Primal-Dual Approximation
for
FEEDBACK VERTEX SET
revisited
GOAL: 2-approximation
using a different LP formulation
Goal
Suppose \(F\) is a FVS of G.
Then we know that \(G\setminus F\) has at most \(|V| - c(G)\) edges,
where \(c(G)\) denotes the number of
connected components of \(G\).
This means that \(F\) must have knocked out at least
\(|E| - \) \(\bigl(|V| - c(G) \bigr)\) edges.
The impact of a single vertex
Suppose we remove a single vertex \(v\).
In \(G \setminus v\), our renewed goal is to remove at least:
\(\bigl(|E|-\) \(d(v)\) \(\bigr)- \bigl(|V\)\(-1\)\(| - c(G\)\(-v\)\() \bigr)\) edges.
Recall that in \(G\) our goal was to remove at least:
\(|E| - \) \(\bigl(|V| - c(G) \bigr)\) edges.
So removing \(v\) from \(G\) reduces our goal by:
\(d(v) - 1 + c(G) - c(G-v)\)
The impact of a single vertex
Suppose we remove a single vertex \(v\).
In \(G \setminus v\), our renewed goal is to remove at least:
\(\bigl(|E|-\) \(d(v)\) \(\bigr)- \bigl(|V\)\(-1\)\(| - c(G\)\(-v\)\() \bigr)\) edges.
Recall that in \(G\) our goal was to remove at least:
\(|E| - \) \(\bigl(|V| - c(G) \bigr)\) edges.
So removing \(v\) from \(G\) reduces our goal by:
\(d(v) - \)\( \bigl(1 + c(G - v) - c(G)\bigr)\)
The impact of a single vertex
Suppose we remove a single vertex \(v\).
In \(G \setminus v\), our renewed goal is to remove at least:
\(\bigl(|E|-\) \(d(v)\) \(\bigr)- \bigl(|V\)\(-1\)\(| - c(G\)\(-v\)\() \bigr)\) edges.
Recall that in \(G\) our goal was to remove at least:
\(|E| - \) \(\bigl(|V| - c(G) \bigr)\) edges.
So removing \(v\) from \(G\) reduces our goal by:
\(d(v) - \)\( \bigl(1 + c(G - v) - c(G)\bigr)\)
The impact of a single vertex
Suppose we remove a single vertex \(v\).
In \(G \setminus v\), our renewed goal is to remove at least:
\(\bigl(|E|-\) \(d(v)\) \(\bigr)- \bigl(|V\)\(-1\)\(| - c(G\)\(-v\)\() \bigr)\) edges.
Recall that in \(G\) our goal was to remove at least:
\(|E| - \) \(\bigl(|V| - c(G) \bigr)\) edges.
So removing \(v\) from \(G\) reduces our goal by:
\(d(v) - \)\(b(v)\)
impacts add up
Claim. If \(F\) is a FVS for \(G\), then:
\(\sum_{v \in F} \bigl(\) \(d(v) - b(v)\) \(\bigr)\) \(\geqslant\) \(|E| - |V| + c(G)\)
impacts add up
Claim. If \(F\) is a FVS for \(G\), then:
\(\sum_{v \in F} \bigl(\) \(d(v) - b(v)\) \(\bigr)\) \(\geqslant\) \(\underbrace{|E| - |V| + c(G)}_{\text{{\color{white}.}}}\)
any FVS needs to
remove these many edges
impacts add up
Claim. If \(F\) is a FVS for \(G\), then:
\(\sum_{v \in F} \bigl(\) \(d(v) - b(v)\) \(\bigr)\) \(\geqslant\) \(\underbrace{|E| - |V| + c(G)}_{\text{{\color{white}.}}}\)
any FVS needs to
remove these many edges
#edges removed by \(F\)
\(\geqslant\)
#edges removed by \(F\) = \(\biggl(\sum_{v \in F} d(v)\biggr) - |E(F)|\)
Claim. \(\sum_{v \in F} b(v) - \bigl(|F| + |E(F)|\bigr) \leqslant c(G - F) - c(G)\)
\( - |E(F)| \leqslant - \sum_{v \in F} b(v) + \biggl(c(G - F) - c(G)\biggr) \) + |F|
claim relating individual to global impact
Claim. \(\sum_{v \in F} b(v) - \bigl(|F| + |E(F)|\bigr) \leqslant c(G - F) - c(G)\)
\(\sum_{v \in F{\color{red}-w}} b(v) - \bigl(|F|{\color{red}-1} + |E(F{\color{red}-w})|\bigr) \leqslant c(G - \bigl(F{\color{red}-w}\bigr)) - c(G)\)
Add \(b(w) - 1\) to both sides.
\(\sum_{v \in F} b(v) - \bigl(|F| + |E(F{\color{red}-w})|\bigr) \leqslant c(G - \bigl(F{\color{red}-w}\bigr)) - c(G) + b(w) - 1\)
claim relating individual to global impact
Claim. \(\sum_{v \in F} b(v) - \bigl(|F| + |E(F)|\bigr) \leqslant c(G - F) - c(G)\)
\(\sum_{v \in F{\color{red}-w}} b(v) - \bigl(|F|{\color{red}-1} + |E(F{\color{red}-w})|\bigr) \leqslant c(G - \bigl(F{\color{red}-w}\bigr)) - c(G)\)
Add \(b(w) - 1\) to both sides.
\(\sum_{v \in F} b(v) - \bigl(|F| + |E(F{\color{red}-w})|\bigr) \leqslant c(G - \bigl(F{\color{red}-w}\bigr)) - c(G) + b(w) - 1\)
Add \(-d_F(w)\) to both sides.
\(\sum_{v \in F} b(v) - \bigl(|F| + |E(F)|\bigr) \leqslant c(G - \bigl(F{\color{red}-w}\bigr)) - c(G) + \bigl(b(w) - 1 - d_F(w)\bigr)\)
\(\sum_{v \in F} b(v) - \bigl(|F| + |E(F)|\bigr) \leqslant \underbrace{c(G - \bigl(F{\color{red}-w}\bigr)) + \bigl(b(w) - 1 - d_F(w)\bigr)} - c(G) \)
\(\leqslant c(G-F)\)
claim relating individual to global impact
Claim. \(\sum_{v \in F} b(v) - \bigl(|F| + |E(F)|\bigr) \leqslant c(G - F) - c(G)\)
\(F - w\)
\(w\)
\(c(G - \bigl(F{\color{red}-w}\bigr))\)
\(F\)
\(w\)
\(\geqslant c(G - \bigl(F{\color{red}-w}\bigr))\)
\(+ \bigl(b(w) - 1 - d_F(w)\bigr)\)
\(c(G - F)\)
\(F\) is the solution that we will be building.
\(\ell\) will track the size of the solution.
\(S\) will (roughly) denote \(V \setminus F\)
For any graph \(G\) such that every vertex \(v \in V\) is contained in some cycle, and for any minimal feedhack feedback vertex set \(F\) for \(G\),
\(\sum_{v \in F}(d(v)-b(v)) \leq 2 f(V)=2(|E|-|V|+c(G))\)
\(\sum_{v \in F^{\prime}} w_v=\sum_{v \in F^{\prime}} \sum_{S: v \in S}\left(d_S(v)-b_S(v)\right) y_S\)
\( = \sum_{S \subseteq V} y_S \sum_{v \in F^{\prime} \cap S}\left(d_S(v)-b_S(v)\right)\)
\(\leqslant 2 \sum_{S \subseteq V} f(S)y_S \leqslant 2 OPT\)
\(\sum_{v \in F}(d(v)-b(v)) \leq 2 f(V)=2(|E|-|V|+c(G))\)
\(\sum_{v \in F}(d(v)-b(v)) - 2|E| \leq 2 f(V)=2(c(G) - |V|)\)
\(\sum_{u \in F}(d(v)-b(v)) - \sum_{u\in V}d(u) \leq 2 f(V)=2(c(G) - |V|)\)
\(\sum_{v \notin F} d(v) \geq 2|V|-\sum_{v \in F} b(v)-2 c(G)\)
\(\sum_{v \notin F} d_{V-F}(v)+|\delta(F)| \geq 2|V|-\sum_{v \in F} b(v)-2 c(G)\)
\(2(|V|-|F|-c(G-F)) +|\delta(F)| \geq 2|V|-\sum_{v \in F} b(v)-2 c(G)\)
\(2|F|+2 c(G-F) \leq|\delta(F)|+\sum_{v \in F} b(v)+2 c(G)\)
Minimal
By Neeldhara Misra
