Parameterized Complexity of Streaming Diameter and Connectivity Problems

TL;DR

The paper investigates the parameterized streaming complexity of Diameter and Connectivity, showing that knowledge of a small vertex cover enables efficient AL-streaming algorithms (including BFS-based distance propagation) for both problems, with either poly$(k)$ passes and $O(k\log n)$ space or a single pass using $O(2^k)$-scaled memory. It simultaneously establishes broad lower bounds, via Disj$_n$-based reductions and permutation-inspired constructions, that many other parameters and modulators do not yield efficient streaming algorithms; some results even yield $\Omega(n^2/p)$ memory for Diameter in AL and $\Omega(n/p)$ memory for Connectivity in AL, and 1-pass $\Omega(n\log n)$ lower bounds for several cases. The work also extends to a streaming kernelization for Vertex Cover[$k$], achieving a $2k$-vertex kernel in AL with poly$(k)$ passes, and discusses extensions to modulators to disjoint unions of cliques. Overall, the results delineate the boundary where small structural parameters empower streaming algorithms for distance and connectivity in graphs, and where they do not. The findings have implications for designing space-efficient graph exploration in streaming, and raise open questions on further kernelization and parameterized boundaries in AL/EA/VA models.

Abstract

We initiate the investigation of the parameterized complexity of Diameter and Connectivity in the streaming paradigm. On the positive end, we show that knowing a vertex cover of size $k$ allows for algorithms in the Adjacency List (AL) streaming model whose number of passes is constant and memory is $O(\log n)$ for any fixed $k$. Underlying these algorithms is a method to execute a breadth-first search in $O(k)$ passes and $O(k \log n)$ bits of memory. On the negative end, we show that many other parameters lead to lower bounds in the AL model, where $Ω(n/p)$ bits of memory is needed for any $p$-pass algorithm even for constant parameter values. In particular, this holds for graphs with a known modulator (deletion set) of constant size to a graph that has no induced subgraph isomorphic to a fixed graph $H$, for most $H$. For some cases, we can also show one-pass, $Ω(n \log n)$ bits of memory lower bounds. We also prove a much stronger $Ω(n^2/p)$ lower bound for Diameter on bipartite graphs. Finally, using the insights we developed into streaming parameterized graph exploration algorithms, we show a new streaming kernelization algorithm for computing a vertex cover of size $k$. This yields a kernel of $2k$ vertices (with $O(k^2)$ edges) produced as a stream in $\text{poly}(k)$ passes and only $O(k \log n)$ bits of memory.

Figures (14)

• Figure 1: VA lower bound for diameter with vertex cover size 3, called 'Simple VA'. The vertices in the middle are indexed $1,\ldots,n$. An edge incident to $a$ ($b$) is present when the entry of Alice (Bob) at the corresponding index is 1. The vertex $c$ ensures the graph is connected.
• Figure 2: VA lower bound for diameter with distance 2 to $1$ clique, called 'Clique VA'. A dashed edge is present when the entry at the corresponding index is 1. The vertices inside the grey area form a clique. Hence, deletion distance to a clique is $2$ (remove $a$ and $b$).
• Figure 3: AL lower bound for diameter, called 'Simple AL'. The edges in the middle are indexed $1,\ldots,n$. An edge incident on $a$ ($b$) is present when the entry of Alice (Bob) at the corresponding index is 1. The vertex $c$ ensures the graph is connected.
• Figure 4: AL lower bound for diameter consisting of a tree, called 'Windmill'. The difference in an entry 1 or 0 is shown on the left. The gadget for index $i$ combines a 0/1-gadget for Alice and a 0/1-gadget for Bob. It makes two 1 entries at this index a path of length 5, and a tree structure of depth at most 4 otherwise. These $n$ gadgets are then identified at $a_{i,1}$ and a tail is added.
• Figure 5: AL lower bound for diameter consisting of a path and 2 vertices, called 'Diamond'. Note that $a$ is connected to $b$ with an edge (indicated with a dashed line here). On the left the gadget for a single index $i$ is shown, where the dotted edges are present when the entry at index $i$ is 0 (for Alice incident on $a$, for Bob incident on $b$). On the right, the construction is sketched in full.

Theorems & Definitions (48)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Proposition 7
• Proposition 8
• Lemma 8
• Lemma 9