The Directed Dominating Set Problem: Generalized Leaf Removal and Belief Propagation

Abstract

A minimum dominating set for a digraph (directed graph) is a smallest set of vertices such that each vertex either belongs to this set or has at least one parent vertex in this set. We solve this hard combinatorial optimization problem approximately by a local algorithm of generalized leaf removal and by a message-passing algorithm of belief propagation. These algorithms can construct near-optimal dominating sets or even exact minimum dominating sets for random digraphs and also for real-world digraph instances. We further develop a core percolation theory and a replica-symmetric spin glass theory for this problem. Our algorithmic and theoretical results may facilitate applications of dominating sets to various network problems involving directed interactions.

Appendix: Mean Field Equations for the GLR Process

The mean field theory for the directed GLR process is a simple extension of the same theory presented in [13] for undirected graphs. Therefore here we only list the main equations of this theory but do not give the derivation details. We denote by $P(k_{+}, k_{-})$ the probability that a randomly chosen vertex of a digraph has in-degree $k_{+}$ and out-degree $k_{-}$. Similarly, the in- and out-degree joint probabilities of the predecessor vertex $i$ and successor vertex $j$ of a randomly chosen arc $(i, j)$ of the digraph are denoted as $Q_{+}(k_{+}, k_{-})$ and $Q_{-}(k_{+}, k_{-})$, respectively. We assume that there is no structural correlation in the digraph, therefore

$$\begin{aligned} Q_{+}(k_{+}, k_{-}) = \frac{ k_{-} P(k_{+}, k_{-})}{\alpha }\,, \quad Q_{-}(k_{+}, k_{-}) = \frac{ k_{+} P(k_{+}, k_{-})}{\alpha }\,, \end{aligned}$$

(5)

where $\alpha \equiv \sum _{k_{+}, \; k_{-}} k_{+} P(k_{+}, k_{-}) = \sum _{k_{+}, \; k_{-}} k_{-} P(k_{+}, k_{-})$ is the arc density.

Consider a randomly chosen arc $(i, j)$ from vertex $i$ to vertex $j$, suppose vertex $i$ is always unobserved, then we denote by $\alpha _t$ the probability that vertex $j$ becomes an unobserved leaf vertex (i.e., it has no unobserved successor and has only a single predecessor) at the $t$-th GLR evolution step, and by $\gamma _{[0,t]}$ the probability that $j$ has been observed at the end of the $t$-th GLR step. Similarly, suppose the successor vertex $j$ of a randomly chosen arc $(i, j)$ is always unobserved, we denote by $\beta _{[0,t]}$ the probability that the predecessor vertex $i$ has been occupied at the end of the $t$-th GLR step, and by $\eta _t$ the probability that at the end of the $t$-th GLR step vertex $i$ becomes observed but unoccupied and having no other unoccupied successors except vertex $j$. These four set of probabilities are related by the following set of iterative equations:

$$\begin{aligned} \alpha _t&= \delta _{t}^{0} Q_{-}(1, 0) + \sum \limits _{k_{+},\; k_{-}} Q_{-}(k_{+}, k_{-}) \biggl [ \delta _{t}^{1} \Bigl [ (\eta _{0})^{k_{+}-1} (\gamma _{[0,0]})^{k_{-}} -\delta _{k_{+}}^{1} \delta _{k_{-}}^{0}\Bigr ] + \nonumber \\&\quad (1-\delta _{t}^{0} -\delta _{t}^{1}) \Bigl [ \bigl (\sum \limits _{t^\prime =0}^{t-1} \eta _{t^\prime }\bigr )^{k_{+}-1} (\gamma _{[0,t-1]})^{k_{-}} -\bigl (\sum \limits _{t^\prime =0}^{t-2} \eta _{t^\prime }\bigr )^{k_{+}-1} (\gamma _{[0,t-2]})^{k_{-}} \Bigr ] \biggr ]\,, \end{aligned}$$

(6a)

$$\begin{aligned} \beta _{[0,t]}&= 1 - \sum \limits _{k_{+}, \; k_{-}}Q_{+}(k_{+}, k_{-}) \biggl [ \delta _{t}^{0} (1- \delta _{k_{+}}^0) (1-\alpha _0)^{k_{-}-1} + \nonumber \\&\quad \quad \quad \quad \quad \quad \quad \quad (1-\delta _{t}^{0}) \Bigl [ 1- \bigl (\sum \limits _{t^\prime =0}^{t-1} \eta _{t^\prime } \bigr )^{k_{+}} \Bigr ] (1-\sum \limits _{t^\prime = 0}^{t} \alpha _{t^\prime } )^{k_{-}-1} \biggr ]\,, \end{aligned}$$

(6b)

$$\begin{aligned} \gamma _{[0,t]}&= 1 - \sum \limits _{k_{+}, \; k_{-}} Q_{-}(k_{+}, k_{-}) (1-\beta _{[0,t]})^{k_{+}-1} \bigl ( 1-\sum \limits _{t^\prime =0}^{t} \alpha _{t^\prime }\bigr )^{k_{-}} \; , \end{aligned}$$

(6c)

$$\begin{aligned} \eta _t&= \delta _{t}^0 \sum \limits _{k_{+}, \; k_{-}} Q_{+}(k_{+}, k_{-}) \bigl (1- (1-\beta _{[0,0]})^{k_{+}}\bigr ) (\gamma _{[0,0]})^{k_{-}-1} + \nonumber \\&\quad \quad (1-\delta _{t}^0) \sum \limits _{k_{+}, \; k_{-}} Q_{+}(k_{+}, k_{-}) \Bigl [ \bigl (1- (1-\beta _{[0,t]})^{k_{+}} \bigr ) (\gamma _{[0,t]})^{k_{-}-1} \nonumber \\&\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad -\bigl (1- (1-\beta _{[0,t-1]})^{k_{+}} \bigr ) (\gamma _{[0,t-1]})^{k_{-}-1}\Bigr ]\,. \end{aligned}$$

(6d)

Let us define $\alpha _{cum}\equiv \sum _{t\ge 0}^{+\infty } \alpha _t$, $\beta _{cum}\equiv \beta _{[0,\infty ]}$, $\gamma _{cum}\equiv \gamma _{[0,\infty ]}$ and $\eta _{cum}\equiv \sum _{t\ge 0}^{\infty } \eta _{t}$ as the cumulative probabilities over the whole GLR process. From Eq. (6) we can verify that these four cumulative probabilities satisfy the following self-consistent equations:

$$\begin{aligned} \alpha _{cum}&= \sum \limits _{k_{+},\; k_{-}} Q_{-}(k_{+},\; k_{-}) (\eta _{cum})^{k_{+}-1} (\gamma _{cum})^{k_{-}} \; , \end{aligned}$$

(7a)

$$\begin{aligned} \beta _{cum}&= 1- \sum \limits _{k_{+},\; k_{-}}Q_{+}(k_{+},\; k_{-}) \bigl [1-(\eta _{cum})^{k_{+}} \bigr ] (1-\alpha _{cum})^{k_{-}-1} \; , \end{aligned}$$

(7b)

$$\begin{aligned} \gamma _{cum}&= 1- \sum \limits _{k_{+},\; k_{-}} Q_{-}(k_{+},\; k_{-}) (1-\beta _{cum})^{k_{+}-1} (1-\alpha _{cum})^{k_{-}} \;, \end{aligned}$$

(7c)

$$\begin{aligned} \eta _{cum}&= \sum \limits _{k_{+},\; k_{-}}Q_{+}(k_{+},\; k_{-}) \bigl [1- (1-\beta _{cum})^{k_{+}}\bigr ] (\gamma _{cum})^{k_{-}-1} \; . \end{aligned}$$

(7d)

The fraction $n_{core}$ of vertices that remain to be unobserved at the end of the GLR process is

$$\begin{aligned} n_{core}= & {} \sum \limits _{k_{+}, \; k_{-}} P(k_{+}, k_{-}) \bigl [ (1-\beta _{cum})^{k_{+}} - (\eta _{cum})^{k_{+}} \bigr ] (1-\alpha _{cum})^{k_{-}} \nonumber \\&- \sum \limits _{k_{+}, \; k_{-}} P(k_{+}, k_{-}) k_{+} (1-\beta _{cum}-\eta _{cum}) (\eta _{cum})^{k_{+}-1} (\gamma _{cum})^{k_{-}} \; . \end{aligned}$$

(8)

The fraction $w$ of vertices that are occupied during the whole GLR process is evaluated through

$$\begin{aligned} w= & {} 1 - \sum \limits _{k_{+}, \; k_{-}} P(k_{+}, \; k_{-}) \bigl [1- (\eta _{cum})^{k_{+}} \bigr ] (1-\alpha _{cum})^{k_{-}} \nonumber \\&- P(1, 0) \eta _{0} -\sum \limits _{t\ge 1} \sum \limits _{k_{+}, \; k_{-}} P(k_{+},\; k_{-}) k_{+} \eta _t \bigl (\sum _{t^\prime =0}^{t-1} \eta _{t^\prime } \bigr )^{k_{+}-1} \bigl (\sum \limits _{t^\prime =0}^{t-1} \gamma _{t^\prime }\bigr )^{k_{-}} \nonumber \\&-\sum \limits _{t\ge 1} \sum \limits _{k_{+}, \; k_{-}} P(k_{+}, \; k_{-}) k_{-} \alpha _t \bigl ( \sum \limits _{t^\prime =0}^{t-1} \gamma _{t^\prime } \bigr )^{k_{-}-1} \Bigl [1-\bigl (1-\sum \limits _{t^\prime =0}^{t-1} \beta _{t^\prime } \bigr )^{k_{+}} \Bigr ] \; . \end{aligned}$$

(9)