A new line of attack on the dichotomy conjecture

The well known dichotomy conjecture of Feder and Vardi states that for every finite family ? of constraints CSP ( ? ) is either polynomially solvable or N P -hard. Bulatov and Jeavons reformulated this conjecture in terms of the properties of the algebra P o l ( ? ) , where the latter is the collection of those n -ary operations ( n = 1 , 2 , ? ) that keep all constraints in ? invariant. We show that the algebraic condition boils down to whether there are arbitrarily resilient functions in P o l ( ? ) . Equivalently, we can express this in terms of the PCP theory: CSP ( ? ) is N P -hard iff every long code test created from ? that passes with zero error admits only juntas.33For us "junta" means that a constant number of the variables have constant influence on the outcome. Also, ? need to be a core (see Definition?8). Then, using this characterization and a result of Dinur, Friedgut and Regev, we give an entirely new and transparent proof to the Hell-Nesetřil theorem, which states that for a simple, connected and undirected graph H , the problem CSP ( H ) is N P -hard if and only if H is non-bipartite.We also introduce another notion of resilience (we call it strong resilience), and we use it in the investigation of CSP problems that 'do not have the ability to count'. We show that CSP problems without the ability to count are exactly the ones with strongly resilient term operations. This gave already a handier tool to attack the conjecture that CSP problems without the ability to count have bounded width, or equivalently, that they can be characterized by existential k -pebble games: Barto and Kozik already proved this conjecture using a variant of our characterization. This is considered a major step towards the resolution of the dichotomy conjecture.Finally, we show that a yet stronger notion of resilience, when the term operation is asymptotically constant, holds for the class of width one CSPs.What emerges from our research, is that certain important algebraic conditions that are usually expressed via identities have equivalent analytic definitions that rely on asymptotic properties of term operations.

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

[2]  Peter Jeavons,et al.  Constraint Satisfaction Problems and Finite Algebras , 2000, ICALP.

[3]  Xuding Zhu,et al.  Duality and Polynomial Testing of Tree Homomorphisms , 1996 .

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

[5]  Albert Atserias,et al.  On digraph coloring problems and treewidth duality , 2005, 20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05).

[6]  Gábor Kun,et al.  Constraints, MMSNP and expander relational structures , 2007, Combinatorica.

[7]  Pawel M. Idziak,et al.  Tractability and learnability arising from algebras with few subpowers , 2007, 22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007).

[8]  M. Valeriote.  A Subalgebra Intersection Property for Congruence Distributive Varieties , 2009, Canadian Journal of Mathematics.

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

[10]  Andrei A. Bulatov,et al.  A dichotomy theorem for constraints on a three-element set , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

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

[12]  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..

[13]  Jaroslav Nesetril,et al.  Forbidden lifts (NP and CSP for combinatorialists) , 2007, Eur. J. Comb..

[14]  Emil W. Kiss,et al.  An Easy Way to Minimal Algebras , 1997, Int. J. Algebra Comput..

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

[16]  Benoît Larose,et al.  Omitting Types, Bounded Width and the Ability to Count , 2009, Int. J. Algebra Comput..

[17]  Andrei A. Bulatov,et al.  A dichotomy theorem for constraint satisfaction problems on a 3-element set , 2006, JACM.

[18]  Libor Barto,et al.  Robust satisfiability of constraint satisfaction problems , 2012, STOC '12.

[19]  Jaroslav Nesetril,et al.  Generalised dualities and maximal finite antichains in the homomorphism order of relational structures , 2008, Eur. J. Comb..

[20]  Emil W. Kiss.  AN INTRODUCTION TO TAME CONGRUENCE THEORY , 1997 .

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

[22]  Phokion G. Kolaitis,et al.  Conjunctive-query containment and constraint satisfaction , 1998, PODS.

[23]  Irit Dinur,et al.  Independent Sets in Graph Powers are Almost Contained in Juntas , 2008 .

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

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

[26]  Andrei A. Krokhin,et al.  Robust Satisfiability for CSPs: Hardness and Algorithmic Results , 2013, TOCT.

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

[28]  Andrei A. Bulatov,et al.  Towards a dichotomy theorem for the counting constraint satisfaction problem , 2007, Inf. Comput..

[29]  Marc Gyssens,et al.  Closure properties of constraints , 1997, JACM.

[30]  P. Jeavons.  Algebraic structures in combinatorial problems , 2001 .

[31]  Jaroslav Nesetril,et al.  Constraint Satisfaction with Countable Homogeneous Templates , 2003, J. Log. Comput..

[32]  László Zádori,et al.  Idempotent Totally Symmetric Operations on Finite Posets , 2001, Order.

[33]  A. Bulatov.  A graph of a relational structure and constraint satisfaction problems , 2004, LICS 2004.

[34]  Andrei A. Bulatov,et al.  A Simple Algorithm for Mal'tsev Constraints , 2006, SIAM J. Comput..

[35]  Jaroslav Nesetril,et al.  A combinatorial constraint satisfaction problem dichotomy classification conjecture , 2010, Eur. J. Comb..

[36]  Benjamin Rossman,et al.  Existential positive types and preservation under homomorphisms , 2005, 20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05).

[37]  Libor Barto,et al.  Constraint Satisfaction Problems of Bounded Width , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[38]  Andrei A. Bulatov,et al.  Tractable conservative constraint satisfaction problems , 2003, 18th Annual IEEE Symposium of Logic in Computer Science, 2003. Proceedings..

[39]  P. P. Pálfy.  Unary polynomials in algebras, I , 1984 .

[40]  Hubie Chen,et al.  Quantified constraint satisfaction and the polynomially generated powers property , 2008, ICALP.

[41]  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..

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

[43]  Andrei A. Bulatov,et al.  Mal'tsev constraints are tractable , 2002, Electron. Colloquium Comput. Complex..

[44]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

[45]  Phokion G. Kolaitis,et al.  A Game-Theoretic Approach to Constraint Satisfaction , 2000, AAAI/IAAI.

[46]  Richard E. Ladner,et al.  On the Structure of Polynomial Time Reducibility , 1975, JACM.

[47]  Jaroslav Nesetril,et al.  Colouring, constraint satisfaction, and complexity , 2008, Comput. Sci. Rev..