Lower Bounds on Dynamic Programming for Maximum Weight Independent Set

Abstract: We prove lower bounds on pure dynamic programming algorithms for maximum weight independent set (MWIS). We model such algorithms as tropical circuits, i.e., circuits that compute with $\max$ and $+$ operations. For a graph $G$, an MWIS-circuit of $G$ is a tropical circuit whose inputs correspond to vertices of $G$ and which computes the weight of a maximum weight independent set of $G$ for any assignment of weights to the inputs. We show that if $G$ has treewidth $w$ and maximum degree $d$, then any MWIS-circuit of $G$ has $2^{\Omega(w/d)}$ gates and that if $G$ is planar, or more generally $H$-minor-free for any fixed graph $H$, then any MWIS-circuit of $G$ has $2^{\Omega(w)}$ gates. An MWIS-formula is an MWIS-circuit where each gate has fan-out at most one. We show that if $G$ has treedepth $t$ and maximum degree $d$, then any MWIS-formula of $G$ has $2^{\Omega(t/d)}$ gates. It follows that treewidth characterizes optimal MWIS-circuits up to polynomials for all bounded degree graphs and $H$-minor-free graphs, and treedepth characterizes optimal MWIS-formulas up to polynomials for all bounded degree graphs.