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:
$$d(v) -$$$$b(v)$$

Claim. If $$F$$ is a FVS for $$G$$, then:
$$\sum_{v \in F} \bigl($$ $$d(v) - b(v)$$ $$\bigr)$$ $$\geqslant$$ $$|E| - |V| + c(G)$$

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

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

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