In low-depth circuit complexity, the polynomial method is a way to prove lower bounds by translating weak circuits into low-degree polynomials, then analyzing properties of these polynomials. Recently, this method found an application to algorithm design: Williams (STOC 2014) used it to compute all-pairs shortest paths in n 3 /2 Ω(√ log n) time on dense n-node graphs. In this paper, we extend this methodology to solve a number of problems in combinatorial pattern matching and Boolean algebra, considerably faster than previously known methods. First, we give an algorithm for BOOLEAN ORTHOGONAL DETECTION, which is to detect among two sets A, B ⊆ {0, 1} d of size n if there is an x ∈ A and y ∈ B such that x, y = 0. For vectors of dimension d = c(n) log n, we solve BOOLEAN ORTHOGONAL DETECTION in n 2−1/O(log c(n)) time by a Monte Carlo randomized algorithm. We apply this as a subroutine in several other new algorithms:

Due to the lack of unconditional polynomial lower bounds, it is now in fashion to prove conditional lower bounds in order to advance our understanding of the class P. The vast majority of these lower bounds are based on one of three famous hypotheses: the 3-SUM conjecture, the APSP conjecture, and the Strong Exponential Time Hypothesis. Only circumstantial evidence is known in support of these hypotheses, and no formal relationship between them is known. In hopes of obtaining "less conditional" and therefore more reliable lower bounds, we consider the conjecture that at least one of the above three hypotheses is true. We design novel reductions from 3-SUM, APSP, and CNF-SAT, and derive interesting consequences of this very plausible conjecture, including:• Tight n 3−o(1) lower bounds for purely-combinatorial problems about the triangles in unweighted graphs.• New n 1−o(1) lower bounds for the amortized update and query times of dynamic algorithms for single-source reachability, strongly connected components, and Max-Flow.• New n 1.5−o(1) lower bound for computing a set of n stmaximum-flow values in a directed graph with n nodes and Õ(n) edges.• There is a hierarchy of natural graph problems on n nodes with complexity n c for c ∈ (2, 3).Only slightly non-trivial consequences of this conjecture were known prior to our work. Along the way we also obtain new conditional lower bounds for the Single-Source-Max-Flow problem.

We consider the problem of solving systems of multivariate polynomial equations of degree k over a finite field. For every integer k ≥ 2 and finite field F q where q = p d for a prime p, we give, to the best of our knowledge, the first algorithms that achieve an exponential speedup over the brute force O(q n ) time algorithm in the worst case. We present two algorithms, a randomized algorithm with running time q n+o (n) • q −n/O(k) time if q ≤ 2 4ekd , and q n+o(n) • ( log q dek ) −dn otherwise, where e = 2.718 . . . is Napier's constant, and a deterministic algorithm for counting solutions with running time q n+o (n) • q −n/O(kq 6/7d ) . For the important special case of quadratic equations in F 2 , our randomized algorithm has running time O(2 0.8765n ).For systems over F 2 we also consider the case where the input polynomials do not have bounded degree, but instead can be efficiently represented as a ΣΠΣ circuit, i.e., a sum of products of sums of variables. For this case we present a deterministic algorithm running in time 2 n−δ n for δ = 1/O(log(s/n)) for instances with s product gates in total and n variables.Our algorithms adapt several techniques recently developed via the polynomial method from circuit complexity. The algorithm for systems of ΣΠΣ polynomials also introduces a new degree reduction method that takes an instance of the problem and outputs a subexponential-sized set of instances, in such a way that feasibility is preserved and every polynomial among the output instances has degree O(log(s/n)).

We present new algorithms for finding induced four-node subgraphs in a given graph, which run in time roughly that of detecting a clique on three nodes (i.e., a triangle).• The best known algorithms for triangle finding in an nnode graph take O(n ω ) time, where ω < 2.373 is the matrix multiplication exponent. We give a general randomized technique for finding any induced four-node subgraph, except for the clique or independent set on 4 nodes, iñ O(n ω ) time with high probability. The algorithm can be derandomized in some cases: we show how to detect a diamond (or its complement) in deterministicÕ(n ω ) time.Our approach substantially improves on prior work. For instance, the previous best algorithm for C 4 detection ran in O(n 3.3 ) time, and for diamond detection in O(n 3 ) time.• For sparse graphs with m edges, the best known triangle finding algorithm runs in O(m 2ω/(ω+1) ) ≤ O(m 1.41 ) time. We give a randomizedÕ(m 2ω/(ω+1) ) time algorithm (analogous to the best known for triangle finding) for finding any induced four-node subgraph other than C 4 , K 4 and their complements. In the case of diamond detection, we also design a deterministicÕ(m 2ω/(ω+1) ) time algorithm. For C 4 or its complement, we give randomized O(m (4ω−1)/(2ω+1) ) ≤ O(m 1.48 ) time finding algorithms. These algorithms substantially improve on prior work. For instance, the best algorithm for diamond detection ran in O(m 1.5 ) time.

Due to the lack of unconditional polynomial lower bounds, it is now in fashion to prove conditional lower bounds in order to advance our understanding of the class P. The vast majority of these lower bounds are based on one of three famous hypotheses: the 3-SUM conjecture, the APSP conjecture, and the Strong Exponential Time Hypothesis. Only circumstantial evidence is known in support of these hypotheses, and no formal relationship between them is known. In hopes of obtaining "less conditional" and therefore more reliable lower bounds, we consider the conjecture that at least one of the above three hypotheses is true. We design novel reductions from 3-SUM, APSP, and CNF-SAT, and derive interesting consequences of this very plausible conjecture, including:• Tight n 3−o(1) lower bounds for purely-combinatorial problems about the triangles in unweighted graphs.• New n 1−o(1) lower bounds for the amortized update and query times of dynamic algorithms for single-source reachability, strongly connected components, and Max-Flow.• New n 1.5−o(1) lower bound for computing a set of n stmaximum-flow values in a directed graph with n nodes andÕ(n) edges.• There is a hierarchy of natural graph problems on n nodes with complexity n c for c ∈ (2, 3).Only slightly non-trivial consequences of this conjecture were known prior to our work. Along the way we also obtain new conditional lower bounds for the Single-Source-Max-Flow problem.