Comparing heuristics for graph edit distance computation

Abstract

Because of its flexibility, intuitiveness, and expressivity, the graph edit distance (GED) is one of the most widely used distance measures for labeled graphs. Since exactly computing GED is NP-hard, over the past years, various heuristics have been proposed. They use techniques such as transformations to the linear sum assignment problem with error correction, local search, and linear programming to approximate GED via upper or lower bounds. In this paper, we provide a systematic overview of the most important heuristics. Moreover, we empirically evaluate all compared heuristics within an integrated implementation.

Appendices

Datasets and edit cost functions

• The datasets aids and muta: Graphs contained in aids and muta represent molecular compounds. The molecules represented by the graphs contained in aids are divided into the class of molecules that do and the class of molecules that do not exhibit activity against HIV. Similarly, the molecules represented by the graphs contained in muta are divided into the class of molecules that do and the class of molecules that do not cause genetic mutation. The nodes of the graphs contained in aids and muta are labeled with chemical symbols, and their edges are labeled with a valence (either 1 or 2). Node edit costs are defined as $c_V(\alpha ,\alpha ^\prime ):=5.5\cdot \delta _{\alpha \ne \alpha ^\prime }$, $c_V(\alpha ,\varepsilon ):=2.75$, and $c_V(\varepsilon ,\alpha ^\prime ):=2.75$, for all $(\alpha ,\alpha ^\prime )\in \varSigma _V\times \varSigma _V$. Edge edit costs are defined as $c_E(\beta ,\beta ^\prime ):=1.65\cdot \delta _{\beta \ne \beta ^\prime }$, $c_E(\beta ,\varepsilon ):=0.825$, and $c_E(\varepsilon ,\beta ^\prime ):=0.825$, for all $(\beta ,\beta ^\prime )\in \varSigma _E\times \varSigma _E$.

• The dataset protein: Graphs contained in protein represent proteins which are annotated with their EC classes (EC1, EC2, EC3, EC4, EC5, and EC6) [60]. Nodes are labeled with tuples (t, s), where t is the node’s type (helix, sheet, or loop) and s is its amino acid sequence. Nodes are connected via structural or sequential edges or both, i.e., edges $(u_i,u_j)$ are labeled with tuples $(t_1,t_2)$, where $t_1$ is the type of the first edge connecting $u_i$ and $u_j$ and $t_2$ is the type of the second edge connecting $u_i$ and $u_j$ (possibly null). Node edit costs are defined as $c_V(\alpha ,\alpha ^\prime ):=16.5\cdot \delta _{\alpha .t\ne \alpha ^\prime .t}+0.75\cdot \delta _{\alpha .t=\alpha ^\prime .t}\cdot \mathrm{LD}(\alpha .s,\alpha ^\prime .s))$, $c_V(\alpha ,\varepsilon ):=8.25$, and $c_V(\varepsilon ,\alpha ^\prime ):=8.25$, for all $(\alpha ,\alpha ^\prime )\in \varSigma _V\times \varSigma _V$, where $\mathrm{LD}(\cdot ,\cdot )$ is Levenshtein’s string edit distance. Edge edit costs are defined as $c_E(\beta ,\beta ^\prime ):=0.25\cdot \mathrm{LSAPE} ({\mathbf{C}} ^{\beta ,\beta ^\prime })$, $c_E(\beta ,\varepsilon ):=0.25\cdot f(\beta )$, and $c_E(\varepsilon ,\beta ^\prime ):=0.25\cdot f(\beta ^\prime )$, for all $(\beta ,\beta ^\prime )\in \varSigma _E\times \varSigma _E$, where $f(\beta ):=1+\delta _{\beta .t_2\ne {\texttt {null}}}$ and ${\mathbf{C}} ^{\beta ,\beta ^\prime }\in {\mathbb {R}} ^{(f(\beta )+1)\times (f(\beta ^\prime )+1)}$ is constructed as $c^{\beta ,\beta ^\prime }_{r,s}:=2\cdot \delta _{\beta .t_r\ne \beta ^\prime .t_s}$ and $c^{\beta ,\beta ^\prime }_{r,f(\beta ^\prime )+1}:=c^{\beta ,\beta ^\prime }_{f(\beta )+1,s}:=1$, for all $(r,s)\in [f(\beta )]\times [f(\beta ^\prime )]$.

• The dataset letter (h): Graphs contained in letter (h) represent highly distorted drawings of the capital letters A, E, F, H, I, K, L, M, N, T, V, W, X, Y, and Z. Nodes are labeled with two-dimensional Euclidean coordinates. Edges are unlabeled. Node edit costs are defined as $c_V(\alpha ,\alpha ^\prime ):=0.75\cdot \left||\alpha -\alpha ^\prime \right||$, $c_V(\alpha ,\varepsilon ):=0.675$, and $c_V(\varepsilon ,\alpha ^\prime ):=0.675$, for all $(\alpha ,\alpha ^\prime )\in \varSigma _V\times \varSigma _V$, where $\left||\cdot \right||$ is the Euclidean norm. The edge edit costs $c_E$ are defined as $c_E(1,\varepsilon ):=c_E(\varepsilon ,1):=0.425$.

• The dataset grec: Graphs contained in grec represent 22 different symbols from electronic and architectural drawings. Nodes are labeled with tuples (t, x, y), where t equals one of four node types and (x, y) is a two-dimensional Euclidean coordinate. Nodes are connected via line or arc edges or both, i.e., edges $(u_i,u_j)$ are labeled with tuples $(t_1,t_2)$, where $t_1$ is the type of the first edge connecting $u_i$ and $u_j$ and $t_2$ is the type of the second edge connecting $u_i$ and $u_j$ (possibly null). Node edit costs are defined as $c_V(\alpha ,\alpha ^\prime ):=0.5\cdot \left||\alpha .(x,y)-\alpha ^\prime .(x,y)\right||\cdot \delta _{\alpha .t=\alpha ^\prime .t}+90\cdot \delta _{\alpha .t\ne \alpha ^\prime .t}$, $c_V(\alpha ,\varepsilon ):=45$, and $c_V(\varepsilon ,\alpha ^\prime ):=45$, for all $(\alpha ,\alpha ^\prime )\in \varSigma _V\times \varSigma _V$. Edge edit costs are defined as $c_E(\beta ,\beta ^\prime ):=0.5\cdot \mathrm{LSAPE} ({\mathbf{C}} ^{\beta ,\beta ^\prime })$, $c_E(\beta ,\varepsilon ):=0.5\cdot f(\beta )$, and $c_E(\varepsilon ,\beta ^\prime ):=0.5\cdot f(\beta ^\prime )$, for all $(\beta ,\beta ^\prime )\in \varSigma _E\times \varSigma _E$, where $f(\beta ):=1+\delta _{\beta .t_2\ne {\texttt {null}}}$ and ${\mathbf{C}} ^{\beta ,\beta ^\prime }\in {\mathbb {R}} ^{(f(\beta )+1)\times (f(\beta ^\prime )+1)}$ is constructed as $c^{\beta ,\beta ^\prime }_{r,s}:=30\cdot \delta _{\beta .t_r\ne \beta ^\prime .t_s}$ and $c^{\beta ,\beta ^\prime }_{r,f(\beta ^\prime )+1}:=c^{\beta ,\beta ^\prime }_{f(\beta )+1,s}:=15$ for all $(r,s)\in [f(\beta )]\times [f(\beta ^\prime )]$.

• The dataset fp: Graphs contained in fp represent fingerprint images which are annotated with their classes (arch, left loop, right loop, and whorl) from the Galton-Henry classification system [35]. Nodes are unlabeled and edges are labeled with an orientation $\beta \in {\mathbb {R}}$ with $-\pi /2<\beta \le \pi /2$. Node edit costs are defined as $c_V(1,\varepsilon ):=c_V(\varepsilon ,1):=0.525$. Edge edit costs are defined as $c_E(\beta ,\beta ^\prime ):=0.5\cdot \min \{|\beta -\beta ^\prime |,\pi -|\beta -\beta ^\prime |\}$, $c_E(\beta ,\varepsilon ):=0.375$, and $c_E(\varepsilon ,\beta ^\prime ):=0.375$, for all $(\beta ,\beta ^\prime )\in \varSigma _E\times \varSigma _E$.

Visualization of experiments via dominance graphs

See Figures 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, and 28.

Figures 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, and 28 visualize the transitive reductions of the dominance graphs induced by $\succ _{\textit{LB}}$ (Figs. 17, 18, 19, 20, 21, 22) and $\succ _{\textit{UB}}$ (Figs. 23, 24, 25, 26, 27, 28) and hence provide more detailed views on the results of the experiments reported in Sects. 9.5 and 9.6. In the dominance graphs, instantiations of LSAPE-GED are displayed black on white, instantiations of LP-GED are displayed black on light gray, instantiations of LS-GED are displayed white on dark gray, and miscellaneous heuristics are displayed white on black. For all algorithms instantiating LSAPE-GED, we display the configuration $(K,\gamma )$ of the extensions MULTI-SOL and CENTRALITIES in addition to the name of the heuristic. Similarly, for all algorithms instantiating LS-GED, we display the configuration $(K,\rho ,L,\eta )$ of the extensions MULTI-START and RANDPOST. Recall that instantiations of LSAPE-GED are run without extensions just in case $(K,\gamma )=(1,0)$ and that instantiations of LS-GED are run without extensions just in case $(K,\rho ,L,\eta )=(1,1,0,0)$ (cf. Sect. 9.2 for more details). For Pareto optimal algorithms, we also show the test metrics $d_{{\textit{LB}} |{\textit{UB}}}$, t, and $c_{{\textit{LB}} |{\textit{UB}}}$, and the joint score $s_{{\textit{LB}} |{\textit{UB}}}$.

As the extensions MULTI-SOL and CENTRALITIES of LSAPE-GED improve the computed upper bounds at the price of increased runtimes but have no effect on the obtained lower bounds, for all instantiations of LSAPE-GED, we only show the baseline configurations $(K,\gamma )=(1,0)$ in the dominance graphs induced by $\succ _{\textit{LB}}$. In the dominance graphs induced by $\succ _{\textit{UB}}$, for each heuristic H, we only display those configurations that are Pareto optimal (i.e., maximal w.r.t. $\succ _{\textit{UB}}$) or have a maximal joint score $s_{\textit{UB}}$ among all tested configurations of H.

In the transitive reduction of the dominance graphs induced by $\succ _{\textit{LB}}$, we draw an arc from ${{\texttt {ALG}}} _1$ to ${{\texttt {ALG}}} _2$ just in case ${{\texttt {ALG}}} _1\succ _{\textit{LB}} {{\texttt {ALG}}} _2$ and there is no algorithm ${{\texttt {ALG}}} _3$ such that ${{\texttt {ALG}}} _1\succ _{\textit{LB}} {{\texttt {ALG}}} _3\succ _{\textit{LB}} {{\texttt {ALG}}} _2$. Arcs are blue if, additionally, ${{\texttt {ALG}}} _1$ yielded a tighter lower bound than ${{\texttt {ALG}}} _2$, red if ${{\texttt {ALG}}} _1$ was faster than ${{\texttt {ALG}}} _2$, and green if ${{\texttt {ALG}}} _1$ had a better classification coefficient than ${{\texttt {ALG}}} _2$. Multicolored arcs indicate that several of these relations holds. The graphs are oriented from left to right, such that an algorithm is Pareto optimal just in case it appears in the leftmost layer. The colored labels $d^\star _{{\textit{LB}}}$, $t^\star _{{\textit{LB}}}$, and $c^\star _{{\textit{LB}}}$ highlight those Pareto optimal algorithms that, respectively, yielded the tightest lower bound, exhibited the best runtime behavior among all heuristics that compute lower bounds, or gave the best lower bound classification coefficient. The dominance graphs induced by $\succ _{\textit{UB}}$ are constructed analogously.

Example 3

Figure 16 exemplifies the visualizations of the dominance graphs induced by $\succ _{\textit{LB}}$ and $\succ _{\textit{UB}}$. It shows a snapshot of the dominance graph induced by $\succ _{\textit{UB}}$ on the dataset fp shown in Fig. 24. We see that IPFP run with the configuration $(K,\rho ,L,\eta )=(40,1,0,0)$ was Pareto optimal on fp. The blue label $d^\star _{{\textit{LB}}}$ indicates that IPFP (40, 1, 0, 0) computed the tightest average upper bound on fp; the green label $c^\star _{{\textit{LB}}}$ tells us that it also yielded the best upper bound classification coefficient. Furthermore, we see that $t(\texttt {IPFP} \,(40,1,0,0))=1.63\cdot 10^{-2}\hbox {s}$, $d_{\textit{UB}} (\texttt {IPFP} \,(40,1,0,0))=3.08$, $c_{\textit{UB}} (\texttt {IPFP} \,(40,1,0,0))=0.11$, and $s_{\textit{UB}} (\texttt {IPFP} \,(40,1,0,0))=0.67$.

The blue–red arc from IPFP (40, 1, 0, 0) to IBP-BEAM (40, 0.5, 1, 0) tells us that, on fp, IPFP with $(K,\rho ,L,\eta )=(40,1,0,0)$ dominated IBP-BEAM with $(K,\rho ,L,\eta )=(40,0.5,1,0)$. More precisely, we have $t(\texttt {IBP-BEAM} \,(40,0.5,1,0))>t(\texttt {IPFP} \,(40,1,0,0))=1.63\cdot 10^{-2}\hbox {s}$, $d_{\textit{UB}} (\texttt {IBP-BEAM} \,(40,0.5,1,0))>d_{\textit{UB}} (\texttt {IPFP} \,(40,1,0,0))=3.08$, and $c_{\textit{UB}} (\texttt {IBP-BEAM} \,(40,0.5,1,0))=c_{\textit{LB}} (\texttt {IPFP} \,(40,1,0,0))=0.11$. As IPFP and IBP-BEAM instantiate LS-GED, they are shown white on dark gray.