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