A Heuristic Approach to the Treedepth Decomposition Problem for Large Graphs

Abstract

In this article, we describe algorithms and techniques used in the method ExTREEm for the treedepth decomposition problem. ExTREEm won the heuristic track of the 5th Parameterized Algorithms and Computational Experiments Challenge (PACE 2020). It searches for a minimum-height treedepth decomposition of a graph via computing graph separators. Among concepts that are incorporated into the approach, we can distinguish a new objective function for evaluating separators, preprocessing based on finding treedepth decompositions in cactus subgraphs and on identification of graphlets, five algorithms for finding separators, a separator minimization method for a refinement of found separators, and a refinement of an obtained treedepth decomposition by merging techniques of tree rotations. This approach enables us to quickly obtain low-depth decompositions of very large graphs.

Appendix A

Let us define

$$\begin{aligned} score_n(S)&= |S| \cdot \frac{ 1 - \beta ^{ \lceil - \frac{\log |V|}{\log \beta } \rceil } }{1 - \beta }, \;\;\; \text { where } \beta = \frac{mn(G,S)}{|V|}, \\ score_e(S)&= |S| \cdot \frac{ 1 - \gamma ^{ \lceil - \frac{\log |E|}{\log \gamma } \rceil } }{1 - \gamma }, \;\;\; \text { where } \gamma = \frac{me(G,S)}{|E|}. \end{aligned}$$

We now show why the proposed objective functions $score_n(S)$ and $score_e(S)$ estimate the height of a treedepth decomposition. We specify only the case of $score_n(S)$, arguments for $score_e(S)$ are analogous.

Let S be a separator of a graph $G=(V,E)$, $\alpha = \frac{|S|}{|V|}$, $\beta = \frac{mn(G,S)}{|V|}$, and let $EH(G,\alpha ,\beta )$ be a function that estimates the height of a sought decomposition T(G). It is calculated on the basis of values $\alpha$ and $\beta$, that is on information that we can obtain knowing only graph G and separator S. For each $C \in C(G,S)$ the decomposition T(C) will be attached to some node from the set $S \cap N(C)$ (see Sect. 3.5), therefore we use the following estimation:

$$\begin{aligned}&EH(G,\alpha ,\beta ) \le |S| + \max \limits _{C \in C(G,S)} h(T(C)) = \alpha \cdot |V| + \max \limits _{C \in C(G,S)} h(T(C)) \end{aligned}$$

Assuming that in each recursive call the values of parameters $\alpha$ and $\beta$ do not change, we can replace h(T(C)) with $EH(C,\alpha , \beta )$ to obtain the following assessment:

$$\begin{aligned} EH(G,\alpha ,\beta )&\le \alpha \cdot |V| + \max \limits _{C \in C(G,S)} h(T(C)) \\&\approx \alpha \cdot |V| + \max \limits _{C \in C(G,S)} EH( C, \alpha , \beta ) \\&\approx \alpha \cdot |V| + \alpha \cdot \beta \cdot |V| + \max \limits _{C' \in C(C,S')}EH(C',\alpha ,\beta ) \\&\approx \alpha \cdot |V| + \alpha \cdot \beta \cdot |V| + \alpha \cdot \beta ^2 \cdot |V| + \ldots \\&\approx \sum \limits _{i=0}^{ \lceil \log _{ \beta ^{-1} }|V| \rceil } \alpha \cdot |V| \cdot \beta ^i = \alpha \cdot |V| \cdot \sum \limits _{i=0}^{ \lceil -\frac{\log |V|}{\log \beta } \rceil } \beta ^i \\&\approx |S| \cdot \frac{ 1 - \beta ^{ \lceil - \frac{\log |V|}{\log \beta } \rceil } }{1 - \beta } \end{aligned}$$

Let us note here that the formulas for the objective functions can be further simplified via the estimation ${\beta ^{ \lceil \log _{\beta ^{-1}}|V| \rceil } \approx \beta ^{ \log _{\beta ^{-1}}|V| } = \frac{1}{|V|} }$. We found, however, test cases where the replacement made a difference to the evaluation of separators.