Largest common subgraph of two forests

Abstract

A common subgraph of two graphs $G_1$ and $G_2$ is a graph that is isomorphic to subgraphs of $G_1$ and $G_2$. In the largest common subgraph problem the task is to determine a common subgraph for two given graphs $G_1$ and $G_2$ that is of maximum possible size ${\rm lcs}(G_1,G_2)$. This natural problem generalizes the well-studied graph isomorphism problem, has many applications, and remains NP-hard even restricted to unions of paths. We present a simple $4$-approximation algorithm for forests, and, for every fixed $ε\in (0,1)$, we show that, for two given forests $F_1$ and $F_2$ of order at most $n$, one can determine in polynomial time a common subgraph $F$ of $F_1$ and $F_2$ with at least ${\rm lcs}(F_1,F_2)-εn$ edges. Restricted to instances with ${\rm lcs}(F_1,F_2)\geq cn$ for some fixed positive $c$, this yields a polynomial time approximation scheme. Our approach relies on the approximation of the given forests by structurally simpler forests that are composed of copies of only $O(\log (n))$ different starlike rooted trees and iterative quantizations of the options for the solutions.

Figures (4)

• Figure 1: A component $K$ with root $r_K$ of an $(\epsilon,\Delta)$-clean forest $F$. Note that we consider the components $L$ of $K-r_K$ as trees rooted in the neighbor of $r_K$ in $V(L)$, which means that we distinguish isomorphic components of $K-r_K$ that are attached differently to the root $r_K$. The figure shows five isomorphic components of $K-r_K$, two of which are attached to $r_k$ at their unique vertex of degree $2$ and three are attached to $r_k$ differently.
• Figure 2: A common subgraph $F$ of $F_1$ and $F_2$ as a spanning subgraph of $F_1$ together with a corresponding bijection $f:V(F_1)\to V(F_2)$. The forest $f(F)$ with vertex set $V(F_2)$ and edge set $\{ f(x)f(y):xy\in E(F)\}$ is a spanning subgraph of $F_2$. Note that $F_f$ contains strictly more edges than $F$; the edge $uf^{-1}(v)$ of $F_1$ belongs to $F_f$, because the edge $f(u)v$ belongs to $F_2$. Up to isomorphism, $F$ is described by the multiplicities of its components; it consists of $4$ copies of $T_p$, $3$ copies of $K_2$, and $3$ copies of $P_3$. Note that there are two non-isomorphic ways to choose a root for $P_3$.
• Figure 3: Subgraphs $A_{0,1}$ and $A_1$ of $T_{j_1}\subseteq F_1$ (on the left) and $A_{0,2}$ and $A_2$ of $T_{j_2}\subseteq F_2$ (on the right) for some $x=(j_1,j_2,A_0,A_1,A_2)$ in $X$. Note that there are different isomorphic copies of $A_0$ within $T_{j_1}$ and $T_{j_2}$ containing their roots, which lead to different possibilities for the subforests $A_1$ and $A_2$, completely specified up to isomorphism by the considered $5$-tupels.
• Figure 4: The figure illustrates part of the information encoded by $M=(s(i_1,i_2,k))_{(i_1,i_2,k)}$ as a bipartite graph $G$. One partite set of $G$ --- shown in the upper half --- corresponds to $F_1$ and contains a vertex for each component of $F_1$. Similarly for the other partite set shown in the lower half corresponding to $F_2$. In the upper partite set, there are $s_{1,i_1}$ vertices corresponding to copies of $S_{i_1}$. The edges of $G$ encode which root vertices of components of $F_1$ can be mapped onto which root vertices of components of $F_2$. Since we aim for a nice common subgraph, the edges of $G$ reflect $c_2=2$.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Lemma 3
• proof
• proof : Proof of Theorem \ref{['theorem1']}
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6