__
TR19-045 | 19th February 2019 22:46
__

#### On the Fine-grained Complexity of Least Weight Subsequence in Graphs

**Abstract:**
Least Weight Subsequence (LWS) is a type of highly sequential optimization problems with form $F(j) = \min_{i < j} [F(i) + c_{i,j}]$. They can be solved in quadratic time using dynamic programming, but it is not known whether these problems can be solved faster than $n^{2-o(1)}$ time. Surprisingly, each such problem is subquadratic time reducible to a highly parallel, non-dynamic programming problem [KPS17]. In other words, if a "static" problem is faster than quadratic time, so is an LWS problem. For many instances of LWS, the sequential versions are equivalent to their static versions by subquadratic time reductions. The previous result applies to LWS on linear structures, and this paper extends this result to LWS on paths in sparse graphs. When the graph is a multitree (i.e. a DAG where any pair vertices can have at most one path) or when the graph is a DAG whose underlying undirected graph has constant treewidth, we show that LWS on this graph is still subquadratically reducible to their corresponding static problems. For many instances, the graph versions are still equivalent to their static versions.

Moreover, this paper shows that on these graphs, property testing of form $\exists x \exists y (\text{TC}_E(x,y) \wedge P(x,y))$ is subquadratically reducible to property testing of form $\exists x \exists y P(x,y)$, where $P$ is a property checkable in time linear to the sizes of $x$ and $y$, and $\text{TC}_E$ is the transitive closure of relation $E$. Furthermore, when $P$ is definable by a first-order logic formula with at most one quantified variable, then the above two problems are equivalent to each other by subquadratic reductions.