arXiv subjects

Maxim Klimenko

Publications and source records attributed to Maxim Klimenko.

2 recordsLinked to original sources

arXiv2026

A Communication Complexity Lower Bound for Nonuniformly Convex Consensus Optimization

We study the communication complexity of convex decentralized optimization over time-varying networks, where $n$ nodes hold private functions and must agree on the global minimizer using only synchronous exchanges with neighbors. The cost is the number of communication rounds to reach accuracy $\varepsilon$ -- a measure akin to round complexity in the LOCAL model, but constrained by nodes sharing only oracle responses. We prove a new lower bound of $\Omega\!\left(\chi_{\mathcal G} \sqrt{\kappa_g}\,\log\frac{n}{\chi_{\mathcal G}}\log\frac1\varepsilon\right)$ communication rounds, where $\chi_{\mathcal G}$ is the condition number of the network Laplacians and $\kappa_g$ that of the global objective, showing the round complexity attainable under uniform regularity cannot be matched in the nonuniform regime. The construction rests on spectral graph theory: we embed time-rotating star gadgets into the edges of an expander and patch them to preserve spectral connectivity.

math.OC↗

arXiv2026

Algorithms for Standard-form ILP Problems via Koml\'os' Discrepancy Setting

We study the standard-form ILP problem $\max\{ c^\top x \colon A x = b,\; x \in Z_{\geq 0}^n \}$, where $A\in Z^{k\times n}$ has full row rank. We obtain refined FPT algorithms parameterized by $k$ and $\Delta$, the maximum absolute value of a $k\times k$ minor of $A$. Our approach combines discrepancy-based dynamic programming with matrix discrepancy bounds in Koml\'os' setting. Let $\kappa_k$ denote the maximum discrepancy over all matrices with $k$ columns whose columns have Euclidean norm at most $1$. Up to polynomial factors in the input size, the optimization problem can be solved in time $O(\kappa_k)^{2k}\Delta^2$, and the corresponding feasibility problem in time $O(\kappa_k)^k\Delta$. Using the best currently known bound $\kappa_k=\widetilde O(\log^{1/4}k)$, this yields running times $O(\log k)^{\frac{k}{2}(1+o(1))}\Delta^2$ and $O(\log k)^{\frac{k}{4}(1+o(1))}\Delta$, respectively. Under the Koml\'os conjecture, the dependence on $k$ in both running times reduces to $2^{O(k)}$.

cs.DS↗