arXiv subjects

Radu Curticapean

Publications and source records attributed to Radu Curticapean.

At least 19 recordsLinked to original sources

arXiv2026

A Dense Weisfeiler-Leman Algorithm for Deciding Bounded-Cliquewidth Homomorphism Indistinguishability

Two graphs $G$ and $H$ are homomorphism indistinguishable over a graph class $\mathcal{F}$ if they admit the same number of homomorphisms from every graph in $\mathcal{F}$. A wide range of relaxations of graph isomorphism arise this way: isomorphism itself over the class of all graphs [Lov\'asz, Acta Math. Hung. 1967], equivalence under the $k$-dimensional Weisfeiler-Leman algorithm over the graphs of treewidth $\leq k$ [Dvo\v{r}\'ak, J. Graph Theory 2010], and quantum isomorphism over planar graphs [Man\v{c}inska-Roberson, FOCS 2020]. Since the class $\mathcal{F}$ is typically infinite, it is not clear a priori whether homomorphism indistinguishability over $\mathcal{F}$ is decidable; for planar graphs it is undecidable. Every class for which decidability was previously known is sparse. We give the first decidability results for dense graph classes: We introduce the dense Weisfeiler-Leman algorithm that decides homomorphism indistinguishability over the class of graphs of cliquewidth $\leq k$, the dense counterpart of treewidth. This relation was not previously known to be decidable. The algorithm colors $k$-tuples of vertex subsets rather than $k$-tuples of vertices. Beyond the class of all graphs of cliquewidth $\leq k$, we prove a general meta-theorem: homomorphism indistinguishability over every $\mathsf{CMSO}_1$-definable graph class of bounded cliquewidth is decidable, in randomized exponential time. For classes of bounded linear cliquewidth the bound improves to $\mathsf{PSPACE}$, and we show this is tight by exhibiting such a class for which the problem is $\mathsf{PSPACE}$-complete. These are the first general algorithms for homomorphism indistinguishability over dense graph classes.

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arXiv2026

Counting Small Induced Subgraphs: Hardness of Symmetry-Based Properties

Jerrum and Meeks (TOCT, JCSS 2015) introduced the counting problems $\text{IndSub}(\Phi)$ for fixed graph properties $\Phi$: Given an input graph $G$ and $k\in\mathbb N$, count the $k$-vertex subsets $S \subseteq V(G)$ such that the induced subgraph $G[S]$ satisfies $\Phi$. For recursively enumerable $\Phi$, it is known that $\text{IndSub}(\Phi)$ is either #W[1]-hard or fixed-parameter tractable. A direct classification depending on $\Phi$ however still remains open. In particular, the status was open for the property of graphs without nontrivial automorphisms, also mentioned in a very recent survey on parameterized counting by Roth (Comput.~Sci.~Rev.~2026). This is a natural property that evades all currently known techniques for proving #W[1]-hardness, including a general toolkit based on Fourier analysis that was very recently introduced by Curticapean and Neuen (SODA~2025). In this paper, we show that counting induced $k$-vertex graphs without nontrivial automorphisms is #W[1]-hard by constructing ``clique scaffolds'', i.e., problem-specific restrictions of the property that enable a reduction from the $k$-clique problem. More generally, we show that for every finite group $Q$, counting $k$-vertex induced subgraphs with automorphism group $Q$ is #W[1]-hard.

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arXiv2026

Planar Perfect Matching Counting is as Hard as Determinants

In the 1960s, Fisher, Kasteleyn and Temperley designed an ingenious algorithm for computing the partition function of the dimer model, or equivalently, for counting perfect matchings in edge-weighted planar graphs (Philos. Mag. 1961; J. Mathematical Phys. 1963). This FKT algorithm later became the foundation for Valiant's holographic algorithms (FOCS 2004; SIAM J. Comput. 2008), which motivated the study of counting problems under the Holant framework. Combined with an algorithm by Yuster (FOCS 2008), the FKT algorithm allows us to count edge-weighted perfect matchings in planar $n$-vertex graphs with $\tilde{O}(n^{ω/2})$ arithmetic operations, where $ω<2.372$ is the matrix multiplication exponent. We prove a corresponding lower bound: Over algebraic circuits and other sufficiently strong computational models, perfect matchings in edge-weighted $n$-vertex planar graphs $G$ cannot be counted in $O(n^{ω/2-ε})$ arithmetic operations. This confirms the optimality of Yuster's algorithm. Our bound holds even when $G$ is an edge-weighted square grid.

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arXiv2026

Beyond Bilinear Complexity: What Works and What Breaks with Many Modes?

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 $d \geq 4$ 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 $d\geq 4$ 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 $d \in \{4,5\}$. 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 $T \otimes U$. 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 ({\em Comput.~Complexity}~28~(2019)~27--56) and [...]

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arXiv2025

Counting Small Induced Subgraphs: Hardness via Fourier Analysis

For a fixed graph property $Φ$ and integer $k \geq 1$, consider the problem of counting the induced $k$-vertex subgraphs satisfying $Φ$ in an input graph $G$. This problem can be solved by brute-force in time $O(n^{k})$. Under ETH, we prove several lower bounds on the optimal exponent in this running time: If $Φ$ is edge-monotone (i.e., closed under deleting edges), then ETH rules out $n^{o(k)}$ time algorithms for this problem. This strengthens a recent lower bound by Döring, Marx and Wellnitz [STOC 2024]. Our result also holds for counting modulo fixed primes. If at most $(2-\varepsilon)^{\binom{k}{2}}$ graphs on $k$ vertices satisfy $Φ$, for some $\varepsilon > 0$, then ETH also rules out an exponent of $o(k)$. This holds even when the graphs in $Φ$ have arbitrary individual weights, generalizing previous results for hereditary properties by Focke and Roth [SIAM J. Comput. 2024]. If $Φ$ is non-trivial and excludes $β_Φ$ edge-densities, then the optimal exponent under ETH is $Ω(β_Φ)$. This holds even when the graphs in $Φ$ have arbitrary individual weights, generalizing previous results by Roth, Schmitt and Wellnitz [SIAM J. Comput. 2024]. In all cases, we also obtain $\mathsf{\#W[1]}$-hardness if $k$ is part of the input and considered as the parameter. We also obtain lower bounds on the Weisfeiler-Leman dimension. As opposed to the nontrivial techniques from combinatorics, group theory, and simplicial topology used before, our results follow from a relatively straightforward ``algebraization'' of the problem in terms of polynomials, combined with applications of simple algebraic facts, which can also be interpreted in terms of Fourier analysis.

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arXiv2025

Which graph motif parameters count?

For a fixed graph H, the function #IndSub(H,*) maps graphs G to the count of induced H-copies in G; this function obviously "counts something" in that it has a combinatorial interpretation. Linear combinations of such functions are called graph motif parameters and have recently received significant attention in counting complexity after a seminal paper by Curticapean, Dell and Marx (STOC'17). We show that, among linear combinations of functions #IndSub(H,*) involving only graphs H without isolated vertices, precisely those with positive integer coefficients maintain a combinatorial interpretation. It is important to note that graph motif parameters can be nonnegative for all inputs G, even when some coefficients are negative. Formally, we show that evaluating any graph motif parameter with a negative coefficient is impossible in an oracle variant of #P, where an implicit graph is accessed by oracle queries. Our proof follows the classification of the relativizing closure properties of #P by Hertrampf, Vollmer, and Wagner (SCT'95) and the framework developed by Ikenmeyer and Pak (STOC'22), but our application of the required Ramsey theorem turns out to be more subtle, as graphs do not have the required Ramsey property. Our techniques generalize from graphs to relational structures, including colored graphs. Vastly generalizing this, we introduce motif parameters over categories that count occurrences of sub-objects in the category. We then prove a general dichotomy theorem that characterizes which such parameters have a combinatorial interpretation. Using known results in Ramsey theory for categories, we obtain a dichotomy for motif parameters of finite vector spaces as well as parameter sets.

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arXiv2025

Counting Small Induced Subgraphs: Scorpions Are Easy but Not Trivial

We consider the parameterized problem $\#$IndSub$(Φ)$ for fixed graph properties $Φ$: Given a graph $G$ and an integer $k$, this problem asks to count the number of induced $k$-vertex subgraphs satisfying $Φ$. Dörfler et al. [Algorithmica 2022] and Roth et al. [SICOMP 2024] conjectured that $\#$IndSub$(Φ)$ is $\#$W[1]-hard for all non-meager properties $Φ$, i.e., properties that are nontrivial for infinitely many $k$. This conjecture has been confirmed for several restricted types of properties, including all hereditary properties [STOC 2022] and all edge-monotone properties [STOC 2024]. In this work, we refute this conjecture by showing that scorpion graphs, certain $k$-vertex graphs which were introduced more than 50 years ago in the context of the evasiveness conjecture, can be counted in time $O(n^4)$ for all $k$. A simple variant of this construction results in graph properties that achieve arbitrary intermediate complexity assuming ETH. We formulate an updated conjecture on the complexity of $\#$IndSub$(Φ)$ that correctly captures the complexity status of scorpions and related constructions.

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arXiv2025

Faster Convolutions: Yates and Strassen Revisited

Given two vectors $u,v \in \mathbb{Q}^D$ over a finite domain $D$ and a function $f : D\times D\to D$, the convolution problem asks to compute the vector $w \in \mathbb{Q}^D$ whose entries are defined by $w(d) = \sum_{\substack{x,y \in D \\ f(x,y)=d}} u(x)v(y).$ In parameterized and exponential-time algorithms, convolutions on product domains are particularly prominent: Here, a finite domain $B$ and a function $h : B \times B \to B$ are fixed, and convolution is done over the product domain $D = B^k$, using the function $h^k :D \times D\to D$ that applies $h$ coordinate-wise to its input tuples. We present a new perspective on product-domain convolutions through multilinear algebra. This viewpoint streamlines the presentation and analysis of existing algorithms, such as those by van Rooij et al. (ESA 2009). Moreover, using established results from the theory of fast matrix multiplication, we derive improved $O^\ast(|B|^{2ω/3 \cdot k}) = O(|D|^{1.582})$ time algorithms, improving upon previous upper bounds by Esmer et al. (Algorithmica 86(1), 2024) of the form $c^k |B|^{2k}$ for $c < 1$. Using the setup described in this note, Strassen's asymptotic rank conjecture from algebraic complexity theory would imply quasi-linear $|D|^{1+o(1)}$ time algorithms. This conjecture has recently gained attention in the algorithms community. (Björklund-Kaski and Pratt, STOC 2024, Björklund et al., SODA 2025) Our paper is intended as a self-contained exposition for an algorithms audience, and it includes all essential mathematical prerequisites with explicit coordinate-based notation. In particular, we assume no knowledge in abstract algebra.

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arXiv2025

Monotone Bounded-Depth Complexity of Homomorphism Polynomials

For every fixed graph $H$, it is known that homomorphism counts from $H$ and colorful $H$-subgraph counts can be determined in $O(n^{t+1})$ time on $n$-vertex input graphs $G$, where $t$ is the treewidth of $H$. On the other hand, a running time of $n^{o(t / \log t)}$ would refute the exponential-time hypothesis. Komarath, Pandey and Rahul (Algorithmica, 2023) studied algebraic variants of these counting problems, i.e., homomorphism and subgraph $\textit{polynomials}$ for fixed graphs $H$. These polynomials are weighted sums over the objects counted above, where each object is weighted by the product of variables corresponding to edges contained in the object. As shown by Komarath et al., the $\textit{monotone}$ circuit complexity of the homomorphism polynomial for $H$ is $Θ(n^{\mathrm{tw}(H)+1})$. In this paper, we characterize the power of monotone $\textit{bounded-depth}$ circuits for homomorphism and colorful subgraph polynomials. This leads us to discover a natural hierarchy of graph parameters $\mathrm{tw}_Δ(H)$, for fixed $Δ\in \mathbb N$, which capture the width of tree-decompositions for $H$ when the underlying tree is required to have depth at most $Δ$. We prove that monotone circuits of product-depth $Δ$ computing the homomorphism polynomial for $H$ require size $Θ(n^{\mathrm{tw}_Δ(H^{\dagger})+1})$, where $H^{\dagger}$ is the graph obtained from $H$ by removing all degree-$1$ vertices. This allows us to derive an optimal depth hierarchy theorem for monotone bounded-depth circuits through graph-theoretic arguments.

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arXiv2025

Can You Link Up With Treewidth?

In a fundamental paper in parameterized complexity theory, Marx [ToC '10] constructed $k$-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. We give 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 colorful 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].

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arXiv2024

Fast Deterministic Chromatic Number under the Asymptotic Rank Conjecture

In this paper we further explore the recently discovered connection by Björklund and Kaski [STOC 2024] and Pratt [STOC 2024] between the asymptotic rank conjecture of Strassen [Progr. Math. 1994] and the three-way partitioning problem. We show that under the asymptotic rank conjecture, the chromatic number of an $n$-vertex graph can be computed deterministically in $O(1.99982^n)$ time, thus giving a conditional answer to a question of Zamir [ICALP 2021], and questioning the optimality of the $2^n\operatorname{poly}(n)$ time algorithm for chromatic number by Björklund, Husfeldt, and Koivisto [SICOMP 2009]. Viewed in the other direction, if chromatic number indeed requires deterministic algorithms to run in close to $2^n$ time, we obtain a sequence of explicit tensors of superlinear rank, falsifying the asymptotic rank conjecture. Our technique is a combination of earlier algorithms for detecting $k$-colorings for small $k$ and enumerating $k$-colorable subgraphs, with an extension and derandomisation of Pratt's tensor-based algorithm for balanced three-way partitioning to the unbalanced case.

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arXiv2023

Count on CFI graphs for #P-hardness

Given graphs $H$ and $G$, possibly with vertex-colors, a homomorphism is a function $f:V(H)\to V(G)$ that preserves colors and edges. Many interesting counting problems (e.g., subgraph and induced subgraph counts) are finite linear combinations $p(\cdot)=\sum_{H}α_{H}\hom(H,\cdot)$ of homomorphism counts, and such linear combinations are known to be hard to evaluate iff they contain a large-treewidth graph $S$. The hardness can be shown in two steps: First, the problems $\hom(S,\cdot)$ for colorful (i.e., bijectively colored) large-treewidth graphs $S$ are shown to be hard. In a second step, these problems are reduced to finite linear combinations of homomorphism counts that contain the uncolored version $S^{\circ}$ of $S$. This step can be performed via inclusion-exclusion in $2^{|E(S)|}\mathrm{poly}(n,s)$ time, where $n$ is the size of the input graph and $s$ is the maximum number of vertices among all graphs in the linear combination. We show that the second step can be performed even in time $4^{Δ(S)}\mathrm{poly}(n,s)$, where $Δ(S)$ is the maximum degree of $S$. Our reduction is based on graph products with Cai-Fürer-Immerman graphs, a novel technique that is likely of independent interest. For colorful graphs $S$ of constant maximum degree, this technique yields a polynomial-time reduction from $\hom(S,\cdot)$ to linear combinations of homomorphism counts involving $S^{\circ}$. Under certain conditions, it actually suffices that a supergraph $T$ of $S^{\circ}$ is contained in the target linear combination. The new reduction yields $\mathsf{\#P}$-hardness results for several counting problems that could previously be studied only under parameterized complexity assumptions. This includes the problems of counting, on input a graph from a restricted graph class and a general graph $G$, the homomorphisms or (induced) subgraph copies from $H$ in $G$.

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arXiv2022

Determinants from homomorphisms

We give a new combinatorial explanation for well-known relations between determinants and traces of matrix powers. Such relations can be used to obtain polynomial-time and poly-logarithmic space algorithms for the determinant. Our new explanation avoids linear-algebraic arguments and instead exploits a classical connection between subgraph and homomorphism counts.

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arXiv2021

Parameterizing the Permanent: Hardness for $K_8$-minor-free graphs

In the 1960s, statistical physicists discovered a fascinating algorithm for counting perfect matchings in planar graphs. Valiant later showed that the same problem is #P-hard for general graphs. Since then, the algorithm for planar graphs was extended to bounded-genus graphs, to graphs excluding $K_{3,3}$ or $K_{5}$, and more generally, to any graph class excluding a fixed minor $H$ that can be drawn in the plane with a single crossing. This stirred up hopes that counting perfect matchings might be polynomial-time solvable for graph classes excluding any fixed minor $H$. Alas, in this paper, we show #P-hardness for $K_{8}$-minor-free graphs by a simple and self-contained argument.

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arXiv2021

Modular counting of subgraphs: Matchings, matching-splittable graphs, and paths

We systematically investigate the complexity of counting subgraph patterns modulo fixed integers. For example, it is known that the parity of the number of $k$-matchings can be determined in polynomial time by a simple reduction to the determinant. We generalize this to an $n^{f(t,s)}$-time algorithm to compute modulo $2^t$ the number of subgraph occurrences of patterns that are $s$ vertices away from being matchings. This shows that the known polynomial-time cases of subgraph detection (Jansen and Marx, SODA 2015) carry over into the setting of counting modulo $2^t$. Complementing our algorithm, we also give a simple and self-contained proof that counting $k$-matchings modulo odd integers $q$ is Mod_q-W[1]-complete and prove that counting $k$-paths modulo $2$ is Parity-W[1]-complete, answering an open question by Björklund, Dell, and Husfeldt (ICALP 2015).

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arXiv2021

A full complexity dichotomy for immanant families

Given an integer $n\geq 1$ and an irreducible character $χ_λ$ of $S_{n}$ for some partition $λ$ of $n$, the immanant $\mathrm{imm}_λ:\mathbb{C}^{n\times n}\to\mathbb{C}$ maps matrices $A\in\mathbb{C}^{n\times n}$ to $\mathrm{imm}_λ(A)=\sum_{π\in S_{n}}χ_λ(π)\prod_{i=1}^{n}A_{i,π(i)}$. Important special cases include the determinant and permanent, which are the immanants associated with the sign and trivial character, respectively. It is known that immanants can be evaluated in polynomial time for characters that are close to the sign character: Given a partition $λ$ of $n$ with $s$ parts, let $b(λ):=n-s$ count the boxes to the right of the first column in the Young diagram of $λ$. For a family of partitions $Λ$, let $b(Λ):=\max_{λ\inΛ}b(λ)$ and write Imm$(Λ)$ for the problem of evaluating $\mathrm{imm}_λ(A)$ on input $A$ and $λ\inΛ$. If $b(Λ)<\infty$, then Imm$(Λ)$ is known to be polynomial-time computable. This subsumes the case of the determinant. On the other hand, if $b(Λ)=\infty$, then previously known hardness results suggest that Imm$(Λ)$ cannot be solved in polynomial time. However, these results only address certain restricted classes of families $Λ$. In this paper, we show that the parameterized complexity assumption FPT $\neq$ #W[1] rules out polynomial-time algorithms for Imm$(Λ)$ for any computationally reasonable family of partitions $Λ$ with $b(Λ)=\infty$. We give an analogous result in algebraic complexity under the assumption VFPT $\neq$ VW[1]. Furthermore, if $b(λ)$ even grows polynomially in $Λ$, we show that Imm$(Λ)$ is hard for #P and VNP. This concludes a series of partial results on the complexity of immanants obtained over the last 35 years.

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arXiv2019

A Fixed-Parameter Perspective on #BIS

The problem of (approximately) counting the independent sets of a bipartite graph (#BIS) is the canonical approximate counting problem that is complete in the intermediate complexity class #RHΠ_1. It is believed that #BIS does not have an efficient approximation algorithm but also that it is not NP-hard. We study the robustness of the intermediate complexity of #BIS by considering variants of the problem parameterised by the size of the independent set. We exhaustively map the complexity landscape for three problems, with respect to exact computation and approximation and with respect to conventional and parameterised complexity. The three problems are counting independent sets of a given size, counting independent sets with a given number of vertices in one vertex class and counting maximum independent sets amongst those with a given number of vertices in one vertex class. Among other things, we show that all of these problems are NP-hard to approximate within any polynomial ratio. (This is surprising because the corresponding problems without the size parameter are complete in #RHΠ_1, and hence are not believed to be NP-hard.) We also show that the first problem is #W[1]-hard to solve exactly but admits an FPTRAS, whereas the other two are W[1]-hard to approximate even within any polynomial ratio. Finally, we show that, when restricted to graphs of bounded degree, all three problems have efficient exact fixed-parameter algorithms.

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arXiv2018

Counting edge-injective homomorphisms and matchings on restricted graph classes

We consider the $\#\mathsf{W}[1]$-hard problem of counting all matchings with exactly $k$ edges in a given input graph $G$; we prove that it remains $\#\mathsf{W}[1]$-hard on graphs $G$ that are line graphs or bipartite graphs with degree $2$ on one side. In our proofs, we use that $k$-matchings in line graphs can be equivalently viewed as edge-injective homomorphisms from the disjoint union of $k$ length-$2$ paths into (arbitrary) host graphs. Here, a homomorphism from $H$ to $G$ is edge-injective if it maps any two distinct edges of $H$ to distinct edges in $G$. We show that edge-injective homomorphisms from a pattern graph $H$ can be counted in polynomial time if $H$ has bounded vertex-cover number after removing isolated edges. For hereditary classes $\mathcal{H}$ of pattern graphs, we complement this result: If the graphs in $\mathcal{H}$ have unbounded vertex-cover number even after deleting isolated edges, then counting edge-injective homomorphisms with patterns from $\mathcal{H}$ is $\#\mathsf{W}[1]$-hard. Our proofs rely on an edge-colored variant of Holant problems and a delicate interpolation argument; both may be of independent interest.

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