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Consider the following problem.

Given a graph $G=(V,E)$, and two positive integers $k$ and $\gamma$, decide if there is a set of new edges to be added such that $|E'|\le k$, and any subset $V'\subseteq V$ of size $\gamma$ is not a vertex cover of $(V, E\cup E')$.

i.e., can we add at most $k$ edges to ensures that the minimum vertex cover has a size at least $\gamma+1$.

I am trying to write this problem in Graph-MSOL (monadic second-order logic).

$$C(V, E, \gamma) := \exists x_1 \exists x_2 \ldots \exists x_\gamma (uv \in E \implies (\lor_{i=1}^\gamma ((u=x_i \lor v=x_i))))$$ which defines a vertex cover of size $\gamma$.

$$ \exists e_1\exists e_2 \ldots \exists e_k (\lnot(C(V, E \cup \{e_1,\ldots,e_k\}, \gamma) )) $$ corresponds to adding $k$ edges so that there does not exist a vertex cover of size $\gamma$.

Substituting the definition of $C$ in the last line, we can write a single large MSOL as

$$ \exists e_1\exists e_2 \ldots \exists e_k \left ( \lnot \left (\vphantom{\int^A} \exists x_1 \exists x_2 \ldots \exists x_\gamma \left( \left (\vphantom{\bar{A^k}}(uv \in E) \lor \left (\lor_{j=1}^k (uv=e_j)\right) \right) \implies \left (\vphantom{\bar{A^k}}\lor_{i=1}^\gamma \left ((u=x_i) \lor (v=x_i)\right)\right) \right) \right ) \right ) $$.

Is the above a valid Graph MSOL? Consequently, does Courcelle's theorem now imply that the above problem is FPT for graphs with fixed treewidth?

Lisa E.
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1 Answers1

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You can use CMSOL which allows cardinality predicates. See 5.2.6 in this report. This is authored by Courcelle himself. You can refer the original paper by Courcelle too, but it is a bit more terse.

In the form you have written above, the length of the formula increases with $k$ and $\gamma$. However, by using cardinality predicate, you can indeed decide all these in linear time, provided the tree width is fixed.

$$ \exists U \left( (\operatorname{card}(U) =k) \bigwedge \left( \neg \left( \exists X \left( (\operatorname{card}(X) = \gamma) \land (\beta(u, v, E, U, X)) \right)\vphantom{\sum^n} \right)\vphantom{\int} \right)\vphantom{\int_x^{x^2}} \right) $$ where $$ \alpha(u,v,E,U) = ((u,v) \in E )\lor ((u,v) \in U)\\ \beta(E,U, X) = \forall u\in V \;\forall v \in V \left( \vphantom{\int}\left( \alpha(u,v,E,U) \right) \to \left((u\in X ) \lor (v\in X) \right)\right) $$. Here,

  • $\alpha(u,v,E,U)$ means, $(u,v)$ is either an edge in $E$ or in $X$.
  • $\beta(E, U, X)$ means $X$ is a vertex cover of a graph with edge set $E\cup U$.
  • The part after the big AND in the main formula ensures that there does not exist a vertex cover of size $\gamma$.
  • The part before the big AND ensures that there is a set $U$ of edges which achieves that.

Because of the above representation, we have FPT w.r.t. tree width.

Sriram
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