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In this work, we show the first worst-case to average-case reduction for the classical $k$-SUM problem. A $k$-SUM instance is a collection of $m$ integers, and the goal of the $k$-SUM problem is to find a subset of $k$ elements that sums to $0$. In the average-case version, the $m$ elements are chosen uniformly at random from some interval $[-u,u]$. We consider the total setting where $m$ is sufficiently large (with respect to $u$ and $k$), so that we are guaranteed (with high probability) that solutions must exist. Much of the appeal of $k$-SUM, in particular connections to problems in computational geometry, extends to the total setting. The best known algorithm in the average-case total setting is due to Wagner (following the approach of Blum-Kalai-Wasserman), and achieves a run-time of $u^{O(1/log k)}$. This beats the known (conditional) lower bounds for worst-case $k$-SUM, raising the natural question of whether it can be improved even further. However, in this work, we show a matching average-case lower-bound, by showing a reduction from worst-case lattice problems, thus introducing a new family of techniques into the field of fine-grained complexity. In particular, we show that any algorithm solving average-case $k$-SUM on $m$ elements in time $u^{o(1/log k)}$ will give a super-polynomial improvement in the complexity of algorithms for lattice problems.
We study the problem of approximating the value of a Unique Game instance in the streaming model. A simple count of the number of constraints divided by $p$, the alphabet size of the Unique Game, gives a trivial $p$-approximation that can be computed
A function $f:[n_1]timesdotstimes[n_d]tomathbb{F}_2$ is a direct sum if it is of the form $fleft(a_1,dots,a_dright) = f_1(a_1)oplusdots oplus f_d (a_d),$ for some $d$ functions $f_i:[n_i]tomathbb{F}_2$ for all $i=1,dots, d$, and where $n_1,dots,n_din
We study the problem of efficiently refuting the k-colorability of a graph, or equivalently certifying a lower bound on its chromatic number. We give formal evidence of average-case computational hardness for this problem in sparse random regular gra
We study the computational complexity of approximating the 2->q norm of linear operators (defined as ||A||_{2->q} = sup_v ||Av||_q/||v||_2), as well as connections between this question and issues arising in quantum information theory and the study o
We study the complexity of Boolean constraint satisfaction problems (CSPs) when the assignment must have Hamming weight in some congruence class modulo M, for various choices of the modulus M. Due to the known classification of tractable Boolean CSPs