Algebra and the Complexity of Digraph CSPs: a Survey

We present a brief survey of some of the key results on the interplay between algebraic and graph-theoretic methods in the study of the complexity of digraph-based constraint satisfaction problems. 1998 ACM Subject Classification A.1. Introductory and Survey; D.3.2 Constraint and Logic Languages; F.2.2 Computations on Discrete Structures.

[1]  R. Willard,et al.  Characterizations of several Maltsev conditions , 2015 .

[2]  Alexandr Kazda,et al.  Maltsev digraphs have a majority polymorphism , 2009, Eur. J. Comb..

[3]  Jaroslav Nesetril,et al.  Complexity of Tree Homomorphisms , 1996, Discret. Appl. Math..

[4]  Libor Barto,et al.  Csp Dichotomy for Special Polyads , 2013, Int. J. Algebra Comput..

[5]  Pavol Hell,et al.  Monotone Proper Interval Digraphs and Min-Max Orderings , 2012, SIAM J. Discret. Math..

[6]  Siam J. Discrete.  CLASSIFICATION OF HOMOMORPHISMS TO ORIENTED CYCLES AND OF k -PARTITE SATISFIABILITY ∗ , 2001 .

[7]  A. Mackay.  On complexity , 2001 .

[8]  Benoît Larose,et al.  List-Homomorphism Problems on Graphs and Arc Consistency , 2012, 2012 IEEE 42nd International Symposium on Multiple-Valued Logic.

[9]  Jaroslav Nesetril,et al.  On the complexity of H-coloring , 1990, J. Comb. Theory, Ser. B.

[10]  Libor Barto,et al.  The collapse of the bounded width hierarchy , 2016, J. Log. Comput..

[11]  Gerhard J. Woeginger,et al.  Polynomial Graph-Colorings , 1989, STACS.

[12]  M. Maróti,et al.  Existence theorems for weakly symmetric operations , 2008 .

[13]  Libor Barto,et al.  The CSP Dichotomy Holds for Digraphs with No Sources and No Sinks (A Positive Answer to a Conjecture of Bang-Jensen and Hell) , 2008, SIAM J. Comput..

[14]  Pavol Hell,et al.  Bi-Arc Digraphs and Conservative Polymorphisms , 2016, ArXiv.

[15]  Andrei A. Bulatov,et al.  Complexity of conservative constraint satisfaction problems , 2011, TOCL.

[16]  Pavol Hell,et al.  Descriptive Complexity of List H-Coloring Problems in Logspace: A Refined Dichotomy , 2015, 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science.

[17]  Benoît Larose,et al.  A polynomial-time algorithm for near-unanimity graphs , 2005, J. Algorithms.

[18]  Gary MacGillivray,et al.  The Complexity of Colouring by Semicomplete Digraphs , 1988, SIAM J. Discret. Math..

[19]  Libor Barto,et al.  The Dichotomy for Conservative Constraint Satisfaction Problems Revisited , 2011, 2011 IEEE 26th Annual Symposium on Logic in Computer Science.

[20]  A. Wires.  A quasi-Mal’cev condition with unexpected application , 2015 .

[21]  Andrei A. Bulatov,et al.  Recent Results on the Algebraic Approach to the CSP , 2008, Complexity of Constraints.

[22]  Claude Tardif,et al.  Graphs Admitting k-NU Operations. Part 1: The Reflexive Case , 2013, SIAM J. Discret. Math..

[23]  Ralph McKenzie.  Monotone clones, residual smallness and congruence distributivity , 1990, Bulletin of the Australian Mathematical Society.

[24]  Narayan Vikas,et al.  Computational Complexity Classification of Partition under Compaction and Retraction , 2004, COCOON.

[25]  Andrei A. Krokhin,et al.  Two new homomorphism dualities and lattice operations , 2011, J. Log. Comput..

[26]  Libor Barto,et al.  THE CONSTRAINT SATISFACTION PROBLEM AND UNIVERSAL ALGEBRA , 2015, The Bulletin of Symbolic Logic.

[27]  D. Hobby,et al.  The structure of finite algebras , 1988 .

[28]  Gustav Nordh,et al.  Retractions to Pseudoforests , 2010, SIAM J. Discret. Math..

[29]  Pavol Hell,et al.  List Homomorphisms to Reflexive Graphs , 1998, J. Comb. Theory, Ser. B.

[30]  Todd Niven,et al.  Complexity and polymorphisms for digraph constraint problems under some basic constructions , 2016, Int. J. Algebra Comput..

[31]  Gary MacGillivray,et al.  The complexity of colouring by locally semicomplete digraphs , 2010, Discret. Math..

[32]  Andrei A. Krokhin,et al.  Retractions onto series-parallel posets , 2008, Discret. Math..

[33]  Benoît Larose,et al.  Algebraic properties and dismantlability of finite posets , 1997, Discret. Math..

[34]  Jørgen Bang-Jensen,et al.  The effect of two cycles on the complexity of colourings by directed graphs , 1989, Discret. Appl. Math..

[35]  Cynthia Loten.  Retractions of chordal and related graphs , 2003 .

[36]  Pascal Tesson,et al.  The Complexity of the List Homomorphism Problem for Graphs , 2011, Theory of Computing Systems.

[37]  Pavol Hell,et al.  The dichotomy of list homomorphisms for digraphs , 2011, SODA '11.

[38]  Pawel M. Idziak,et al.  Tractability and Learnability Arising from Algebras with Few Subpowers , 2010, SIAM J. Comput..

[39]  Pavol Hell,et al.  Semilattice polymorphisms and chordal graphs , 2014, Eur. J. Comb..

[40]  Neil Immerman,et al.  The complexity of satisfiability problems: Refining Schaefer's theorem , 2009, J. Comput. Syst. Sci..

[41]  Libor Barto,et al.  Near Unanimity Constraints Have Bounded Pathwidth Duality , 2012, 2012 27th Annual IEEE Symposium on Logic in Computer Science.

[42]  R. McKenzie,et al.  Varieties with few subalgebras of powers , 2009 .

[43]  D. Stanovský.  POLYMORPHISMS OF SMALL DIGRAPHS , 2010 .

[44]  T. Feder.  List homomorphisms and retractions to reflexive digraphs , 2007 .

[45]  Libor Barto,et al.  Absorbing Subalgebras, Cyclic Terms, and the Constraint Satisfaction Problem , 2012, Log. Methods Comput. Sci..

[46]  Tomás Feder,et al.  The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory , 1999, SIAM J. Comput..

[47]  A. Lemaître.  Complexité des homomorphismes de graphes avec listes , 2012 .

[48]  Miklós Maróti,et al.  Reflexive digraphs with near unanimity polymorphisms , 2012, Discret. Math..

[49]  Alexandr Kazda,et al.  CSP for binary conservative relational structures , 2011, 1112.1099.

[50]  Claude Tardif,et al.  A discrete homotopy theory for binary reflexive structures , 2004 .

[51]  B. Barak,et al.  Computational Complexity: Complexity of counting , 2009 .

[52]  R. McKenzie,et al.  Maltsev families of varieties closed under join or Maltsev product , 2017 .

[53]  Peter Jeavons,et al.  Classifying the Complexity of Constraints Using Finite Algebras , 2005, SIAM J. Comput..

[54]  Pascal Tesson,et al.  Universal algebra and hardness results for constraint satisfaction problems , 2007, Theor. Comput. Sci..

[55]  Justin Pearson,et al.  Closure Functions and Width 1 Problems , 1999, CP.

[56]  Xuding Zhu,et al.  The Existence of Homomorphisms to Oriented Cycles , 1995, SIAM J. Discret. Math..

[57]  Jaroslav Nesetril,et al.  Duality Theorems for Finite Structures (Characterising Gaps and Good Characterisations) , 2000, J. Comb. Theory, Ser. B.

[58]  M. Siggers.  DISTRIBUTIVE LATTICE POLYMORPHISM ON REFLEXIVE GRAPHS , 2014, 1411.7879.

[59]  Libor Barto,et al.  CSP dichotomy for special triads , 2009 .

[60]  Richard C. Brewster,et al.  Near-Unanimity Functions and Varieties of Reflexive Graphs , 2008, SIAM J. Discret. Math..

[61]  Benoît Larose,et al.  The Complexity of the Extendibility Problem for Finite Posets , 2003, SIAM J. Discret. Math..

[62]  Andrei A. Bulatov.  H-Coloring dichotomy revisited , 2005, Theor. Comput. Sci..

[63]  Víctor Dalmau,et al.  Maltsev + Datalog --≫ Symmetric Datalog , 2008, 2008 23rd Annual IEEE Symposium on Logic in Computer Science.

[64]  Alexandr Kazda,et al.  $n$-permutability and linear Datalog implies symmetric Datalog , 2015, Log. Methods Comput. Sci..

[65]  Peter Jeavons,et al.  On the Algebraic Structure of Combinatorial Problems , 1998, Theor. Comput. Sci..

[66]  B. Larose,et al.  Bounded width problems and algebras , 2007 .

[67]  Pavol Hell,et al.  Space complexity of list H-colouring: a dichotomy , 2013, SODA.

[68]  Todd Niven,et al.  A finer reduction of constraint problems to digraphs , 2014, Log. Methods Comput. Sci..

[69]  Jakub Bulin.  2 O ct 2 01 7 On the complexity of H-coloring for special oriented trees , 2018 .

[70]  Claude Tardif,et al.  A Characterisation of First-Order Constraint Satisfaction Problems , 2006, 21st Annual IEEE Symposium on Logic in Computer Science (LICS'06).

[71]  Libor Barto,et al.  Constraint Satisfaction Problems Solvable by Local Consistency Methods , 2014, JACM.

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

[73]  Hubie Chen,et al.  Asking the Metaquestions in Constraint Tractability , 2016, TOCT.

[74]  Claude Tardif,et al.  Graphs Admitting k-NU Operations. Part 2: The Irreflexive Case , 2014, SIAM J. Discret. Math..

[75]  Andrei A. Bulatov,et al.  Dualities for Constraint Satisfaction Problems , 2008, Complexity of Constraints.

[76]  Mario Szegedy,et al.  A new line of attack on the dichotomy conjecture , 2009, STOC '09.

[77]  Pavol Hell,et al.  Bi-arc graphs and the complexity of list homomorphisms , 2003, J. Graph Theory.

[78]  Andrei A. Krokhin,et al.  First-Order Definable Retraction Problems for Posets and Reflexive Graph , 2004, LICS.