Primitive normalisers in quasipolynomial time

The normaliser problem has as input two subgroups H and K of the symmetric group $$\mathrm {S}_n$$ S n , and asks for a generating set for $$N_K(H)$$ N K ( H ) : it is not known to have a subexponential time solution. It is proved in Roney-Dougal and Siccha (Bull Lond Math Soc 52(2):358–366, 2020) that if H is primitive, then the normaliser problem can be solved in quasipolynomial time. We show that for all subgroups H and K of $$\mathrm {S}_n$$ S n , in quasipolynomial time, we can decide whether $$N_{\mathrm {S}_n}(H)$$ N S n ( H ) is primitive, and if so, compute $$N_K(H)$$ N K ( H ) . Hence we reduce the question of whether one can solve the normaliser problem in quasipolynomial time to the case where the normaliser in $$\mathrm {S}_n$$ S n is known not to be primitive.

Quantum algorithms and approximating polynomials for composed functions with shared inputs

We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let f be an m-bit Boolean function and consider an n-bit function F obtained by applying f to conjunctions of possibly overlapping subsets of n variables. If f has quantum query complexity Q(f), we give an algorithm for evaluating F using O~(Q(f)⋅n) quantum queries. This improves on the bound of O(Q(f)⋅n) that follows by treating each conjunction independently, and our bound is tight for worst-case choices of f. Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of f.By recursively applying our composition theorems, we obtain a nearly optimal O~(n1−2−d) upper bound on the quantum query complexity and approximate degree of linear-size depth-d AC0 circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC0 circuits.As an additional consequence, we show that AC0∘⊕ circuits of depth d+1 require size Ω~(n1/(1−2−d))≥ω(n1+2−d) to compute the Inner Product function even on average. The previous best size lower bound was Ω(n1+4−(d+1)) and only held in the worst case (Cheraghchi et al., JCSS 2018).

On the induced matching problem in hamiltonian bipartite graphs

In this paper, we study the parameterized complexity of the induced matching problem in hamiltonian bipartite graphs and the inapproximability of the maximum induced matching problem in hamiltonian bipartite graphs. We show that, given a hamiltonian bipartite graph, the induced matching problem is W[1]-hard and cannot be solved in time n o ⁢ ( k ) {n^{o(\sqrt{k})}} , where n is the number of vertices in the graph, unless the 3SAT problem can be solved in subexponential time. In addition, we show that unless NP = P {\operatorname{NP}=\operatorname{P}} , a maximum induced matching in a hamiltonian bipartite graph cannot be approximated within a ratio of n 1 / 4 - ϵ {n^{1/4-\epsilon}} , where n is the number of vertices in the graph.

Matching Cut in Graphs with Large Minimum Degree

In a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be $${\mathsf {NP}}$$ NP -complete. While Matching Cut is trivial for graphs with minimum degree at most one, it is $${\mathsf {NP}}$$ NP -complete on graphs with minimum degree two. In this paper, we show that, for any given constant $$c>1$$ c > 1 , Matching Cut is $${\mathsf {NP}}$$ NP -complete in the class of graphs with minimum degree c and this restriction of Matching Cut has no subexponential-time algorithm in the number of vertices unless the Exponential-Time Hypothesis fails. We also show that, for any given constant $$\epsilon >0$$ ϵ > 0 , Matching Cut remains $${\mathsf {NP}}$$ NP -complete in the class of n-vertex (bipartite) graphs with unbounded minimum degree $$\delta >n^{1-\epsilon }$$ δ > n 1 - ϵ . We give an exact branching algorithm to solve Matching Cut for graphs with minimum degree $$\delta \ge 3$$ δ ≥ 3 in $$O^*(\lambda ^n)$$ O ∗ ( λ n ) time, where $$\lambda$$ λ is the positive root of the polynomial $$x^{\delta +1}-x^{\delta }-1$$ x δ + 1 - x δ - 1 . Despite the hardness results, this is a very fast exact exponential-time algorithm for Matching Cut on graphs with large minimum degree; for instance, the running time is $$O^*(1.0099^n)$$ O ∗ ( 1 . 0099 n ) on graphs with minimum degree $$\delta \ge 469$$ δ ≥ 469 . Complementing our hardness results, we show that, for any two fixed constants $$1< c <4$$ 1 < c < 4 and $$c^{\prime }\ge 0$$ c ′ ≥ 0 , Matching Cut is solvable in polynomial time for graphs with large minimum degree $$\delta \ge \frac{1}{c}n-c^{\prime }$$ δ ≥ 1 c n - c ′ .

A subexponential-time, polynomial quantum space algorithm for inverting the CM group action

We present a quantum algorithm which computes group action inverses of the complex multiplication group action on isogenous ordinary elliptic curves, using subexponential time, but only polynomial quantum space. One application of this algorithm is that it can be used to find the private key from the public key in the isogeny-based CRS and CSIDH cryptosystems. Prior claims by Childs, Jao, and Soukharev of such a polynomial quantum space algorithm for this problem are false; our algorithm (along with contemporaneous, independent work by Biasse, Iezzi, and Jacobson) is the first such result.