graph algorithm engineers

Abstract: Given a mixed graph $G$ and an ordered pair $(u,v)$ of vertices in $V(G)$, we say that an orientation $\vec{G}$ of $G$ satisfies $(u,v)$ if $v$ is reachable from $u$ in $\vec{G}$. We consider problems in which we seek orientations that satisfy a maximum number of pairs. We distinguish between two different problems. In the most general problem called Fixed Pairs Maximum Mixed Reachability Orientation (FPRO), a multiset $P$ of ordered pairs of vertices in $V(G)$ comes with the input. In the more restricted problem All Pairs Maximum Mixed Reachability Orientation (APRO), $P$ contains exactly one copy of every ordered pair of distinct vertices in $V(G)$. We compare the computational complexity of FPRO and APRO in two ways: First, we show that both FPRO and APRO behave similarly when it comes to approximation. Namely, they both admit an asymptotic approximation, but are unlikely to admit a constant factor approximation. Then, we consider the parameterization with respect to the treewidth of the underlying graph of the input mixed graph. We show that here the complexities of FPRO and APRO differ significantly. While it is well-known that FPRO is NP-hard even when this parameter is 1, we show that APRO is XP with respect to this parameter. On the other hand, we give a $W[1]$-hardness result for APRO.

Abstract: In this paper, we study the theoretical computational complexity of counting trees and graphs that are classified positively by a graph neural networks, or said equivalently that satisfy a given formula of modal logic with counting. We also study convergence.

Abstract: A potential maximal clique (PMC) of a graph $G$ is a nonempty vertex set that forms a maximal clique in some minimal triangulation of $G$. Listing the PMCs of a graph is a key step in exact algorithms for computing treewidth, minimum fill-in, and related graph parameters. We present a deterministic algorithm that enumerates all PMCs of cardinality at most $k$ in an $n$-vertex graph $G$, without repetition, with delay $2^{O(k^3)}n^4\log n$. This bound also covers the preprocessing time before the first output, the time from the last output to termination, and the total running time when no such PMC exists. At the core of our enumeration algorithm is a procedure that solves the following constrained search problem within the same time bound: given sets of required and excluded vertices $I,O\subseteq V(G)$, find a PMC $Ω$ of size at most $k$ satisfying $I\subseteqΩ$ and $Ω\cap O=\emptyset$, or determine that none exists. Our enumeration algorithm directly supports these inclusion and exclusion constraints as well. Our approach builds on the \emph{recursive understanding} technique of Chitnis et al.~\cite{chitnis2016} and the reduction to unbreakable graphs of Lokshtanov et al.~\cite{lokshtanov2018}. A graph is $(q,k)$-unbreakable if no removal of at most $k$ vertices leaves two disjoint unions of connected components that each contain more than $q$ vertices. We first solve constrained PMC search on $(q,k)$-unbreakable graphs, and then apply this procedure to replace separated subgraphs with graphs of size $2^{O(k^2)}$ that preserve the existence of a constrained solution for every graph attached along the shared boundary.

Abstract: Given a finite planar point set $X=\{x_1,\ldots,x_n\}$, let $a_1<a_2< \cdots < a_s$ denote the $s \leq {n \choose 2}$ different (Euclidean) distances between points in $X$. The aim is to retain as many of the points as possible such that certain imposed inter-pair distances are avoided. For example, we show that: (A) Given a set $X$ of $n$ points in the plane, an independent set of size $0.2293 n$ in the unit distance graph determined by $X$ can be found in $O(n^2)$ time. For any fixed $k \in \mathbf{N}$ there exists a constant $c_k>0$ such that for any set $X$ of $n$ points in the plane, and set $D$ of size $k$, there exists a subset $X' \subseteq X$ of size $|X'| \geq c_k |X|$ in which no pair of points is at some distance in $D$ (we say that $X'$ avoids distances in $D$). Such a set can be found efficiently. (B) (i) There is a subset of $X$ of size $n/2$ that avoids the maximum distance; this bound cannot be improved. (ii) There is a subset of $X$ of size $8n/47$ that avoids both the minimum and maximum distance; on the other hand, there exists $X$ such that at most $5n/16$ points can be retained with this property. (iii) There is a subset of $X$ of size $n/7$ whose diameter is at most half the original diameter; on the other hand, there exists $X$ such that at most $n/6$ points can be retained with this property. (C) (i) For any $\varepsilon>0$, a $(1-\varepsilon)$-approximation for \textsc{MIS} in the minimum distance graph of $n$ points in the plane can be found in $2^{O(1/\varepsilon)} n^{O(1)}$ time. (ii) Given a set $X$ of $n$ points in the plane, a maximum independent set in the graph of diameters can be computed in $O(n^2)$ time. The techniques used to handle the planar case can be extended to $\mathbf{R}^d$ for any fixed dimension $d$.

Abstract: Exact forward triangle listers orient each edge of a graph by a vertex order. The order does not change the output, but it changes the work. For the adaptive lister, which scans the smaller of the two forward neighborhoods, the number of candidate tests is a known sum of minima of forward degrees, but how to optimize it had not been characterized. We show that this work is a concave function of the forward degrees, so choosing an order is concave minimization over the orientation polytope, whose vertices are the forward-degree vectors of orders. A single sort therefore rounds any point of the polytope, including the out-degree vector of any orientation, to an order with no more work, with a computable lower bound on the improvement. Minimizing the work is nevertheless NP-hard and APX-hard on triangle-free graphs of maximum degree three, and under the Unique Games Conjecture it has no polynomial-time constant-factor approximation. We also analyze logistic score relaxations and the large-graph limit of degree-based orders; on random cubic graphs, the expected work of every degree-based order is asymptotically at least $2.43$ times the expected optimum. Tangent refinement followed by iterated local search reduces the work by \HeldTIKMean\% on average on nine held-out networks fixed by a pre-registered protocol, and by \NSFrozenTILSMean\% in a supplementary rescoring, specified in advance, against a stronger reference that adds the published Neigh local search. The gains are in candidate tests; we observe no reproducible wall-clock speedup.

Abstract: String graphs are intersection graphs of continuous curves in the plane. In 2010, Fox and Pach proved that string graphs with $m$ edges admit balanced vertex separators of size $\mathcal{O}(m^{3/4}\sqrt{\log m})$ and conjectured that this bound can be improved to $\mathcal{O}(\sqrt{m})$, which would be optimal. After partial progress, Lee claimed in 2017 to prove this conjecture, and more generally to prove analogous separator theorems for region intersection graphs over minor-closed graph families. A key intermediate step in Lee's approach was a padded decomposition scheme for region intersection graphs. In 2024, Korhonen and Lokshtanov extended Lee's approach to the broader setting of graph classes excluding a fixed graph as an induced minor. These results have since found applications in a wide range of works. Later in 2024, a flaw was discovered in the construction of Lee's padded decomposition scheme, invalidating Lee's as well as Korhonen and Lokshtanov's separator results, and many works relying on them. In this work, we prove that string graphs with $m$ edges admit balanced separators of size $\mathcal{O}(\sqrt{m})$, and more generally obtain analogous $\mathcal{O}(\sqrt{m})$ bounds for region intersection graphs over fixed minor-closed graph families and for graphs excluding a fixed graph as an induced minor, thereby repairing the vast majority of the affected works. Furthermore, our balanced separators can be found in polynomial time. Our proof uses a different approach from Lee's and does not rely on padded decompositions. Since padded decompositions have also found applications independent of the separator theorems, particularly for string graphs, we also prove a padded decomposition result for region intersection graphs that is optimal for string graphs. We have formalized both the balanced separator and the padded decomposition results in Lean 4 with mathlib.

Abstract: We study \textsc{Bounded-Density Vertex Deletion} (BDVD), where one asks whether at most \(k\) vertices can be deleted so that every remaining subgraph has density at most a rational input threshold \(τ_ρ\). We address the twin-cover and distance-to-cluster cases left open by Raupach, Breitkopf, Herrmann, and Nichterlein (MFCS 2026). We prove fixed-parameter tractability parameterized solely by the twin-cover number, for unrestricted input density. For distance to cluster, we establish fixed-parameter tractability parameterized by \(\operatorname{dc}(G)+\lfloor2τ_ρ+1\rfloor\), and complement this with \(\mathsf{W[1]}\)-hardness parameterized by \(\operatorname{dc}(G)\) alone when the target density is part of the input, even for integral targets. Thus distance to cluster alone is fixed-parameter tractable throughout \(τ_ρ<1\), while it is insufficient for unrestricted input density. Finally, we settle the combined parameter treewidth plus solution size, an open problem posed by Bazgan, Nichterlein, and Vazquez Alferez (JCSS 2025). We prove that BDVD is \(\mathsf{W[1]}\)-hard parameterized by \(\operatorname{td}(G)+k\), and hence also by \(\operatorname{tw}(G)+k\), already for targets strictly between \(1\) and \(2\).

Abstract: Madiman and Tetali established strong and weak fractional inequalities that subsume classical entropy subset inequalities such as Han's inequality and Shearer's lemma. In this work, we derive a decomposition of the gap in the strong fractional inequality and use it to establish the necessary and sufficient conditions for equality in the strong form of Han's inequality. As a consequence, we characterize the equality conditions for a strong Loomis-Whitnet-type projection inequality of Balister and Bollobas (2012). Finally, we apply the weak fractional inequality to an extremal graph theory problem, recovering and extending results of Sason (2022) and Boucheron et al. (2013), whose approaches rely on Han's inequality.

Abstract: A landmark result states that the expressive power of graph neural networks (GNNs) is bounded from above by the 1-dimensional Weisfeiler-Leman (1-WL) test. We ask whether a constant-depth GNN can match the expressive power of 1-WL while using only rational weights. We prove that this is the case when Schanuel's conjecture, a famous open problem from transcendental number theory, is taken to be true. Our GNN construction works even with integer weights, and more generally with a broad range of rational weights. As the activation function, it may use logistic sigmoid, hyperbolic tangent, or related functions. It is uniform across all graph sizes. We also prove a partial converse: if a GNN with exponentiation, hyperbolic sine or hyperbolic cosine as activation matches the expressive power of the 1-WL test in a slightly more expressive setting where graphs may contain certain auxiliary vertices, then this implies Schanuel's conjecture restricted to the numbers that arise in computations of the GNN. We also conduct experiments showing that the constructed GNNs indeed match the 1-WL test.

Abstract: Many exact Hamiltonian-cycle solvers use the same edge-level representation for every input, even when graph structure admits a smaller set of global decisions. We present StructHCP, a SAT--CEGAR pipeline that selects a representation through inexpensive structural recognizers. Specialized searches handle high-degree clusters and paired chains, while complete directed and undirected encodings handle cubic and general graphs. Every returned cycle is reconstructed when necessary and verified in the input graph. On the 1,001-instance FHCP Challenge Set, StructHCP solves 826 instances, compared with 741 when the same general SAT--CEGAR engine is applied directly to every unreduced input graph. Relative to this ablation, StructHCP reduces PAR-2 by 32.9 percent. Formulation selection solves 82 more instances than the contracted general encoding; high-degree-cluster and cubic graphs account for 81 of this difference. StructHCP also solves 34 of the 51 instances solved by neither HCup nor Cut-set CEGAR, raising the combined coverage of the three solvers from 950 to 984. Twelve of its solutions lie outside the union of published per-instance SAT results analyzed here, and six also lie outside the published exact result sets in our comparison. These results show that selecting the representation can complement connectivity refinement on structured HCP instances.

Abstract: The small blocking set detection and construction problem asks whether a given connected simple graph contains a nonempty independent set whose cardinality equals the size of its neighborhood and is strictly less than half the number of vertices, and, if so, how to construct such a set. Previous work characterized restless explorability in vertex-permuted temporal graphs using small blocking sets, but left the computational complexity of detecting their existence as an open problem. We give a polynomial-time algorithm for detection and construction. Specifically, we first compute a maximum-cardinality critical independent set and use its difference and cardinality to handle the cases that admit an immediate solution. In the remaining boundary case, where the critical difference is zero and the selected independent set contains exactly half the vertices, we establish a projection theorem using a perfect matching and neighborhood counting: the graph contains a small blocking set if and only if the selected independent set contains a nonempty proper tight subset. We then use matching contraction to translate the tight-subset condition into the absence of arcs leaving a vertex subset in an auxiliary digraph. This yields a strong-connectivity test for the existence of a small blocking set and, when one exists, a construction based on a sink strongly connected component. The result resolves the open complexity question for small blocking set detection and provides an explicit construction algorithm.

Abstract: A classical well-quasi-ordering result guarantees the existence of non-uniform linear-time algorithms for all problems in Strict NP on relational structures of bounded treedepth; however, this provides neither a procedure for constructing these algorithms nor computable bounds on their parameter dependence. We turn this existential result into a uniform algorithmic metatheorem. Given a Strict NP sentence $\varphi$ and a relational structure $\mathcal{R}$, our algorithm decides whether $\mathcal{R}\models\varphi$ in time $f(|\varphi|, td(\mathcal{R})) \cdot |\mathcal{R}|$ for a computable function $f$, where the treedepth of $\mathcal{R}$ is measured on the Gaifman graph. The algorithm also constructs witness relations, with the polynomial exponent depending on their arity, and provides a unified framework for settling hereditary graph problems parameterized by treedepth. We also present several applications - among others, our result resolves open questions on the fixed-parameter tractability of computing the stack number, queue number, track number and twin-width parameterized by treedepth.

Abstract: Topology is, by its nature and design, suited to structure that is nonlinear, multiscale, and nonstationary - however, within machine learning, its use remains largely confined to topological data analysis. We advocate that tools from low-dimensional topology which have remained almost exclusively contained within the domain of pure mathematics (such as Morse theory) offer a strong, complementary, and yet virtually unexplored perspective on the hidden structure of data-generating processes and learning tasks built upon them. Here we introduce concepts from cobordism theory and harness tools from discrete Morse theory to improve the performance of graph diffusion models through our pipeline MG-Diff. Further, we derive theoretical guarantees and sufficient conditions so that under a positive decision-gap, the Morse-theoretic tools and their application for induced diffusion guidance are stable under small perturbations. Finally, we illustrate the utility of discrete Morse theory in application to graph diffusion models for spatio-temporal graph forecasting and graph regeneration, and argue that these applications are only a small window into the part of what low-dimensional topology can offer to the field of machine learning.

Abstract: Subgraph detection asks whether an $n$-vertex graph, accessed through queries to its adjacency matrix, contains a copy of a fixed graph $H$. We prove the first unconditional superlinear lower bounds on the bounded-error quantum query complexity of this problem, answering a longstanding open question. A copy of $H$ is a certificate of constant size, so the adversary method with nonnegative weights cannot prove superlinear lower bounds. For every fixed $r\ge 4$, detecting the clique $K_r$ requires $n^{λ_r-o(1)}$ queries, where $λ_4=19/18$, the exponents $λ_r$ increase strictly with $r$, and $λ_r\ge 2-4\sqrt{2/r}+O(1/r)$. More generally, we prove superlinear lower bounds for detecting every fixed connected graph $H$ with chromatic number $c\ge 4$. These bounds approach quadratic as $c$ grows: for sufficiently large $c$, detection requires $n^{2-O(\sqrt{\log\log c/c})-o(1)}$ queries. Chromatic number alone does not characterize the quantum query complexity of subgraph detection: we show that detecting the complete bipartite graph $K_{r,r}$ requires $n^{β_r-o(1)}$ queries, where $β_{10}=181/180$ and $β_r\ge 2-O(1/\sqrt{r})$. Our main technical result is a lower bound for finding an all-ones certificate from a known family when the input bits are sampled independently. Its proof combines Zhandry's compressed oracle (CRYPTO 2019) with conditioning on a randomly planted certificate, adapting an argument of Belovs (FOCS 2026). Our hard instances are built from graphs containing many copies of the desired subgraph with limited overlap. For cliques, we use a construction of Gowers and Janzer (CPC 2021); for complete bipartite graphs, we use a random construction.

Abstract: We prove the first non-trivial upper bound for the quantum query complexity of Boolean Matrix Product Verification ($\mathsf{BMPV}$), answering a longstanding open question in quantum query complexity. For $n\times n$ matrices, our upper bound is $\widetilde O(n^{17/12})$, improving on the standard $O(n^{3/2})$ bound obtained using Grover search by Buhrman and Špalek (SODA 2006). We complement this result by showing an $Ω(n^{5/4})$ lower bound, which improves over the previous best known lower bound of $\widetildeΩ(n^{19/18})$ by Childs, Kimmel, and Kothari (ESA 2012). Our approach centers on a connection with Orthogonal Vectors ($\mathsf{OV}$), which asks whether an indexed list of $n$ Boolean vectors of dimension $n$ contains two vectors with disjoint supports. In particular, we prove equivalences between $\mathsf{OV}$ and $\mathsf{BMPV}$ and establish the above bounds for $\mathsf{OV}$. We also prove a tight $\widetilde Θ(n^{3/2})$ bound for a variant of $\mathsf{BMPV}$ that asks whether the product contains a given row vector. Together, these results imply a polynomial separation between the quantum query complexities of deciding whether a graph has radius at most two and whether it has diameter at most two.

Abstract: Graph foundation models need a discrete token representation, but casting a graph as a generatable token sequence faces a structural obstacle: edges spanning beyond the serialization window cannot be emitted in one pass--so one-pass autoregressive generators systematically under-produce cycles--and a single global condition cannot tell candidate edges apart. GraphVQ removes both obstacles: node contexts--features plus a local edge mask under multi-order breadth-first serialization--are quantized into a shared codebook by a VQ-VAE with BCE-calibrated Bernoulli edge decoding, and a second-stage structure-aware decoder emits the global adjacency conditioned on token-derived pair features, whose necessity over any global-summary condition is formalized in a scoped impossibility result. The tokenizer reconstructs node features at 0.86--0.99 accuracy and decodes local edges at AUROC >= 0.89 (ECE <= 0.007). Under one same-split protocol on four datasets, pair conditioning improves orbit MMD 0.248 -> 0.174 on PROTEINS and 3.4x on a ring stress test, and vanishes on a random-label control--the signature of attribute--topology coupling--so the gain is claimed exactly where attributes carry edge-relevant signal. GraphVQ ranks first among learned generators on PROTEINS, ties for first on SYN-COMM, and improves orbit MMD 2.7--17x over one-stage generation on three datasets, with seed-level bootstrap intervals confirming the rankings are not seed noise; on MUTAG the unweighted edge target under-generates and is reported as such. These results locate the structural control of autoregressive graph generation in the granularity of the condition: pair-level token context turns a quantized vocabulary into a usable capacity axis for distribution-faithful graph generation and future token-level pretraining.

Abstract: In the Bipartite One-sided Vertex Explosion problem (BOVE), the input is a bipartite graph $G = (T, B, E)$ and a nonnegative integer $k$. The goal is to decide whether there is a set $S \subseteq B$ of size at most $k$ such that exploding every vertex in $S$ results in a graph with pathwidth at most one. The restricted Pathwidth One Vertex Explosion problem (restricted POVE) is a generalization of BOVE. The input to this problem is a graph $G$, a nonnegative integer $k$, and a set of vertices $X \subseteq V($ The goal is to decide whether there is a set $S \subseteq X$ of size at most $k$ such that exploding every vertex in $S$ results in a graph with pathwidth at most one. In this paper, we show parameter-preserving reductions in both directions between BOVE and Co-Path Set, and between restricted POVE and restricted Co-Path Packing. These reductions yield improved parameterized algorithms for BOVE and restricted POVE and an improved kernel for BOVE.

Abstract: The Minimum Sum Vertex Cover (MSVC) problem asks for an ordering of the vertices of a graph that minimizes the sum, over all edges, of the time at which each edge is first covered. We study the problem through the structure of vertex covers and obtain new approximation and exact algorithms, together with conditional lower bounds. For graphs of maximum degree $Δ$, we show that a simple ordering algorithm based on a minimum vertex cover achieves approximation ratio $R_Δ\le {(\sqrtΔ+1)}/{2}$. For $d$-regular graphs, we give a polynomial-time $1.184$-approximation by combining Max-$k$-Vertex-Cover approximation with a structural bound on optimal prefixes. On the exact side, we give an algorithm parameterized by the vertex cover number $k$ running in $2^{O(k\log k)} + O(n+m)$ time, improving the previous dependence on $k$, where $n$ and $m$ are the number of vertices and edges in the graph, respectively. We also develop a separator-based exact algorithm running in $ 2^{O(\sqrt n \log n)}$ time on planar, bounded-genus, and fixed-minor-free graph classes. Finally, we prove that Minimum Sum Vertex Cover is NP-hard on planar graphs and, assuming ETH, admits no $2^{o(\sqrt n)}$-time exact algorithm on $n$-vertex planar graphs. Thus our planar upper bound is tight up to logarithmic factors in the exponent.

Abstract: Graph-based Android malware classifiers can lose accuracy under malware-type or family shifts. We test whether mesoscopic organization in function-call graphs provides shift-stable information beyond local degree profiles (LDP), global statistics, lightweight metadata, and size-matched random partitions. Using 15,000 MalNet-Tiny, Common, and Distinct graphs, six Leiden descriptors specified before evaluation, and five optimizer seeds, communities raise Tiny macro F1 from 78.7% to 81.3% but yield 29.4-point source-only Common degradation. One size-matched random partition yields 27.8-point degradation. Across five random partitions, mean degradation is 27.9 points; the 95% two-level bootstrap interval for random minus community degradation is [-4.2, 1.2] points. With metadata, the corresponding difference is -0.2 points with interval [-1.1, 0.6]. Removing modularity raises structure-only Common macro F1 from 51.9% to 53.6%. The tested signature adds IID signal but shows no repeatable shift-stability advantage, demonstrating why mesoscopic graph claims need partition-matched controls and repeated null draws.

Abstract: We show that isomorphism of $F$-free tournaments can be solved in FPT time $f(k) \cdot n^{O(1)}$, where $k$ denotes the size of $F$, and $n$ denotes the size of the input tournaments. Our result extends on a previous FPT isomorphism test for tournaments of bounded twin-width [Grohe, Neuen 2024], as well as XP isomorphism tests parameterized by the VC dimension or the chromatic number [Raßmann, Schweitzer 2026]. It also implies that every non-trivial hereditary class of tournaments admits a polynomial-time isomorphism test. Our algorithm builds on a novel combination of spectral, geometric, combinatorial and group-theoretic tools.

Abstract: In the 1960s, Erdős and Hajnal conjectured that every graph with sufficiently large chromatic number contains a subgraph of large girth (size of a smallest cycle) and large chromatic number. In this paper, we prove that every tournament with sufficiently large dichromatic number contains a subdigraph of large digirth (size of a smallest directed cycle) and large dichromatic number. We investigate the same statement when replacing digirth by girth (of the underlying graph). We show that it implies the conjecture of Erdős and Hajnal, and prove it for a particular family of tournaments.