Algorithm 4 Matroid Intersection

Input:Two matroids ${\mathcal{M}}_1=(V,{\mathcal{I}}_1)$, ${\mathcal{M}}_2=(V,{\mathcal{I}}_2)$, distance metric d, initial set $S_0\subseteq V$ Output: A maximum cardinality set $S\subseteq V$ in ${\mathcal{I}}_1\cap {\mathcal{I}}_2$ 1:Initialize $S\leftarrow S_0$, $V_1=\{x\in V\setminus S:S\cup \left\{x\right\}\in {\mathcal{I}}_1\}$, and $V_2=\{x\in V\setminus S:S\cup \left\{x\right\}\in {\mathcal{I}}_2\}$ 2:while $V_1\cap V_2\ne \varnothing$ do 3:    $x^{*}\leftarrow {arg\; max}_{x\in V_1\cap V_2}d(x,S)$ and $S\leftarrow S\cup \left\{x^{*}\right\}$ 4:    for all $x\in V_1$ do 5:        $V_1\leftarrow V_1\setminus \left\{x\right\}$ if $S\cup \left\{x\right\}\notin {\mathcal{I}}_1$ 6:    for all $x\in V_2$ do 7:        $V_2\leftarrow V_2\setminus \left\{x\right\}$ if $S\cup \left\{x\right\}\notin {\mathcal{I}}_2$ 8:Build an augmentation graph G for S 9:while there is a directed path from a to b in G do 10:    Let $P^{*}$ be a shortest path from a to b in G 11:    for all $x\in P^{*}\setminus \{a,b\}$ do 12:        $S\leftarrow S\cup \left\{x\right\}$ if $x\notin S$ 13:        $S\leftarrow S\setminus \left\{x\right\}$ if $x\in S$ 14:    Rebuild G for the updated S 15:returnS