Local Density and its Distributed Approximation

TL;DR

The definition of the local out-degree of a vertex of a graph is introduced, and it is shown to be equal to the local density, which is a previously unknown fact: that existing algorithms already dynamically approximate the local density.

Abstract

The densest subgraph problem is a classic problem in combinatorial optimisation. Danisch, Chan, and Sozio propose a definition for \emph{local density} that assigns to each vertex $v$ a value $ρ^*(v)$. This local density is a generalisation of the maximum subgraph density of a graph. I.e., if $ρ(G)$ is the subgraph density of a finite graph $G$, then $ρ(G)$ equals the maximum local density $ρ^*(v)$ over vertices $v$ in $G$. They approximate the local density of each vertex with no theoretical (asymptotic) guarantees. We provide an extensive study of this local density measure. Just as with (global) maximum subgraph density, we show that there is a dual relation between the local out-degrees and the minimum out-degree orientations of the graph. We introduce the definition of the local out-degree $g^*(v)$ of a vertex $v$, and show it to be equal to the local density $ρ^*(v)$. We consider the local out-degree to be conceptually simpler, shorter to define, and easier to compute. Using the local out-degree we show a previously unknown fact: that existing algorithms already dynamically approximate the local density. Next, we provide the first distributed algorithms that compute the local density with provable guarantees: given any $\varepsilon$ such that $\varepsilon^{-1} \in O(poly \, n)$, we show a deterministic distributed algorithm in the LOCAL model where, after $O(\varepsilon^{-2} \log^2 n)$ rounds, every vertex $v$ outputs a $(1 + \varepsilon)$-approximation of their local density $ρ^*(v)$. In CONGEST, we show a deterministic distributed algorithm that requires $\text{poly}(\log n,\varepsilon^{-1}) \cdot 2^{O(\sqrt{\log n})}$ rounds, which is sublinear in $n$. As a corollary, we obtain the first deterministic algorithm running in a sublinear number of rounds for $(1+\varepsilon)$-approximate densest subgraph detection in the CONGEST model.

Figures (2)

• Figure 1: Given a graph $G$, we arbitrarily orient $G$. This allows us to partition the vertices of $G$ into levels$L_1, \ldots L_6$ based on their current out-degree. We say that an edge $(u, v)$ is violating whenever $\textsl{g}(u \!\to\! v) > 0$, $u \in L_i$, $v \in L_j$ and $i > j+1$. We show violating edges in red. Our algorithm iterates over an integer $h$ from high to low, and tries to flip all violating edges from level $L_h$.
• Figure 2: $(4 : 0 : 0)$ - at the first minute start in hour$4$, we consider all violating out-edges $\overline{uv}$ from level $4$ (red). Per definition, these edges point to level $2$ or lower. $(4 : 0 : 0)$ - at the first minute end, either $u$ has dropped a level (pink), $v$ increased their level to $h - 1$ (green) or the edge $\overline{uv}$ is flipped (blue). We consider violating in-edges to pink vertices (orange) $(4 : 1 : 0)$ - at the first second start, we construct a DAG $D_0$ where the edges are the orange edges plus black edges. The vertex set are all $u$ with a directed path to a pink vertex ($v \in T_1$). The yellow vertices are $S_0$. $(4 : 1 : 0)$ - at the first second end, edges in the DAG may have flipped (blue), vertices in $S_0$ may have dropped a level or vertices in $T_1$ may have increased a level (making some edges no longer violating -- purple). $(4 : 1 : 1)$ - at the second second start, we construct a DAG $D_1$. Note that $D_1$ is a subgraph of $D_0$.

Theorems & Definitions (52)

• Theorem 1: Theorem 1 in charikar2003greedy
• Lemma 1
• Definition 2
• Definition 3
• Definition 4: Definition 2.2 in Prac2
• Definition 5: Definition 2.3 in Prac2
• Definition 6: Definition 2.3 in Prac2
• Definition 7
• Theorem 8
• Corollary 8