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Yu Cong
2025-11-21 00:07:46 +08:00
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@@ -30,7 +30,7 @@ for $e\in E-F^*$:\\
Consider this process on the lattice of flats. The set of flats of $M$ forms a geometric lattice. Take two flats $A,B$ that $A\subset B$ in the lattice and consider the sublattice between $A$ and $B$. This sublattice is exactly the lattice of flats of matroid $(M/A)\setminus (E-B)$. Consider this process on the lattice of flats. The set of flats of $M$ forms a geometric lattice. Take two flats $A,B$ that $A\subset B$ in the lattice and consider the sublattice between $A$ and $B$. This sublattice is exactly the lattice of flats of matroid $(M/A)\setminus (E-B)$.
\begin{lemma} \begin{lemma}
Consider the ideal utilizations $u^*(e)$. Consider the ideal utilizations $u^*(e)$ assigned by the above algorithm.
\begin{enumerate} \begin{enumerate}
\item $\sigma(M)\geq \sigma(M|F^*)$ \item $\sigma(M)\geq \sigma(M|F^*)$
\item $u^*(e)$ is unique even though the $F^*$ may not be unique. \item $u^*(e)$ is unique even though the $F^*$ may not be unique.
@@ -54,6 +54,7 @@ the last inequality follows from $\frac{c(E-F')}{r(E)-r(F')}\geq \frac{c(E-F^*)}
\frac{c(E)-c(F_1)-c(F_2)}{r(E)-r(F_1)-r(F_2)} \frac{c(E)-c(F_1)-c(F_2)}{r(E)-r(F_1)-r(F_2)}
\leq \sigma \leq \sigma
\] \]
...
\end{enumerate} \end{enumerate}
\end{proof} \end{proof}