Parameterized Complexity of Streaming Diameter and Connectivity Problems - Oostveen, Jelle J.; van Leeuwen, Erik Jan (Algorithmica, 2024)
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 2 k vertices (with O ( k 2 ) edges) produced as a stream in poly ( k ) passes and only O ( k log n ) bits of memory.