A polynomial Time Algorithm to Solve The Max-atom Problem
| Autor: | Lahlou, Chams, Truffet, Laurent |
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| Rok vydání: | 2021 |
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| Druh dokumentu: | Working Paper |
| Popis: | In this paper we consider $m$ ($m \geq 1$)conjunctions of Max-atoms that is atoms of the form $\max(z,y) + r \geq x$, where the offset $r$ is a real constant and $x,y,z$ are variables. We show that the Max-atom problem (MAP) belongs to $\textsf{P}$. Indeed, we provide an algorithm which solves the MAP in $O(n^{6} m^{2} + n^{4} m^{3} + n^{2} m^{4})$ operations, where $n$ is the number of variables which compose the max-atoms. As a by-product other problems also known to be in $\textsf{NP} \cap \textsf{co-NP}$ are in $\textsf{P}$. P1: the problem to know if a tropical cone is trivial or not. P2: problem of tropical rank of a tropical matrix. P3: parity game problem. P4: scheduling problem with AND/OR precedence constraints. P5: problem on hypergraph (shortest path). P6: problem in model checking and $\mu$-calculus. Comment: We thank Tom Van Dijk who pointed out errors in the proposed algorithm. We have proposed another approach in the paper entitled "Looking for all solutions of the Max Atom Problem (MAP)" see: arXiv:2408.14256 |
| Databáze: | arXiv |
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