Approximately counting locally-optimal structures

In general, constructing a locally-optimal structure is a little harder than constructing an arbitrary structure, but significantly easier than constructing a globally-optimal structure. A similar situation arises in listing. In counting, most problems are #P-complete, but in approximate counting we observe an interesting reversal of the pattern. Assuming that #BIS is not equivalent to #SAT under AP-reductions, we show that counting maximal independent sets in bipartite graphs is harder than counting maximum independent sets. Motivated by this, we show that various counting problems involving minimal separators are #SAT-hard to approximate. These problems have applications for constructing triangulations and phylogenetic trees. Approximately counting maximal independent sets in a bipartite graph is #SAT-hard.Replacing "maximal" by "maximum" yields #MAX-BIS, which is conjectured to be easier.Normally "maximum" problems are harder to solve than "maximal" problems, not easier.Approximately counting minimal separators in a bipartite graph is also #SAT-hard.How often is the usual pattern reversed in approximate counting?

[1]  Shuji Tsukiyama,et al.  A New Algorithm for Generating All the Maximal Independent Sets , 1977, SIAM J. Comput..

[2]  Rob Gysel.  Unique Perfect Phylogeny Characterizations via Uniquely Representable Chordal Graphs , 2013, ArXiv.

[3]  R. Diestel.  Extremal Graph Theory , 2017 .

[4]  Yoshio Okamoto,et al.  Dominating Set Counting in Graph Classes , 2011, COCOON.

[5]  Swastik Kopparty,et al.  TO PLANAR GRAPHS , 2010 .

[6]  Rob Gysel,et al.  Minimal Triangulation Algorithms for Perfect Phylogeny Problems , 2014, LATA.

[7]  Fedor V. Fomin,et al.  Tree decompositions with small cost , 2002, Discret. Appl. Math..

[8]  Fedor V. Fomin,et al.  Exact algorithms for treewidth and minimum fill-in ∗ † , 2006 .

[9]  Leslie Ann Goldberg,et al.  Approximately Counting Locally-Optimal Structures , 2015, ICALP.

[10]  Dimitrios M. Thilikos,et al.  On exact algorithms for treewidth , 2006, TALG.

[11]  Patrick Cousot,et al.  Theories, solvers and static analysis by abstract interpretation , 2012, JACM.

[12]  Fedor V. Fomin,et al.  Finding Induced Subgraphs via Minimal Triangulations , 2009, STACS.

[13]  Salil P. Vadhan,et al.  The Complexity of Counting in Sparse, Regular, and Planar Graphs , 2002, SIAM J. Comput..

[14]  Leslie Ann Goldberg,et al.  Approximating the partition function of the ferromagnetic Potts model , 2010, JACM.

[15]  Henning Bruhn,et al.  The graph formulation of the union-closed sets conjecture , 2015, Eur. J. Comb..

[16]  Leslie Ann Goldberg,et al.  Approximating the Partition Function of the Ferromagnetic Potts Model , 2010, ICALP.

[17]  Anne Berry,et al.  Generating All the Minimal Separators of a Graph , 1999, Int. J. Found. Comput. Sci..

[18]  Ioan Todinca,et al.  Treewidth and Minimum Fill-in: Grouping the Minimal Separators , 2001, SIAM J. Comput..

[19]  Martin E. Dyer,et al.  The Relative Complexity of Approximate Counting Problems , 2000, Algorithmica.

[20]  Dieter Kratsch,et al.  Listing All Minimal Separators of a Graph , 1998, SIAM J. Comput..

[21]  Mihalis Yannakakis,et al.  How easy is local search? , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[22]  Weifa Liang,et al.  Efficient Enumeration of all Minimal Separators in a Graph , 1997, Theor. Comput. Sci..

[23]  Fedor V. Fomin,et al.  Treewidth computation and extremal combinatorics , 2008, Comb..

[24]  Kate Cummings,et al.  Introduction to the Theory , 2015 .

[25]  Ken Takata,et al.  Space-optimal, backtracking algorithms to list the minimal vertex separators of a graph , 2010, Discret. Appl. Math..

[26]  Ioan Todinca,et al.  Listing all potential maximal cliques of a graph , 2000, Theor. Comput. Sci..

[27]  Daniel Lokshtanov.  On the complexity of computing treelength , 2010, Discret. Appl. Math..

[28]  Koichi Yamazaki,et al.  A revisit of the scheme for computing treewidth and minimum fill-in , 2014, Theor. Comput. Sci..

[29]  Leslie Ann Goldberg,et al.  The Complexity of Approximately Counting Tree Homomorphisms , 2013, TOCT.

[30]  Sumio Masuda,et al.  Generation of Maximum Independent Sets of a Bipartite Graph and Maximum Cliques of a Circular-Arc Graph , 1992, J. Algorithms.

[31]  Pierluigi Crescenzi,et al.  A short guide to approximation preserving reductions , 1997, Proceedings of Computational Complexity. Twelfth Annual IEEE Conference.

[32]  David Zuckerman,et al.  On Unapproximable Versions of NP-Complete Problems , 1996, SIAM J. Comput..

[33]  Reinhard Diestel,et al.  Graph Theory, 4th Edition , 2012, Graduate texts in mathematics.

[34]  Arie M. C. A. Koster,et al.  Combinatorial Optimization on Graphs of Bounded Treewidth , 2008, Comput. J..

[35]  C. Papadimitriou,et al.  Introduction to the Theory of Computation , 2018 .

[36]  Leslie Ann Goldberg,et al.  Efficient algorithms for listing combinatorial structures , 1993 .

[37]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[38]  Leslie Ann Goldberg,et al.  The Complexity of Ferromagnetic Ising with Local Fields , 2006, Combinatorics, Probability and Computing.

[39]  Mihalis Yannakakis,et al.  Simple Local Search Problems That are Hard to Solve , 1991, SIAM J. Comput..

[40]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[41]  Michael Sipser,et al.  Introduction to the Theory of Computation , 1996, SIGA.