For every polynomial $q$, we present worst-case to average-case (almost-linear-time) reductions for a class of problems in $\cal P$ that are widely conjectured not to be solvable in time $q$.
These classes contain, for example, the problems of counting the number of $k$-cliques in a graph, for any fixed $k\geq3$.
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One of the important challenges in circuit complexity is proving strong
lower bounds for constant-depth circuits. One possible approach to
this problem is to use the framework of Karchmer-Wigderson relations:
Karchmer and Wigderson (SIDMA 3(2), 1990) observed that for every Boolean
function $f$ there is a corresponding communication problem $\mathrm{KW}_{f}$,
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The universal relation is the communication problem in which Alice and Bob get as inputs two distinct strings, and they are required to find a coordinate on which the strings differ. The study of this problem is motivated by its connection to Karchmer-Wigderson relations, which are communication problems that are ... more >>>