Representing polynomial of ST-CONNECTIVITY
We show that the coefficients of the representing polynomial of any monotone Boolean function are the values of the M\"obius function of an atomistic lattice related to this function. Using this we determine the representing polynomial of any Boolean function corresponding to a ST-CONNECTIVITY...
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| Vydané v: | Discrete mathematics and theoretical computer science Ročník 25:2; číslo Combinatorics |
|---|---|
| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Discrete Mathematics & Theoretical Computer Science
29.04.2024
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| ISSN: | 1365-8050, 1365-8050 |
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| Abstract | We show that the coefficients of the representing polynomial of any monotone
Boolean function are the values of the M\"obius function of an atomistic
lattice related to this function. Using this we determine the representing
polynomial of any Boolean function corresponding to a ST-CONNECTIVITY problem
in acyclic quivers (directed acyclic multigraphs). Only monomials corresponding
to unions of paths have non-zero coefficients which are $(-1)^D$ where $D$ is
an easily computable function of the quiver corresponding to the monomial (it
is the number of plane regions in the case of planar graphs). We determine that
the number of monomials with non-zero coefficients for the two-dimensional $n
\times n$ grid connectivity problem is $2^{\Omega(n^2)}$. |
|---|---|
| AbstractList | We show that the coefficients of the representing polynomial of any monotone Boolean function are the values of the M\"obius function of an atomistic lattice related to this function. Using this we determine the representing polynomial of any Boolean function corresponding to a ST-CONNECTIVITY problem in acyclic quivers (directed acyclic multigraphs). Only monomials corresponding to unions of paths have non-zero coefficients which are $(-1)^D$ where $D$ is an easily computable function of the quiver corresponding to the monomial (it is the number of plane regions in the case of planar graphs). We determine that the number of monomials with non-zero coefficients for the two-dimensional $n \times n$ grid connectivity problem is $2^{\Omega(n^2)}$. We show that the coefficients of the representing polynomial of any monotone Boolean function are the values of the M\"obius function of an atomistic lattice related to this function. Using this we determine the representing polynomial of any Boolean function corresponding to a ST-CONNECTIVITY problem in acyclic quivers (directed acyclic multigraphs). Only monomials corresponding to unions of paths have non-zero coefficients which are $(-1)^D$ where $D$ is an easily computable function of the quiver corresponding to the monomial (it is the number of plane regions in the case of planar graphs). We determine that the number of monomials with non-zero coefficients for the two-dimensional $n \times n$ grid connectivity problem is $2^{\Omega(n^2)}$. |
| Author | Iraids, Jānis Smotrovs, Juris |
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| Snippet | We show that the coefficients of the representing polynomial of any monotone
Boolean function are the values of the M\"obius function of an atomistic
lattice... We show that the coefficients of the representing polynomial of any monotone Boolean function are the values of the M\"obius function of an atomistic lattice... |
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| Title | Representing polynomial of ST-CONNECTIVITY |
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