Fine-grained reductions, introduced by Vassilevska-Williams and Williams, preserve any improvement in the known algorithms. These have been used very successfully in relating the exact complexities of a wide range of problems, from NP-complete problems like SAT to important quadratic time solvable problems within P such as Edit Distance. However, until ... more >>>

We develop a technique for proving lower bounds in the setting of asymmetric communication, a model that was introduced in the famous works of Miltersen (STOC'94) and Miltersen, Nisan, Safra and Wigderson (STOC'95). At the core of our technique is a novel simulation theorem: Alice gets a $p \times n$ ... more >>>

We show that popular hardness conjectures about problems from the field of fine-grained complexity theory imply structural results for resource-based complexity classes. Namely, we show that if either k-Orthogonal Vectors or k-CLIQUE requires $n^{\epsilon k}$ time, for some constant $\epsilon > 1/2$, to count (note that these conjectures are significantly ... more >>>

A line of work has shown how nontrivial uniform algorithms for analyzing circuits can be used to derive non-uniform circuit lower bounds. We show how the non-existence of nontrivial circuit-analysis algorithms can also imply non-uniform circuit lower bounds. Our connections yield new win-win circuit lower bounds, and suggest a potential ... more >>>

We prove lower bounds for $k$-OV, $k$-XOR, and $k$-SUM in nonuniform AC$^0$, tracking how the circuit-size exponent scales with $k$ and using no running-time hypothesis. Our framework gives depth-zero projections from colored subgraph isomorphism to the three targets at dimension, row count, or bit width $O(k\log n)$, without increasing depth ... more >>>