Algorithms Parameterized by Vertex Cover and Modular Width, Through Potential Maximal Cliques

Abstract

In this paper we give upper bounds on the number of minimal separators and potential maximal cliques of graphs w.r.t. two graph parameters, namely vertex cover (${\text {vc}}$) and modular width (${\text {mw}}$). We prove that for any graph, the number of its minimal separators is ${\mathcal {O}}^*(3^{{\text {vc}}})$ and ${\mathcal {O}}^*(1.6181^{{\text {mw}}})$, and the number of potential maximal cliques is ${\mathcal {O}}^*(4^{{\text {vc}}})$ and ${\mathcal {O}}^*(1.7347^{{\text {mw}}})$, and these objects can be listed within the same running times (The ${\mathcal {O}}^*$ notation suppresses polynomial factors in the size of the input). Combined with known applications of potential maximal cliques, we deduce that a large family of problems, e.g., Treewidth, Minimum Fill-in, Longest Induced Path, Feedback vertex set and many others, can be solved in time ${\mathcal {O}}^*(4^{{\text {vc}}})$ or ${\mathcal {O}}^*(1.7347^{{\text {mw}}})$. With slightly different techniques, we prove that the Treedepth problem can be also solved in single-exponential time, for both parameters.

Appendices

Appendix 1: More Applications

We give in this Appendix several problems that are all known to be particular cases of Max Induced Subgraph of ${\text {tw}}\le t$ satisfiying $\varphi$ (see [18] proofs and more applications). Proposition 9 also extends to the weighted version and the annotated version of the problems (in the annotated version, a fixed vertex subset must be part of the solution F).

Let ${\mathcal {F}}_m$ be the set of cycles of length . Let $\ell \ge 0$ be an integer. Our first example is the following problem.

Maximum Induced Subgraph with $\le \ell$ copies of ${\mathcal {F}}_m$ -cycles encompasses several interesting problems. For example, when $\ell =0$, the problem is to find a maximum induced subgraph without cycles divisible by m. For $\ell =0$ and $m=1$ this is Maximum Induced Forest.

For integers $\ell \ge 0$ and $p\ge 3$, the problem related to Maximum Induced Subgraph with $\le \ell$ copies of ${\mathcal {F}}_m$ -cycles is the following.

Next example concerns properties described by forbidden minors. Graph H is a minor of graph G if H can be obtained from a subgraph of G by a (possibly empty) sequence of edge contractions. A model M of minor H in G is a minimal subgraph of G, where the edge set E(M) is partitioned into c-edges (contraction edges) and m-edges (minor edges) such that the graph resulting from contracting all c-edges is isomorphic to H. Thus, H is isomorphic to a minor of G if and only if there exists a model of H in G. For an integer $\ell$ and a finite set of graphs ${{\mathcal {F}}_{plan}}$ containing a planar graph we define he following generic problem.

Even the special case with $\ell =0$, this problem and its complementary version called the Minimum ${\mathcal {F}}$ -Deletion, encompass many different problems.

Let $t\ge 0$ be an integer and $\varphi$ be a CMSO-formula. Let ${\mathcal {G}}(t,\varphi )$ be a class of connected graphs of treewidth at most t and with property expressible by $\varphi$. Our next example is the following problem.

As natural sub cases studied in the literature we can cite Independent Triangle Packing or Independent Cycle Packing.

The next problem is an example of annotated version of optimization problem Max Induced Subgraph of ${\text {tw}}\le t$ satisfiying $\varphi$.

Many variants of $k$ -in-a-Graph From ${\mathcal {G}}(t,\varphi )$ can be found in the literature, like $k$ -in-a-Path, $k$ -in-a-Tree, $k$ -in-a-Cycle.

Appendix 2: Monadic Second-Order Logic

We use Counting Monadic Second Order Logic (${\text {CMSO}}_2$), an extension of ${\text {MSO}}_2$, as a basic tool to express properties of vertex and edge sets in graphs.

The syntax of Monadic Second Order Logic (${\text {MSO}}_2$) of graphs includes the logical connectives $\vee ,$ $\wedge ,$ $\lnot ,$ $\Leftrightarrow ,$ $\Rightarrow ,$ variables for vertices, edges, sets of vertices, and sets of edges, the quantifiers $\forall ,$ $\exists$ that can be applied to these variables, and the following five binary relations:

• 1. $u\in U$ where u is a vertex variable and U is a vertex set variable;

• 2. $d \in D$ where d is an edge variable and D is an edge set variable;

• 3. $\mathbf {inc}(d,u),$ where d is an edge variable, u is a vertex variable, and the interpretation is that the edge d is incident with the vertex u;

• 4. $\mathbf {adj}(u,v),$ where u and v are vertex variables and the interpretation is that u and v are adjacent;

• 5. Equality of variables representing vertices, edges, sets of vertices, and sets of edges.

The ${\text {MSO}}_1$ is a restriction of ${\text {MSO}}_2$ in which one cannot use edge set variables (in particular the incidence relation becomes unnecessary). For example Hamiltonicity is expressible in ${\text {MSO}}_2$ but not in ${\text {MSO}}_1$.

In addition to the usual features of monadic second-order logic, if we have atomic sentences testing whether the cardinality of a set is equal to q modulo r,  where q and r are integers such that $0\le q<r$ and $r\ge 2,$ then this extension of the ${\text {MSO}}_2$ (resp. ${\text {MSO}}_1$) is called the counting monadic second-order logic ${\text {CMSO}}_2$ (resp. ${\text {CMSO}}_1$). So essentially ${\text {CMSO}}_2$ (resp. ${\text {CMSO}}_1$) is ${\text {MSO}}_2$ (resp. ${\text {MSO}}_1$) with the following atomic sentence for a set S:

$\mathbf {card}_{q,r}(S) = \mathbf {true}$ if and only if $|S| \equiv q \pmod r.$

We refer to [1, 9] and the book of Courcelle and Engelfriet [10] for a detailed introduction on different types of logic.