The complexity of bilinear maps (equivalently, of 3-mode tensors) has been studied extensively, most notably in the context of matrix multiplication. While circuit complexity and tensor rank coincide asymptotically for 3-mode tensors, this correspondence breaks down for d ≥ 4 modes. As a result, the complexity of d-mode tensors for larger fixed d remains poorly understood, despite its relevance, e.g., in fine-grained complexity. Our paper explores this intermediate regime. First, we give a "graph-theoretic" proof of Strassen’s 2ω/3 bound on the asymptotic rank exponent of 3-mode tensors. Our proof directly generalizes to an upper bound of (d-1)ω/3 for d-mode tensors. Using refined techniques available only for d ≥ 4 modes, we improve this bound beyond the current state of the art for ω. We also obtain a bound of d/2+1 on the asymptotic exponent of circuit complexity of generic d-mode tensors and optimized bounds for d ∈ {4,5}. To the best of our knowledge, asymptotic circuit complexity (rather than rank) of tensors has not been studied before. To obtain a robust theory, we first ask whether low complexity of T and U imply low complexity of their Kronecker product T ⊗ U. While this crucially holds for rank (and thus for circuit complexity in 3 modes), we show that assumptions from fine-grained complexity rule out such a submultiplicativity for the circuit complexity of tensors with many modes. In particular, assuming the Hyperclique Conjecture, this failure occurs already for d = 8 modes. Nevertheless, we can salvage a restricted notion of submultiplicativity. From a technical perspective, our proofs heavily make use of the graph tensors T_H, as employed by Christandl and Zuiddam (Comput. Complexity 28 (2019) 27-56) and Christandl, Vrana and Zuiddam (Comput. Complexity 28 (2019) 57-111), whose modes correspond to the vertices of undirected graphs H. We make the simple but conceptually crucial observation that Kronecker products T_G ⊗ T_H are isomorphic to T_{G+H}, and that G and H may also be fractional graphs. By asymptotically converting generic tensors to specific graph tensors, we can use nontrivial results from algorithmic graph theory to study the rank and complexity of d-mode tensors for fixed d.

A central result by Marx [ToC '10] constructs k-vertex graphs H of maximum degree 3 such that n^o(k/log k) time algorithms for detecting colorful H-subgraphs would refute the Exponential-Time Hypothesis (ETH). This result is widely used to obtain almost-tight conditional lower bounds for parameterized problems under ETH. Our first contribution is a new and fully self-contained proof of this result that further simplifies a recent work by Karthik et al. [SOSA 2024]. In our proof, we introduce a novel graph parameter of independent interest, the linkage capacity γ(H), and show that detecting colorful H-subgraphs in time n^o(γ(H)) refutes ETH. Then, we use a simple construction of communication networks credited to Beneš to obtain k-vertex graphs of maximum degree 3 and linkage capacity Ω(k/log k), avoiding arguments involving expander graphs, which were required in previous papers. We also show that every graph H of treewidth t has linkage capacity Ω(t/log t), thus recovering a stronger result shown by Marx [ToC '10] with a simplified proof. Additionally, we obtain new tight lower bounds on the complexity of subgraph detection for certain types of patterns by analyzing their linkage capacity: We prove that almost all k-vertex graphs of polynomial average degree Ω(k^β) for β > 0 have linkage capacity Θ(k), which implies tight lower bounds for finding such patterns H. As an application of these results, we also obtain tight lower bounds for counting small induced subgraphs having a fixed property Φ, improving bounds from, e.g., [Roth et al., FOCS 2020].

For sink-free orientations in graphs of minimum degree at least $3$, we show that there is a deterministic approximate counting algorithm that runs in time $O((n^{73}/\varepsilon^{72})\log(n/\varepsilon))$, a near-linear time sampling algorithm, and a randomised approximate counting algorithm that runs in time $O((n/\varepsilon)^2\log(n/\varepsilon))$, where $n$ denotes the number of vertices of the input graph and $0<\varepsilon<1$ is the desired accuracy. All three algorithms are based on a local implementation of the sink popping method (Cohn, Pemantle, and Propp, 2002) under the partial rejection sampling framework (Guo, Jerrum, and Liu, 2019).

We show that the flip chain for non-crossing spanning trees of n+1 points in convex position mixes in time O(n⁸log n). We use connections between Fuss-Catalan structures to construct a comparison argument with a chain similar to Wilson’s lattice path chain (Wilson 2004).

A central result by Marx [ToC '10] constructs k-vertex graphs H of maximum degree 3 such that n^o(k/log k) time algorithms for detecting colorful H-subgraphs would refute the Exponential-Time Hypothesis (ETH). This result is widely used to obtain almost-tight conditional lower bounds for parameterized problems under ETH. Our first contribution is a new and fully self-contained proof of this result that further simplifies a recent work by Karthik et al. [SOSA 2024]. In our proof, we introduce a novel graph parameter of independent interest, the linkage capacity γ(H), and show that detecting colorful H-subgraphs in time n^o(γ(H)) refutes ETH. Then, we use a simple construction of communication networks credited to Beneš to obtain k-vertex graphs of maximum degree 3 and linkage capacity Ω(k/log k), avoiding arguments involving expander graphs, which were required in previous papers. We also show that every graph H of treewidth t has linkage capacity Ω(t/log t), thus recovering a stronger result shown by Marx [ToC '10] with a simplified proof. Additionally, we obtain new tight lower bounds on the complexity of subgraph detection for certain types of patterns by analyzing their linkage capacity: We prove that almost all k-vertex graphs of polynomial average degree Ω(k^β) for β > 0 have linkage capacity Θ(k), which implies tight lower bounds for finding such patterns H. As an application of these results, we also obtain tight lower bounds for counting small induced subgraphs having a fixed property Φ, improving bounds from, e.g., [Roth et al., FOCS 2020].

Abstract. We present a new framework to derandomize certain Markov chain Monte Carlo (MCMC) algorithms. As in MCMC, we first reduce counting problems to sampling from a sequence of marginal distributions. For the latter task, we introduce a method called coupling toward the past that can, in logarithmic time, evaluate one or a constant number of variables from a stationary Markov chain state. Since there are at most logarithmic random choices, this leads to very simple derandomization. As an application, we provide an efficient deterministic approximate counting algorithm for hypergraph independent sets, under local lemma type conditions matching, up to lower-order factors, their state-of-the-art randomized counterparts.

We present two randomised approximate counting algorithms with Oe(n2−c/ε2) running time for some constant c > 0 and accuracy ε: 1. for the hard-core model with fugacity λ on graphs with maximum degree ∆ when λ = O(∆−1.5−c1) where c1 = c/(2 − 2c); 2. for spin systems with strong spatial mixing (SSM) on planar graphs with quadratic growth, such as Z2. For the hard-core model, Weitz’s algorithm (STOC, 2006) achieves sub-quadratic running time when correlation decays faster than the neighbourhood growth, namely when λ = o(∆−2). Our first algorithm does not require this property and extends the range where sub-quadratic algorithms exist. Our second algorithm appears to be the first to achieve sub-quadratic running time up to the SSM threshold, albeit on a restricted family of graphs. It also extends to (not necessarily planar) graphs with polynomial growth, such as Zd, but with a running time of the form O (n2ε−2/2c(log n)1/d) where d is the exponent of the polynomial growth and c > 0 is some constant.

It is shown in this note that approximating the number of independent sets in a k-uniform linear hypergraph with maximum degree at most Δ is NP-hard if Δ≥5⋅2k−1+1. This confirms that for the relevant sampling and approximate counting problems, the regimes on the maximum degree where the state-of-the-art algorithms work are tight, up to some small factors. These algorithms include: the approximate sampler and randomised approximation scheme by Hermon, Sly and Zhang (RSA, 2019), the perfect sampler by Qiu, Wang and Zhang (ICALP, 2022), and the deterministic approximation scheme by Feng, Guo, Wang, Wang and Yin (FOCS, 2023).

We present a new framework to derandomise certain Markov chain Monte Carlo (MCMC) algorithms. As in MCMC, we first reduce counting problems to sampling from a sequence of marginal distributions. For the latter task, we introduce a method called coupling towards the past that can, in logarithmic time, evaluate one or a constant number of variables from a stationary Markov chain state. Since there are at most logarithmic random choices, this leads to very simple derandomisation. We provide two applications of this framework, namely efficient deterministic approximate counting algorithms for hypergraph independent sets and hypergraph colourings, under local lemma type conditions matching, up to lower order factors, their state-of-the-art randomised counterparts.

We study the sampling problem for the ferromagnetic Ising model with consistent external fields, and in particular, Swendsen-Wang dynamics on this model. We introduce a new grand model unifying two closely related models: the subgraph world and the random cluster model. Through this new viewpoint, we show: polynomial mixing time bounds for Swendsen-Wang dynamics and (edge-flipping) Glauber dynamics of the random cluster model, generalising the bounds and simplifying the proofs for the no-field case by Guo and Jerrum (2018); near linear mixing time for the two dynamics above if the maximum degree is bounded and all fields are (consistent and) bounded away from 1.

We study the problem of sampling almost uniform proper q-colourings in k-uniform simple hypergraphs with maximum degree Δ. For any δ>0, if k≥20(1+δ)δ and q≥100Δ2+δk−4/δ−4, the running time of our algorithm is O~(poly(Δk)⋅n1.01), where n is the number of vertices. Our result requires fewer colours than previous results for general hypergraphs (Jain, Pham, and Voung, 2021; He, Sun, and Wu, 2021), and does not require Ω(logn) colours unlike the work of Frieze and Anastos (2017).