A tight (non-combinatorial) conditional lower bound for Klee’s Measure Problem in 3D

We revisit the classic geometric problem of computing the volume of the union of n 3-dimensional axis-parallel boxes (Klee’s measure problem in $3 D$). It is well known that the problem can be solved in time $O\left(n^{3 / 2}\right)$ (Overmars, Yap SICOMP’91; Chan FOCS’13). Can we justify this 30-year old barrier of $n^{3 / 2 \pm o(1)}$ under plausible fine-grained complexity assumptions? The only previous conditional lower bound (Chan Comp. Geom.’10) shows that this barrier holds for purely combinatorial algorithms, i.e., algorithms avoiding algebraic techniques for fast matrix multiplication. This leaves open an algorithmic improvement exploiting algebraic techniques, and does not give any superlinear bound if the matrix multiplication exponent $\omega$ turns out to be equal to 2. We resolve this issue by giving a tight conditional lower bound for general algorithms, based on the 3-uniform hyperclique hypothesis. Specifically, we prove that an $O\left(n^{3 / 2-\epsilon}\right)$ algorithm for Klee’s measure problem in 3D would give a $O\left(n^{k-\epsilon^{\prime}}\right)$-time algorithm for counting k-cliques in 3-uniform hypergraphs - this in turn would give a novel $O\left(\left(2-\epsilon^{\prime \prime}\right)^{n}\right)$-algorithm for Max-3SAT. Our lower bound can be generalized to $n^{\frac{d}{3-3 / d}}-o(1)$, which matches the upper bound up to a factor of $n^{\frac{d-3}{6-6 / d}+o(1)}$ and separates the general problem from popular special cases: For all $d \geq 3$, known $\tilde{O}\left(n^{\frac{d+1}{3}}\right)$ algorithms (Bringmann Comp. Geom.’12; Chan FOCS’13) compute the problem for arbitrary hypercubes polynomially faster than our lower bound for the general problem.