Constraint Satisfaction Problems over Numeric Domains

We present a survey of complexity results for constraint satisfaction problems (CSPs) over the integers, the rationals, the reals, and the complex numbers. Examples of such problems are feasibility of linear programs, integer linear programming, the max-atoms problem, Hilbert’s tenth problem, and many more. Our particular focus is to identify those CSPs that can be solved in polynomial time, and to distinguish them from CSPs that are NP-hard. A very helpful tool for obtaining complexity classifications in this context is the concept of a polymorphism from universal algebra. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems

[1]  Manuel Bodirsky,et al.  The Complexity of Equality Constraint Languages , 2006, CSR.

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

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

[4]  James Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part II: The General Decision Problem. Preliminaries for Quantifier Elimination , 1992, J. Symb. Comput..

[5]  Mikhail N. Vyalyi,et al.  Semidefinite programming and arithmetic circuit evaluation , 2005, Discret. Appl. Math..

[6]  Peter Jonsson,et al.  Essential Convexity and Complexity of Semi-Algebraic Constraints , 2012, Log. Methods Comput. Sci..

[7]  Peter Jonsson,et al.  Computational complexity of linear constraints over the integers , 2013, Artif. Intell..

[8]  V. Klee,et al.  HOW GOOD IS THE SIMPLEX ALGORITHM , 1970 .

[9]  A. Ehrenfeucht,et al.  Positional strategies for mean payoff games , 1979 .

[10]  L. G. H. Cijan.  A polynomial algorithm in linear programming , 1979 .

[11]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, STOC '84.

[12]  Pascal Koiran.  Hilbert's Nullstellensatz Is in the Polynomial Hierarchy , 1996, J. Complex..

[13]  James P. Jones.  Universal Diophantine Equation , 1982, J. Symb. Log..

[14]  Dean Gillette,et al.  9. STOCHASTIC GAMES WITH ZERO STOP PROBABILITIES , 1958 .

[15]  Yuri Matiyasevich,et al.  Hilbert’s tenth problem , 2019, 100 Years of Math Milestones.

[16]  Hendrik W. Lenstra,et al.  Integer Programming with a Fixed Number of Variables , 1983, Math. Oper. Res..

[17]  Sergei G. Vorobyov,et al.  Cyclic games and linear programming , 2008, Discret. Appl. Math..

[18]  Ya'acov Peterzil,et al.  Reducts of some structures over the reals , 1993, Journal of Symbolic Logic.

[19]  Claus Scheiderer,et al.  Semidefinite Representation for Convex Hulls of Real Algebraic Curves , 2012, SIAM J. Appl. Algebra Geom..

[20]  Jeffrey C. Lagarias,et al.  The computational complexity of simultaneous Diophantine approximation problems , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[21]  Micha Sharir,et al.  A subexponential bound for linear programming , 1992, SCG '92.

[22]  Oliver Friedmann,et al.  A Subexponential Lower Bound for Zadeh's Pivoting Rule for Solving Linear Programs and Games , 2011, IPCO.

[23]  Peter Jonsson,et al.  Constraint satisfaction and semilinear expansions of addition over the rationals and the reals , 2015, J. Comput. Syst. Sci..

[24]  Christian Glaßer,et al.  Circuit satisfiability and constraint satisfaction around Skolem Arithmetic , 2016, Theor. Comput. Sci..

[25]  George E. Collins,et al.  Algorithms for the Solution of Systems of Linear Diophantine Equations , 1982, SIAM J. Comput..

[26]  J. Helton,et al.  Linear matrix inequality representation of sets , 2003, math/0306180.

[27]  Leonard M. Adleman,et al.  NP-Complete Decision Problems for Binary Quadratics , 1978, J. Comput. Syst. Sci..

[28]  Manuel Bodirsky,et al.  The complexity of temporal constraint satisfaction problems , 2008, STOC.

[29]  Manuel Bodirsky,et al.  Constraint Satisfaction Problems over Numeric Domains , 2017 .

[30]  Nimrod Megiddo,et al.  Towards a Genuinely Polynomial Algorithm for Linear Programming , 1983, SIAM J. Comput..

[31]  Éva Tardos,et al.  A Strongly Polynomial Algorithm to Solve Combinatorial Linear Programs , 1986, Oper. Res..

[32]  J. William Helton,et al.  Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and Sets , 2007, SIAM J. Optim..

[33]  Anand Pillay,et al.  Reducts of (C, +, *) which Contain + , 1990, J. Symb. Log..

[34]  Rolf H. Möhring,et al.  Scheduling with AND/OR Precedence Constraints , 2004, SIAM J. Comput..

[35]  S. Lippman,et al.  Stochastic Games with Perfect Information and Time Average Payoff , 1969 .

[36]  William Y. Sit,et al.  Worst case behavior of the steepest edge simplex method , 1979, Discret. Appl. Math..

[37]  Barnaby Martin,et al.  Constraint Satisfaction Problems over the Integers with Successor , 2015, ICALP.

[38]  Uri Zwick,et al.  The Complexity of Mean Payoff Games on Graphs , 1996, Theor. Comput. Sci..

[39]  Uri Zwick,et al.  An Improved Version of the Random-Facet Pivoting Rule for the Simplex Algorithm , 2015, STOC.

[40]  Henry A. Kautz,et al.  Constraint propagation algorithms for temporal reasoning: a revised report , 1989 .

[41]  Manuel Bodirsky,et al.  Reducts of finitely bounded homogeneous structures, and lifting tractability from finite-domain constraint satisfaction , 2016, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[42]  Marcus Schaefer,et al.  Realizability of Graphs and Linkages , 2013 .

[43]  J. William Helton,et al.  Semidefinite representation of convex sets , 2007, Math. Program..

[44]  James Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part I: Introduction. Preliminaries. The Geometry of Semi-Algebraic Sets. The Decision Problem for the Existential Theory of the Reals , 1992, J. Symb. Comput..

[45]  M. H. Wright.  The interior-point revolution in optimization: History, recent developments, and lasting consequences , 2004 .

[46]  Martin C. Cooper,et al.  Tractable Constraints on Ordered Domains , 1995, Artif. Intell..

[47]  Anne Condon,et al.  The Complexity of Stochastic Games , 1992, Inf. Comput..

[48]  Bernhard Nebel,et al.  Reasoning about temporal relations: a maximal tractable subclass of Allen's interval algebra , 1994, JACM.

[49]  Xin-She Yang,et al.  Introduction to Algorithms , 2021, Nature-Inspired Optimization Algorithms.

[50]  James Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part III: Quantifier Elimination , 1992, J. Symb. Comput..

[51]  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..

[52]  Jeanne Ferrante,et al.  A Decision Procedure for the First Order Theory of Real Addition with Order , 1975, SIAM J. Comput..

[53]  Manolis Koubarakis,et al.  Tractable disjunctions of linear constraints: basic results and applications to temporal reasoning , 2001, Theor. Comput. Sci..

[54]  Uri Zwick,et al.  Subexponential lower bounds for randomized pivoting rules for the simplex algorithm , 2011, STOC '11.

[55]  P. Cameron.  Transitivity of permutation groups on unordered sets , 1976 .

[56]  W. Böge,et al.  Quantifier Elimination for Real Closed Fields , 1985, AAECC.

[57]  Joseph Naor,et al.  Simple and Fast Algorithms for Linear and Integer Programs With Two Variables per Inequality , 1994, SIAM J. Comput..

[58]  Manuel Bodirsky,et al.  Constraint satisfaction tractability from semi-lattice operations on infinite sets , 2013, TOCL.

[59]  Ju. V. Matijasevic,et al.  ENUMERABLE SETS ARE DIOPHANTINE , 2003 .

[60]  Ronald L. Graham,et al.  Some NP-complete geometric problems , 1976, STOC '76.

[61]  A. Tarski.  A Decision Method for Elementary Algebra and Geometry , 2023 .

[62]  Gil Kalai,et al.  A subexponential randomized simplex algorithm (extended abstract) , 1992, STOC '92.

[63]  Yu. V. Matijasevič.  Some Purely Mathematical Results Inspired by Mathematical Logic , 1977 .

[64]  Ya'acov Peterzil,et al.  Additive reducts of real closed fields , 1992, Journal of Symbolic Logic (JSL).

[65]  Alexandra Shlapentokh.  Hilbert's Tenth Problem: Diophantine Classes and Extensions to Global Fields , 2006 .

[66]  Manuel Bodirsky,et al.  Max-Closed Semilinear Constraint Satisfaction , 2016, CSR.

[67]  J. Kratochvil,et al.  Intersection Graphs of Segments , 1994, J. Comb. Theory, Ser. B.

[68]  Barnaby Martin,et al.  On the Scope of the Universal-Algebraic Approach to Constraint Satisfaction , 2010, LICS.

[69]  David K. Smith.  Theory of Linear and Integer Programming , 1987 .

[70]  John F. Canny,et al.  Some algebraic and geometric computations in PSPACE , 1988, STOC '88.

[71]  Robert G. Jeroslow,et al.  The simplex algorithm with the pivot rule of maximizing criterion improvement , 1973, Discret. Math..

[72]  Motakuri V. Ramana,et al.  An exact duality theory for semidefinite programming and its complexity implications , 1997, Math. Program..

[73]  Albert Atserias,et al.  Mean-Payoff Games and the Max-Atom Problem , 2009 .

[74]  Igor Klep,et al.  An Exact Duality Theory for Semidefinite Programming Based on Sums of Squares , 2012, Math. Oper. Res..

[75]  Gary A. Martin.  Definability in Reducts of Algebraically Closed Fields , 1988, J. Symb. Log..

[76]  Ravi Kannan,et al.  Polynomial Algorithms for Computing the Smith and Hermite Normal Forms of an Integer Matrix , 1979, SIAM J. Comput..

[77]  Marcus Schaefer,et al.  Fixed Points, Nash Equilibria, and the Existential Theory of the Reals , 2017, Theory of Computing Systems.

[78]  Nir Halman,et al.  Simple Stochastic Games, Parity Games, Mean Payoff Games and Discounted Payoff Games Are All LP-Type Problems , 2007, Algorithmica.

[79]  J. M. Bilbao,et al.  Contributions to the Theory of Games , 2005 .

[80]  Christer Bäckström,et al.  A Unifying Approach to Temporal Constraint Reasoning , 1998, Artif. Intell..

[81]  Dima Grigoriev,et al.  Tropical Effective Primary and Dual Nullstellensätze , 2014, Discret. Comput. Geom..

[82]  Nimrod Megiddo,et al.  Linear Programming in Linear Time When the Dimension Is Fixed , 1984, JACM.

[83]  Charles Steinhorn,et al.  Tame Topology and O-Minimal Structures , 2008 .