parameterized (Approximate) Defective Coloring , 2018, STACS.

[141]  J. Sgall,et al.  An Approximation Algorithm for Bounded Degree Deletion∗ , 2009 .

[142]  Avner Magen,et al.  Robust Algorithms for on Minor-Free Graphs Based on the Sherali-Adams Hierarchy , 2009, APPROX-RANDOM.

[143]  James R. Lee,et al.  Improved approximation algorithms for minimum-weight vertex separators , 2005, STOC '05.

[144]  Sariel Har-Peled,et al.  On coresets for k-means and k-median clustering , 2004, STOC '04.

[145]  Avi Wigderson,et al.  New direct-product testers and 2-query PCPs , 2009, STOC '09.

[146]  Yuval Rabani,et al.  ON THE HARDNESS OF APPROXIMATING MULTICUT AND SPARSEST-CUT , 2005, 20th Annual IEEE Conference on Computational Complexity (CCC'05).

[147]  Dan Feldman,et al.  A PTAS for k-means clustering based on weak coresets , 2007, SCG '07.

[148]  Jacques Stern,et al.  The hardness of approximate optima in lattices, codes, and systems of linear equations , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[149]  Fedor V. Fomin,et al.  Planar F-Deletion: Approximation, Kernelization and Optimal FPT Algorithms , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[150]  Wenxing Lai The Inapproximability of k-DominatingSet for Parameterized AC 0 Circuits † , 2019, Algorithms.

[151]  Pasin Manurangsi,et al.  Parameterized Approximation Algorithms for Directed Steiner Network Problems , 2017, ESA.

[152]  Ken-ichi Kawarabayashi,et al.  Algorithmic graph minor theory: Decomposition, approximation, and coloring , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[153]  Ke Chen,et al.  On k-Median clustering in high dimensions , 2006, SODA '06.

[154]  Subhash Khot,et al.  On the power of unique 2-prover 1-round games , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[155]  Jesper Nederlof Fast Polynomial-Space Algorithms Using Möbius Inversion: Improving on Steiner Tree and Related Problems , 2009, ICALP.

[156]  Michael Dinitz,et al.  Characterizing Demand Graphs for (Fixed-Parameter) Shallow-Light Steiner Network , 2018, FSTTCS.

[157]  Andreas Wiese A ( 1 + )-Approximation for Unsplittable Flow on a Path in Fixed-Parameter Running Time ∗ , 2017 .

[158]  Vijay V. Vazirani,et al.  Finding k-cuts within twice the optimal , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[159]  Laurence A. Wolsey,et al.  Worst-Case and Probabilistic Analysis of Algorithms for a Location Problem , 1980, Oper. Res..

[160]  David S. Johnson,et al.  The Complexity of Computing Steiner Minimal Trees , 1977 .

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

[162]  Magnus Wahlström,et al.  Directed Multicut is W[1]-hard, Even for Four Terminal Pairs , 2015, SODA.

[163]  Rolf Niedermeier,et al.  Parameterized Complexity of Arc-Weighted Directed Steiner Problems , 2009, SIAM J. Discret. Math..

[164]  Andreas Björklund,et al.  Narrow sieves for parameterized paths and packings , 2010, J. Comput. Syst. Sci..

[165]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[166]  Luca Trevisan,et al.  Mildly Exponential Time Approximation Algorithms for Vertex Cover, Balanced Separator and Uniform Sparsest Cut , 2018, APPROX-RANDOM.

[167]  Joseph Naor,et al.  A 2-Approximation Algorithm for the Directed Multiway Cut Problem , 2001, SIAM J. Comput..

[168]  Amit Kumar,et al.  Tight FPT Approximations for $k$-Median and k-Means , 2019, ICALP.

[169]  Kevin Pratt,et al.  Waring Rank, Parameterized and Exact Algorithms , 2018, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS).

[170]  Mikkel Thorup,et al.  Minimum k-way cuts via deterministic greedy tree packing , 2008, STOC.

[171]  Pasin Manurangsi,et al.  On Closest Pair in Euclidean Metric: Monochromatic is as Hard as Bichromatic , 2018, Combinatorica.

[172]  Philip N. Klein,et al.  Embedding Planar Graphs into Low-Treewidth Graphs with Applications to Efficient Approximation Schemes for Metric Problems , 2019, SODA.

[173]  Michael R. Fellows,et al.  Parameterized approximation via fidelity preserving transformations , 2012, J. Comput. Syst. Sci..

[174]  Mam Riess Jones Color Coding , 1962, Human factors.

[175]  Saket Saurabh,et al.  Parameterized Complexity and Approximability of Directed Odd Cycle Transversal , 2017, SODA.

[176]  Michael R. Fellows,et al.  On the Complexity of Some Colorful Problems Parameterized by Treewidth , 2007, COCOA.

[177]  Jirí Fiala,et al.  Geometric separation and exact solutions for the parameterized independent set problem on disk graphs , 2002, J. Algorithms.

[178]  Marcin Pilipczuk,et al.  Approximation and Kernelization for Chordal Vertex Deletion , 2016, SODA.

[179]  Faisal N. Abu-Khzam,et al.  A kernelization algorithm for d-Hitting Set , 2010, J. Comput. Syst. Sci..

[180]  Nisheeth K. Vishnoi,et al.  Coresets for clustering in Euclidean spaces: importance sampling is nearly optimal , 2020, STOC.

[181]  Russell Impagliazzo,et al.  Which problems have strongly exponential complexity? , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[182]  Fabrizio Grandoni,et al.  Oblivious dimension reduction for k-means: beyond subspaces and the Johnson-Lindenstrauss lemma , 2019, STOC.

[183]  Robert Krauthgamer,et al.  Coresets for Clustering in Excluded-minor Graphs and Beyond , 2020, SODA.

[184]  Karthekeyan Chandrasekaran,et al.  Improving the Integrality Gap for Multiway Cut , 2019, IPCO.

[185]  Sanjeev Arora,et al.  Subexponential Algorithms for Unique Games and Related Problems , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[186]  Rajesh Chitnis,et al.  FPT Inapproximability of Directed Cut and Connectivity Problems , 2019, IPEC.

[187]  Ding-Zhu Du,et al.  The k-Steiner Ratio in Graphs , 1997, SIAM J. Comput..

[188]  Marek Cygan,et al.  Deterministic Parameterized Connected Vertex Cover , 2012, SWAT.

[189]  Ge Xia,et al.  Improved Parameterized Upper Bounds for Vertex Cover , 2006, MFCS.

[190]  Ran Raz,et al.  A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP , 1997, STOC '97.

[191]  Refael Hassin,et al.  Approximation Schemes for the Restricted Shortest Path Problem , 1992, Math. Oper. Res..

[192]  Pasin Manurangsi,et al.  Parameterized Intractability of Even Set and Shortest Vector Problem from Gap-ETH , 2018, Electron. Colloquium Comput. Complex..

[193]  Piotr Indyk,et al.  Approximate clustering via core-sets , 2002, STOC '02.

[194]  Miklós Ajtai,et al.  The shortest vector problem in L2 is NP-hard for randomized reductions (extended abstract) , 1998, STOC '98.

[195]  Luca Trevisan,et al.  From Gap-ETH to FPT-Inapproximability: Clique, Dominating Set, and More , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[196]  Dániel Marx,et al.  Complexity results for minimum sum edge coloring , 2009, Discret. Appl. Math..

[197]  Amit Kumar,et al.  New Approximation Schemes for Unsplittable Flow on a Path , 2015, SODA.

[198]  Samuel Fiorini,et al.  Hitting Diamonds and Growing Cacti , 2009, IPCO.

[199]  Erik D. Demaine,et al.  Equivalence of local treewidth and linear local treewidth and its algorithmic applications , 2004, SODA '04.

[200]  Arnaud de Mesmay,et al.  A Near-Linear Approximation Scheme for Multicuts of Embedded Graphs with a Fixed Number of Terminals , 2018, SODA.

[201]  Mihalis Yannakakis,et al.  Approximate Max-Flow Min-(Multi)Cut Theorems and Their Applications , 1996, SIAM J. Comput..

[202]  Liming Cai,et al.  On fixed-parameter tractability and approximability of NP-hard optimization problems , 1993, [1993] The 2nd Israel Symposium on Theory and Computing Systems.

[203]  Dániel Marx,et al.  Minimum Sum Multicoloring on the Edges of Planar Graphs and Partial k-Trees , 2004, WAOA.

[204]  Prasad Raghavendra,et al.  Rounding Semidefinite Programming Hierarchies via Global Correlation , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[205]  Harald Räcke,et al.  Optimal hierarchical decompositions for congestion minimization in networks , 2008, STOC.

[206]  Michal Pilipczuk,et al.  Linear kernels for edge deletion problems to immersion-closed graph classes , 2016, ICALP.

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

[208]  Fahad Panolan,et al.  Lossy kernelization , 2016, STOC.

[209]  Luca Trevisan,et al.  On the Efficiency of Polynomial Time Approximation Schemes , 1997, Inf. Process. Lett..

[210]  Fabrizio Grandoni,et al.  Steiner Tree Approximation via Iterative Randomized Rounding , 2013, JACM.

[211]  Mohammad Taghi Hajiaghayi,et al.  Fixed-Parameter and Approximation Algorithms: A New Look , 2013, IPEC.

[212]  V. Sós,et al.  On a problem of K. Zarankiewicz , 1954 .

[213]  Johan Hstad Clique is hard to approximate within n1-epsilon , 1997 .

[214]  Derek G. Corneil,et al.  Complexity of finding embeddings in a k -tree , 1987 .

[215]  Rajiv Gandhi,et al.  Bi-Covering: Covering Edges with Two Small Subsets of Vertices , 2016, SIAM J. Discret. Math..

[216]  David P. Williamson,et al.  The Design of Approximation Algorithms , 2011 .

[217]  Brenda S. Baker,et al.  Approximation algorithms for NP-complete problems on planar graphs , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[218]  Ken-ichi Kawarabayashi,et al.  Polylogarithmic Approximation for Minimum Planarization (Almost) , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[219]  Dániel Marx,et al.  Parameterized Complexity and Approximation Algorithms , 2008, Comput. J..

[220]  Ran Raz,et al.  A parallel repetition theorem , 1995, STOC '95.

[221]  Sanjeev Arora,et al.  Inapproximabilty of Densest κ-Subgraph from Average Case Hardness , 2011 .

[222]  Michal Pilipczuk,et al.  Hardness of Approximation for H-free Edge Modification Problems , 2016, APPROX-RANDOM.

[223]  Vangelis Th. Paschos,et al.  New Results on Polynomial Inapproximability and Fixed Parameter Approximability of edge dominating set , 2012, IPEC.

[224]  Nikhil Bansal,et al.  New Tools and Connections for Exponential-Time Approximation , 2017, Algorithmica.

[225]  Erez Petrank The hardness of approximation: Gap location , 2005, computational complexity.

[226]  David Haussler,et al.  Decision Theoretic Generalizations of the PAC Model for Neural Net and Other Learning Applications , 1992, Inf. Comput..

[227]  Dušan Knop,et al.  Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices , 2018, STACS.

[228]  Henning Fernau Saving on Phases: Parameterized Approximation for Total Vertex Cover , 2012, IWOCA.

[229]  Uriel Feige,et al.  Finding small balanced separators , 2006, STOC '06.

[230]  Michael Lampis,et al.  Parameterized Approximation Schemes Using Graph Widths , 2013, ICALP.

[231]  Anupam Gupta,et al.  The number of minimum k-cuts: improving the Karger-Stein bound , 2019, STOC.

[232]  Dániel Marx,et al.  Fixed-parameter tractability of multicut parameterized by the size of the cutset , 2010, STOC '11.

[233]  Richard Ryan Williams,et al.  Distributed PCP Theorems for Hardness of Approximation in P , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[234]  Erik D. Demaine,et al.  The Bidimensionality Theory and Its Algorithmic Applications , 2008, Comput. J..

[235]  Andreas Björklund,et al.  Determinant Sums for Undirected Hamiltonicity , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[236]  S. KarthikC.,et al.  Inapproximability of Clustering in Lp Metrics , 2019, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS).

[237]  Yehoshua Bar-Hillel,et al.  The Intrinsic Computational Difficulty of Functions , 1969 .

[238]  J. Edmonds Paths, Trees, and Flowers , 1965, Canadian Journal of Mathematics.

[239]  Philip N. Klein,et al.  Polynomial-Time Approximation Schemes for k-center, k-median, and Capacitated Vehicle Routing in Bounded Highway Dimension , 2018, ESA.

[240]  Thore Husfeldt,et al.  Extensor-coding , 2018, STOC.

[241]  Dániel Marx,et al.  The Parameterized Hardness of the k-Center Problem in Transportation Networks , 2018, Algorithmica.

[242]  Yicheng Xu,et al.  A constant parameterized approximation for hard-capacitated k-means , 2019, ArXiv.

[243]  Ken-ichi Kawarabayashi,et al.  Contraction decomposition in h-minor-free graphs and algorithmic applications , 2011, STOC '11.

[244]  Carsten Lund,et al.  The Approximation of Maximum Subgraph Problems , 1993, ICALP.

[245]  Oded Goldreich,et al.  Computational complexity: a conceptual perspective , 2008, SIGA.

[246]  Shi Li,et al.  Constant Approximation for Capacitated k-Median with (1+epsilon)-Capacity Violation , 2016, ICALP.

[247]  Rolf H. Möhring,et al.  A constant FPT approximation algorithm for hard-capacitated k-means , 2019 .

[248]  Anupam Gupta,et al.  Faster Exact and Approximate Algorithms for k-Cut , 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[249]  Pasin Manurangsi,et al.  Almost-polynomial ratio ETH-hardness of approximating densest k-subgraph , 2016, STOC.

[250]  Euiwoong Lee Improved Hardness for Cut, Interdiction, and Firefighter Problems , 2017, ICALP.

[251]  Stefan Kratsch,et al.  Safe Approximation and Its Relation to Kernelization , 2011, IPEC.

[252]  Pim van 't Hof,et al.  Parameterized complexity of vertex deletion into perfect graph classes , 2011, Theor. Comput. Sci..

[253]  Shi Li,et al.  Constant approximation for k-median and k-means with outliers via iterative rounding , 2017, STOC.

[254]  Andreas Emil Feldmann,et al.  Fixed-Parameter Approximations for k-Center Problems in Low Highway Dimension Graphs , 2015, Algorithmica.

[255]  Eun Jung Kim,et al.  Erdős-Pósa property of chordless cycles and its applications , 2018, SODA.

[256]  Pasin Manurangsi,et al.  Losing tree-width by separating subsets , 2019, SODA 2019.

[257]  Anupam Gupta,et al.  An FPT Algorithm Beating 2-Approximation for k-Cut , 2017, SODA.

[258]  Fahad Panolan,et al.  Lossy Kernels for Connected Dominating Set on Sparse Graphs , 2017, STACS.

[259]  Hooyeon Lee,et al.  Approximating low-dimensional coverage problems , 2011, SoCG '12.

[260]  Michael Langberg,et al.  A unified framework for approximating and clustering data , 2011, STOC '11.

[261]  Satish Rao,et al.  Expander flows, geometric embeddings and graph partitioning , 2004, STOC '04.

[262]  Michal Pilipczuk,et al.  A ck n 5-Approximation Algorithm for Treewidth , 2016, SIAM J. Comput..

[263]  Chandra Chekuri,et al.  Polynomial bounds for the grid-minor theorem , 2013, J. ACM.

[264]  Dimitrios M. Thilikos,et al.  Recent techniques and results on the Erdős-Pósa property , 2016, Discret. Appl. Math..

[265]  Marek Kubale,et al.  Edge-chromatic sum of trees and bounded cyclicity graphs , 2000, Inf. Process. Lett..

[266]  Jochen Könemann,et al.  A (1+ε)-Embedding of Low Highway Dimension Graphs into Bounded Treewidth Graphs , 2015, ICALP.

[267]  B. Jansen,et al.  A Turing Kernelization Dichotomy for Structural Parameterizations of $\mathcal{F}$-Minor-Free Deletion , 2019 .

[268]  Shi Li,et al.  Approximating capacitated k-median with (1 + ∊)k open facilities , 2014, SODA.

[269]  Noga Alon,et al.  Improved approximation for directed cut problems , 2007, STOC '07.

[270]  Johan Håstad,et al.  Clique is hard to approximate within n/sup 1-/spl epsiv// , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[271]  Mohammad R. Salavatipour,et al.  Approximability of Packing Disjoint Cycles , 2007, Algorithmica.

[272]  Pasin Manurangsi Tight Running Time Lower Bounds for Strong Inapproximability of Maximum k-Coverage, Unique Set Cover and Related Problems (via t-Wise Agreement Testing Theorem) , 2020, SODA.

[273]  Aravind Srinivasan,et al.  An Improved Approximation for k-Median and Positive Correlation in Budgeted Optimization , 2014, SODA.

[274]  Larry Stockmeyer,et al.  Planar 3-colorability is polynomial complete , 1973, SIGA.

[275]  Irit Dinur,et al.  The PCP theorem by gap amplification , 2006, STOC.

[276]  Guy Kortsarz,et al.  Steiner Forest Orientation Problems , 2012, ESA.

[277]  Mihir Bellare,et al.  Free Bits, PCPs, and Nonapproximability-Towards Tight Results , 1998, SIAM J. Comput..

[278]  Michael Dinitz,et al.  Minimizing the Union: Tight Approximations for Small Set Bipartite Vertex Expansion , 2016, SODA.

[279]  Ge Xia,et al.  Polynomial time approximation schemes and parameterized complexity , 2007, Discret. Appl. Math..

[280]  Christos H. Papadimitriou,et al.  The Euclidean Traveling Salesman Problem is NP-Complete , 1977, Theor. Comput. Sci..

[281]  Saket Saurabh,et al.  On the Approximate Compressibility of Connected Vertex Cover , 2019, Algorithmica.