Maximum H-colourable subdigraphs and constraint optimization with arbitrary weights
Journal of Computer and System Sciences
The approximability of MAX CSP with fixed-value constraints
Journal of the ACM (JACM)
Introduction to the Maximum Solution Problem
Complexity of Constraints
Hard constraint satisfaction problems have hard gaps at location 1
Theoretical Computer Science
Note: The expressive power of binary submodular functions
Discrete Applied Mathematics
An algebraic theory of complexity for valued constraints: establishing a Galois connection
MFCS'11 Proceedings of the 36th international conference on Mathematical foundations of computer science
Min CSP on four elements: moving beyond submodularity
CP'11 Proceedings of the 17th international conference on Principles and practice of constraint programming
A new honeybee optimization for constraint reasoning: case of max-CSPs
KES'11 Proceedings of the 15th international conference on Knowledge-based and intelligent information and engineering systems - Volume Part II
An algebraic characterisation of complexity for valued constraint
CP'06 Proceedings of the 12th international conference on Principles and Practice of Constraint Programming
Approximability of integer programming with generalised constraints
CP'06 Proceedings of the 12th international conference on Principles and Practice of Constraint Programming
Journal of Computer and System Sciences
The complexity of surjective homomorphism problems-a survey
Discrete Applied Mathematics
On the complexity of submodular function minimisation on diamonds
Discrete Optimization
Ruling out polynomial-time approximation schemes for hard constraint satisfaction problems
CSR'07 Proceedings of the Second international conference on Computer Science: theory and applications
CP'12 Proceedings of the 18th international conference on Principles and Practice of Constraint Programming
A characterisation of the complexity of forbidding subproblems in binary Max-CSP
CP'12 Proceedings of the 18th international conference on Principles and Practice of Constraint Programming
The complexity of finite-valued CSPs
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Robust Satisfiability for CSPs: Hardness and Algorithmic Results
ACM Transactions on Computation Theory (TOCT)
| Hi-index | 0.00 |
In the maximum constraint satisfaction problem (MAX CSP), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given domain to the variables so as to maximize the number (or the total weight, for the weighted case) of satisfied constraints. This problem is NP-hard in general, and, therefore, it is natural to study how restricting the allowed types of constraints affects the approximability of the problem. It is known that every Boolean (that is, two-valued) MAX CSP with a finite set of allowed constraint types is either solvable exactly in polynomial time or else APX-complete (and hence can have no polynomial-time approximation scheme unless P=NP). It has been an open problem for several years whether this result can be extended to non-Boolean MAX CSP, which is much more difficult to analyze than the Boolean case. In this paper, we make the first step in this direction by establishing this result for MAX CSP over a three-element domain. Moreover, we present a simple description of all polynomial-time solvable cases of our problem. This description uses the well-known algebraic combinatorial property of supermodularity. We also show that every hard three-valued MAX CSP contains, in a certain specified sense, one of the two basic hard MAX CSPs which are the Maximum k-Colorable Subgraph problems for k=2,3.