Half-integrality, LP-branching, and FPT Algorithms

A recent trend in parameterized algorithms is the application of polytope tools to fixed-parameter tractable (FPT) algorithms [e.g., Cygan et al., FOCS 2011, 52nd Annual Symposium on Foundations of Computer Science, IEEE, 2011, pp. 150--159; Narayanaswamy et al., STACS 2012, Symposium on Theoretical Aspects of Computer Science, 2012, pp. 338--349]. Although this approach has yielded significant speedups for a range of important problems, it requires the underlying polytope to have very restrictive properties, including half-integrality and Nemhauser--Trotter-style persistence properties. To date, these properties are essentially known to hold only for two classes of polytopes, covering the cases of Vertex Cover [Nemhauser and Trotter, Math. Program., 8 (1975), pp. 232--248] and Node Multiway Cut [Garg et al., J. Alg., 50 (2004), pp. 49--61]. Taking a slightly different approach, we view half-integrality as a discrete relaxation of a problem, e.g., a relaxation of the search space from $\{0,1\}^V$ to $\{0,...

[1]  Anna Huber,et al.  Towards Minimizing k-Submodular Functions , 2012, ISCO.

[2]  Gyula Pap,et al.  Packing non-returning A-paths algorithmically , 2008, Discret. Math..

[3]  Michal Pilipczuk,et al.  On Group Feedback Vertex Set Parameterized by the Size of the Cutset , 2012, WG.

[4]  Dorit S. Hochbaum,et al.  Solving integer programs over monotone inequalities in three variables: A framework for half integrality and good approximations , 2002, Eur. J. Oper. Res..

[5]  Stanislav Zivny,et al.  The Expressive Power of Binary Submodular Functions , 2009, MFCS.

[6]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

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

[8]  Saket Saurabh,et al.  LP can be a cure for Parameterized Problems , 2012, STACS.

[9]  Gyula Pap,et al.  Packing Non-Returning A-Paths* , 2007, Comb..

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

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

[12]  Saket Saurabh,et al.  Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts , 2014, SODA.

[13]  David P. Williamson,et al.  A primal-dual interpretation of two 2-approximation algorithms for the feedback vertex set problem in undirected graphs , 1998, Oper. Res. Lett..

[14]  Maurice Queyranne,et al.  On the structure of all minimum cuts in a network and applications , 1982, Math. Program..

[15]  Paul D. Seymour,et al.  Packing Non-Zero A-Paths In Group-Labelled Graphs , 2006, Comb..

[16]  Mihalis Yannakakis,et al.  Primal-dual approximation algorithms for integral flow and multicut in trees , 1997, Algorithmica.

[17]  Sylvain Guillemot,et al.  FPT algorithms for path-transversal and cycle-transversal problems , 2011, Discret. Optim..

[18]  Bruce A. Reed,et al.  Computing crossing number in linear time , 2007, STOC '07.

[19]  David P. Williamson,et al.  Iterative rounding 2-approximation algorithms for minimum-cost vertex connectivity problems , 2006, J. Comput. Syst. Sci..

[20]  Mihalis Yannakakis,et al.  Multiway cuts in node weighted graphs , 2004, J. Algorithms.

[21]  Saket Saurabh,et al.  Faster Parameterized Algorithms Using Linear Programming , 2012, ACM Trans. Algorithms.

[22]  Alexander Schrijver,et al.  A Combinatorial Algorithm Minimizing Submodular Functions in Strongly Polynomial Time , 2000, J. Comb. Theory B.

[23]  Martin C. Cooper,et al.  Constraints, Consistency and Closure , 1998, Artif. Intell..

[24]  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.

[25]  Stefan Kratsch,et al.  Representative Sets and Irrelevant Vertices: New Tools for Kernelization , 2011, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[26]  Polish Ministry Subset feedback vertex set is fixed-parameter tractable , 2011 .

[27]  Maria Chudnovsky,et al.  An algorithm for packing non-zero A-paths in group-labelled graphs , 2008, Comb..

[28]  Subhash Khot On the power of unique 2-prover 1-round games , 2002, STOC '02.

[29]  J. Oxley Matroid Theory (Oxford Graduate Texts in Mathematics) , 2006 .

[30]  Vladimir Kolmogorov,et al.  The Power of Linear Programming for Finite-Valued CSPs: A Constructive Characterization , 2013, ICALP.

[31]  Vladimir Kolmogorov,et al.  Generalized roof duality and bisubmodular functions , 2010, Discret. Appl. Math..

[32]  André Bouchet,et al.  Coverings and Delta-Coverings , 1995, IPCO.

[33]  André Bouchet,et al.  Multimatroids I. Coverings by Independent Sets , 1997, SIAM J. Discret. Math..

[34]  Stéphan Thomassé,et al.  A 4k2 kernel for feedback vertex set , 2010, TALG.

[35]  Satoru Iwata,et al.  A combinatorial strongly polynomial algorithm for minimizing submodular functions , 2001, JACM.

[36]  André Bouchet,et al.  Multimatroids III. Tightness and Fundamental Graphs , 2001, Eur. J. Comb..

[37]  André Bouchet,et al.  Multimatroids IV. Chain-group representations , 1998 .

[38]  Yoichi Iwata,et al.  Linear-Time FPT Algorithms via Network Flow , 2013, SODA.

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

[40]  Hans L. Bodlaender,et al.  A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC.

[41]  Stanislav Zivny,et al.  The Power of Linear Programming for Valued CSPs , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[42]  Peter Jonsson,et al.  Min CSP on Four Elements: Moving beyond Submodularity , 2011, CP.

[43]  William H. Cunningham,et al.  Delta-Matroids, Jump Systems, and Bisubmodular Polyhedra , 1995, SIAM J. Discret. Math..

[44]  Kamal Jain,et al.  A Factor 2 Approximation Algorithm for the Generalized Steiner Network Problem , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[45]  Paul Wollan,et al.  Non-zero disjoint cycles in highly connected group labeled graphs , 2005, Electron. Notes Discret. Math..

[46]  André Bouchet,et al.  Multimatroids II. Orthogonality, minors and connectivity , 1997, Electron. J. Comb..

[47]  Michal Pilipczuk,et al.  On Multiway Cut Parameterized above Lower Bounds , 2011, IPEC.

[48]  Satoru Iwata,et al.  Bisubmodular Function Minimization , 2001, IPCO.

[49]  Satoru Iwata,et al.  Submodular Function Minimization under Covering Constraints , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[50]  Yutaro Yamaguchi,et al.  MATHEMATICAL ENGINEERING TECHNICAL REPORTS Packing A-paths in Group-Labelled Graphs via Linear Matroid Parity , 2013 .

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

[52]  Martin C. Cooper,et al.  Characterising Tractable Constraints , 1994, Artif. Intell..

[53]  Prasad Raghavendra,et al.  Optimal algorithms and inapproximability results for every CSP? , 2008, STOC.

[54]  Johan Thapper Linear Programming and the Complexity of Finite-valued CSPs ∗ , 2013 .