Partial matching width and its application to lower bounds for branching programs

We introduce a new structural graph parameter called \emph{partial matching width}. For each (sufficiently large) integer $k \geq 1$, we introduce a class $\mathcal{G}_k$ of graphs of treewidth at most $k$ and max-degree $7$ such that for each $G \in \mathcal{G}_k$ and each (sufficiently large) $V \subseteq V(G)$, the partial matching width of $V$ is $\Omega(k \log |V|)$. We use the above lower bound to establish a lower bound on the size of non-deterministic read-once branching programs (NROBPs). In particular, for each sufficiently large ineteger $k$, we introduce a class ${\bf \Phi}_k$ of CNFs of (primal graph) treewidth at most $k$ such that for any $\varphi \in {\bf \Phi}_k$ and any Boolean function $F \subseteq \varphi$ and such that $|\varphi|/|F| \leq 2^{\sqrt{n}}$ (here the functions are regarded as sets of assignments on which they are true), a NROBP implementing $F$ is of size $n^{\Omega(k)}$. This result significantly generalises an earlier result of the author showing a non-FPT lower bound for NROBPs representing CNFs of bounded treewidth. Intuitively, we show that not only those CNFs but also their arbitrary one side approximations with an exponential ratio still attain that lower bound. The non-trivial aspect of this approximation is that due to a small number of satisfying assignments for $F$, it seems difficult to establish a large bottleneck: the whole function can `sneak' through a single rectangle corresponding to just \emph{one} vertex of the purported bottleneck. We overcome this problem by simultaneously exploring $\sqrt{n}$ bottlenecks and showing that at least one of them must be large. This approach might be useful for establishing other lower bounds for branching programs.