#### On the Complexity of Exact Pattern Matching in Graphs: Determinism and Zig-Zag Matching

##### Massimo Equi, Roberto Grossi, Alexandru I. Tomescu, Veli Mäkinen

Exact pattern matching in labeled graphs is the problem of searching paths of a graph $G=(V,E)$ that spell the same string as the given pattern $P[1..m]$. This basic problem can be found at the heart of more complex operations on variation graphs in computational biology, query operations in graph databases, and analysis of heterogeneous networks, where the nodes of some paths must match a sequence of labels or types. In our recent work we described a conditional lower bound stating that the exact pattern matching problem in labeled graphs cannot be solved in less than quadratic time, namely, $O(|E|^{1 - \epsilon} \, m)$ time or $O(|E| \, m^{1 - \epsilon})$ time for any constant $\epsilon>0$, unless the Strong Exponential Time Hypothesis (SETH) is false. The result holds even if node labels and pattern $P$ are drawn from a binary alphabet, and $G$ is restricted to undirected graphs of maximum degree three or directed acyclic graphs of maximum sum of indegree and outdegree three. It was left open what happens on undirected graphs of maximum degree two, i.e., when the pattern can have a zig-zag match in a (cyclic) bidirectional string. Also, the reduction created a non-determistic directed acyclic graph, and it was left open if determinism would make the problem easier. In this work, we show through the Orthogonal Vectors hypothesis (OV) that the same conditional lower bound holds even for these restricted cases.

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