Polynomial-length ascents for bounded-treedepth VCSPs

Prove that in every valued constraint satisfaction problem whose constraint graph has bounded treedepth, every strictly improving path formed by single-variable flips from any initial assignment to a local optimum (i.e., every ascent) has length polynomial in the number of variables.

Background

The paper proves that for a VCSP with a treedepth-decomposition of height d, any ordered step-steepest ascent has length at most 2^{d+1}·n, implying that VCSPs whose constraint graphs have logarithmic treedepth always admit some polynomial-length ascent. It also shows tightness via constructions where all ascents are long for polylogarithmic treedepth, and provides quasipolynomially long ascents for loglog treedepth.

In the conclusion, the author highlights a stronger goal: moving beyond the existence of short ascents to the assertion that all ascents are short when treedepth is bounded by a constant. This is known to hold for treedepth 0 (smooth landscapes) and 1 (stars), but remains unresolved for general bounded treedepth ≥ 2. The author notes that existing techniques for proving short ascents (e.g., span arguments and encouragement paths) fail in the presence of cycles with treedepth ≥ 2, suggesting new methods are required.