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In recent years, combinatorial reconfiguration problems have attracted great attention due to their connection to various topics such as optimization, counting, enumeration, or sampling. One of the most intriguing open questions concerns the exchange ...

We consider the sparsification of sums F : ℝⁿ → ℝ⁺ where F(x) = f₁(⟨ a₁,x⟩) + ⋯ + f_m(⟨ a_m,x⟩) for vectors a₁,…,a_m ∈ ℝⁿ and functions f₁,…,f_m : ℝ → ℝ⁺. We show that (1+ε)-approximate sparsifiers of F with support size n/ε² (logn/ε)^O(1) exist whenever the ...

We prove that a polynomial fraction of the set of k-component forests in the m × n grid graph have equal numbers of vertices in each component, for any constant k. This resolves a conjecture of Charikar, Liu, Liu, and Vuong, and establishes the first ...

We prove that the single-site Glauber dynamics for sampling proper q-colorings mixes in O_Δ(nlogn) time on line graphs with n vertices and maximum degree Δ when q>(1+o(1))Δ. The main tool in our proof is the matrix trickle-down theorem developed by ...

The hypergraph unreliability problem asks for the probability that a hypergraph gets disconnected when every hyperedge fails independently with a given probability. For graphs, the unreliability problem has been studied over many decades, and multiple ...