The paper **Lower Bounds on Dynamic Programming for Maximum Weight Independent Set** by Tuukka Korhonen of the Constraint Reasoning and Optimization group has been accepted for publication in the proceedings of the 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021), a top conference in its field.

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^{Ω(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^{Ω(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^{Ω(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.