Introduction to the theory of complexity
Introduction to the theory of complexity
Probabilistic checking of proofs and hardness of approximation problems
Probabilistic checking of proofs and hardness of approximation problems
The hardness of approximate optima in lattices, codes, and systems of linear equations
Journal of Computer and System Sciences - Special issue: papers from the 32nd and 34th annual symposia on foundations of computer science, Oct. 2–4, 1991 and Nov. 3–5, 1993
Some optimal inapproximability results
Journal of the ACM (JACM)
The Approximability of Constraint Satisfaction Problems
SIAM Journal on Computing
The complexity of solving equations over finite groups
Information and Computation
Satisfiability of Systems of Equations over Finite Monoids
MFCS '01 Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science
Inapproximability results for equations over finite groups
Theoretical Computer Science - Special issue on automata, languages and programming
The inapproximability of lattice and coding problems with preprocessing
Journal of Computer and System Sciences - Special issue on computational complexity 2002
Computational complexity questions related to finite monoids and semigroups
Computational complexity questions related to finite monoids and semigroups
Introduction to the Maximum Solution Problem
Complexity of Constraints
Approximability of the Maximum Solution Problem for Certain Families of Algebras
CSR '09 Proceedings of the Fourth International Computer Science Symposium in Russia on Computer Science - Theory and Applications
Approximability of integer programming with generalised constraints
CP'06 Proceedings of the 12th international conference on Principles and Practice of Constraint Programming
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In the maximum solution equation problem a collection of equations are given over some algebraic structure. The objective is to find an assignment to the variables in the equations such that all equations are satisfied and the sum of the variables is maximised. We give tight approximability results for the maximum solution equation problem when the equations are given over groups of the form Zp, where p is prime. We also prove that the weighted and unweighted versions of this problem have equal approximability thresholds. Furthermore, we show that the problem is equally hard to solve even if each equation is restricted to contain at most three variables and solvable in polynomial time if the equations are restricted to contain at most two variables. All of our results also hold for a generalised version of maximum solution equation where the elements of the group are mapped arbitrarily to non-negative integers in the objective function.