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Gaurav Rattan

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12 papers

2 author rows

JAIR Journal 2026 Journal Article

Satsuma: Structure-Based Symmetry Breaking in SAT

Markus Anders
Sofia Brenner
Gaurav Rattan

Symmetry reduction is crucial for solving many interesting SAT instances in practice. Numerous approaches have been proposed, which try to strike a balance between symmetry reduction and computational overhead. Arguably the most readily applicable method is the computation of static symmetry breaking constraints: a constraint restricting the search-space to non-symmetrical solutions is added to a given SAT instance. A distinct advantage of static symmetry breaking is that the SAT solver itself is not modified. A disadvantage is that the strength of symmetry reduction is usually limited. In order to boost symmetry reduction, the state-of-the-art tool BreakID (Devriendt et al., 2016) pioneered the identification and tailored breaking of a particular substructure of symmetries, the so-called row interchangeability groups. In this paper, we propose a new symmetry breaking tool called satsuma. The core principle of our tool is to exploit more diverse but frequently occurring symmetry structures. This is enabled by new practical detection algorithms for row interchangeability, row-column symmetry, Johnson symmetry, and various combinations. Based on the resulting structural description, we then produce symmetry breaking constraints. We provide benchmarks testing the effectiveness of our new implementation in conjunction with the state-of-the-art SAT solver Kissat. To this end, we compare satsuma, BreakID, and using no symmetry breaking. We find that satsuma successfully speeds up Kissat on last year’s SAT competition instances. Compared to BreakID, we observe significantly better breaking performance on instances with Johnson symmetry, and lower computational overhead across all tested families.

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SAT Conference 2024 Conference Paper

Satsuma: Structure-Based Symmetry Breaking in SAT

Markus Anders
Sofia Brenner
Gaurav Rattan

Symmetry reduction is crucial for solving many interesting SAT instances in practice. Numerous approaches have been proposed, which try to strike a balance between symmetry reduction and computational overhead. Arguably the most readily applicable method is the computation of static symmetry breaking constraints: a constraint restricting the search-space to non-symmetrical solutions is added to a given SAT instance. A distinct advantage of static symmetry breaking is that the SAT solver itself is not modified. A disadvantage is that the strength of symmetry reduction is usually limited. In order to boost symmetry reduction, the state-of-the-art tool BreakID [Devriendt et. al] pioneered the identification and tailored breaking of a particular substructure of symmetries, the so-called row interchangeability groups. In this paper, we propose a new symmetry breaking tool called satsuma. The core principle of our tool is to exploit more diverse but frequently occurring symmetry structures. This is enabled by new practical detection algorithms for row interchangeability, row-column symmetry, Johnson symmetry, and various combinations. Based on the resulting structural description, we then produce symmetry breaking constraints. We compare this new approach to BreakID on a range of instance families exhibiting symmetry. Our benchmarks suggest improved symmetry reduction in the presence of Johnson symmetry and comparable performance in the presence of row-column symmetry. Moreover, our implementation runs significantly faster, even though it identifies more diverse structures.

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SODA Conference 2023 Conference Paper

Weisfeiler-Leman and Graph Spectra

Gaurav Rattan
Tim Seppelt

Two simple undirected graphs are cospectral if their respective adjacency matrices have the same multiset of eigenvalues. Cospectrality yields an equivalence relation on the family of graphs which is provably weaker than isomorphism. In this paper, we study cospectrality in relation to another well-studied relaxation of isomorphism, namely k -dimensional Weisfeiler-Leman ( k -WL) indistinguishability. Cospectrality with respect to standard graph matrices such as the adjacency or the Laplacian matrix yields a strictly finer equivalence relation than 2-WL indistinguishability. We show that individualising one vertex plus running 1-WL already subsumes cospectrality with respect to all such graph matrices. Building on this result, we resolve an open problem of Fürer (2010) about spectral invariants and strengthen a result of Godsil (1981) about commute distances. Looking beyond 2-WL, we devise a hierarchy of graph matrices generalising the adjacency matrix such that k -WL indistinguishability after a fixed number of iterations can be captured as a spectral condition on these matrices. Precisely, we provide a spectral characterisation of k -WL indistinguishability after d iterations, for k, d ∈ ℕ. Our results can be viewed as characterisations of homomorphism indistinguishability over certain graph classes in terms of matrix equations. The study of homomorphism indistinguishability is an emerging field, to which we contribute by extending the algebraic framework of Mančinska and Roberson (2020) and Grohe et al. (2022). * The full version [45] of the paper can be accessed at https: //arxiv. org/abs/2103. 02972

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NeurIPS Conference 2022 Conference Paper

Ordered Subgraph Aggregation Networks

Chendi Qian
Gaurav Rattan
Floris Geerts
Mathias Niepert
Christopher Morris

Numerous subgraph-enhanced graph neural networks (GNNs) have emerged recently, provably boosting the expressive power of standard (message-passing) GNNs. However, there is a limited understanding of how these approaches relate to each other and to the Weisfeiler-Leman hierarchy. Moreover, current approaches either use all subgraphs of a given size, sample them uniformly at random, or use hand-crafted heuristics instead of learning to select subgraphs in a data-driven manner. Here, we offer a unified way to study such architectures by introducing a theoretical framework and extending the known expressivity results of subgraph-enhanced GNNs. Concretely, we show that increasing subgraph size always increases the expressive power and develop a better understanding of their limitations by relating them to the established $k\mathsf{\text{-}WL}$ hierarchy. In addition, we explore different approaches for learning to sample subgraphs using recent methods for backpropagating through complex discrete probability distributions. Empirically, we study the predictive performance of different subgraph-enhanced GNNs, showing that our data-driven architectures increase prediction accuracy on standard benchmark datasets compared to non-data-driven subgraph-enhanced graph neural networks while reducing computation time.

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ICML Conference 2022 Conference Paper

SpeqNets: Sparsity-aware permutation-equivariant graph networks

Christopher Morris 0001
Gaurav Rattan
Sandra Kiefer
Siamak Ravanbakhsh

While message-passing graph neural networks have clear limitations in approximating permutation-equivariant functions over graphs or general relational data, more expressive, higher-order graph neural networks do not scale to large graphs. They either operate on $k$-order tensors or consider all $k$-node subgraphs, implying an exponential dependence on $k$ in memory requirements, and do not adapt to the sparsity of the graph. By introducing new heuristics for the graph isomorphism problem, we devise a class of universal, permutation-equivariant graph networks, which, unlike previous architectures, offer a fine-grained control between expressivity and scalability and adapt to the sparsity of the graph. These architectures lead to vastly reduced computation times compared to standard higher-order graph networks in the supervised node- and graph-level classification and regression regime while significantly improving standard graph neural network and graph kernel architectures in terms of predictive performance.

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NeurIPS Conference 2020 Conference Paper

Weisfeiler and Leman go sparse: Towards scalable higher-order graph embeddings

Christopher Morris
Gaurav Rattan
Petra Mutzel

Graph kernels based on the $1$-dimensional Weisfeiler-Leman algorithm and corresponding neural architectures recently emerged as powerful tools for (supervised) learning with graphs. However, due to the purely local nature of the algorithms, they might miss essential patterns in the given data and can only handle binary relations. The $k$-dimensional Weisfeiler-Leman algorithm addresses this by considering $k$-tuples, defined over the set of vertices, and defines a suitable notion of adjacency between these vertex tuples. Hence, it accounts for the higher-order interactions between vertices. However, it does not scale and may suffer from overfitting when used in a machine learning setting. Hence, it remains an important open problem to design WL-based graph learning methods that are simultaneously expressive, scalable, and non-overfitting. Here, we propose local variants and corresponding neural architectures, which consider a subset of the original neighborhood, making them more scalable, and less prone to overfitting. The expressive power of (one of) our algorithms is strictly higher than the original algorithm, in terms of ability to distinguish non-isomorphic graphs. Our experimental study confirms that the local algorithms, both kernel and neural architectures, lead to vastly reduced computation times, and prevent overfitting. The kernel version establishes a new state-of-the-art for graph classification on a wide range of benchmark datasets, while the neural version shows promising performance on large-scale molecular regression tasks.

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MFCS Conference 2019 Conference Paper

The Complexity of Homomorphism Indistinguishability

Jan Böker
Yijia Chen 0001
Martin Grohe
Gaurav Rattan

For every graph class {F}, let HomInd({F}) be the problem of deciding whether two given graphs are homomorphism-indistinguishable over {F}, i. e. , for every graph F in {F}, the number hom(F, G) of homomorphisms from F to G equals the corresponding number hom(F, H) for H. For several natural graph classes (such as paths, trees, bounded treewidth graphs), homomorphism-indistinguishability over the class has an efficient structural characterization, resulting in polynomial time solvability [H. Dell et al. , 2018]. In particular, it is known that two non-isomorphic graphs are homomorphism-indistinguishable over the class {T}_k of graphs of treewidth k if and only if they are not distinguished by k-dimensional Weisfeiler-Leman algorithm, a central heuristic for isomorphism testing: this characterization implies a polynomial time algorithm for HomInd({T}_k), for every fixed k in N. In this paper, we show that there is a polynomial-time-decidable class {F} of undirected graphs of bounded treewidth such that HomInd({F}) is undecidable. Our second hardness result concerns the class {K} of complete graphs. We show that HomInd({K}) is co-NP-hard, and in fact, we show completeness for the class C_=P (under P-time Turing reductions). On the algorithmic side, we show that HomInd({P}) can be solved in polynomial time for the class {P} of directed paths. We end with a brief study of two variants of the HomInd({F}) problem: (a) the problem of lexographic-comparison of homomorphism numbers of two graphs, and (b) the problem of computing certain distance-measures (defined via homomorphism numbers) between two graphs.

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AAAI Conference 2019 Conference Paper

Weisfeiler and Leman Go Neural: Higher-Order Graph Neural Networks

Christopher Morris
Martin Ritzert
Matthias Fey
William L. Hamilton
Jan Eric Lenssen
Gaurav Rattan
Martin Grohe

In recent years, graph neural networks (GNNs) have emerged as a powerful neural architecture to learn vector representations of nodes and graphs in a supervised, end-to-end fashion. Up to now, GNNs have only been evaluated empirically—showing promising results. The following work investigates GNNs from a theoretical point of view and relates them to the 1-dimensional Weisfeiler-Leman graph isomorphism heuristic (1-WL). We show that GNNs have the same expressiveness as the 1-WL in terms of distinguishing non-isomorphic (sub-)graphs. Hence, both algorithms also have the same shortcomings. Based on this, we propose a generalization of GNNs, so-called k-dimensional GNNs (k-GNNs), which can take higher-order graph structures at multiple scales into account. These higher-order structures play an essential role in the characterization of social networks and molecule graphs. Our experimental evaluation confirms our theoretical findings as well as confirms that higher-order information is useful in the task of graph classification and regression.

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MFCS Conference 2018 Conference Paper

Graph Similarity and Approximate Isomorphism

Martin Grohe
Gaurav Rattan
Gerhard J. Woeginger

The graph similarity problem, also known as approximate graph isomorphism or graph matching problem, has been extensively studied in the machine learning community, but has not received much attention in the algorithms community: Given two graphs G, H of the same order n with adjacency matrices A_G, A_H, a well-studied measure of similarity is the Frobenius distance dist(G, H): =min_{pi}|A_G^{pi}-A_H|_F, where pi ranges over all permutations of the vertex set of G, where A_G^pi denotes the matrix obtained from A_G by permuting rows and columns according to pi, and where |M |_F is the Frobenius norm of a matrix M. The (weighted) graph similarity problem, denoted by GSim (WSim), is the problem of computing this distance for two graphs of same order. This problem is closely related to the notoriously hard quadratic assignment problem (QAP), which is known to be NP-hard even for severely restricted cases. It is known that GSim (WSim) is NP-hard; we strengthen this hardness result by showing that the problem remains NP-hard even for the class of trees. Identifying the boundary of tractability for WSim is best done in the framework of linear algebra. We show that WSim is NP-hard as long as one of the matrices has unbounded rank or negative eigenvalues: hence, the realm of tractability is restricted to positive semi-definite matrices of bounded rank. Our main result is a polynomial time algorithm for the special case where the associated (weighted) adjacency graph for one of the matrices has a bounded number of twin equivalence classes. The key parameter underlying our algorithm is the clustering number of a graph; this parameter arises in context of the spectral graph drawing machinery.

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I&C Journal 2018 Journal Article

On the complexity of noncommutative polynomial factorization

V. Arvind
Pushkar Joglekar
Gaurav Rattan

In this paper we study the complexity of factorization of polynomials in the free noncommutative ring F 〈 x 1, x 2, …, x n 〉 of polynomials over the field F and noncommuting variables x 1, x 2, …, x n. Our main results are the following: • Although F 〈 x 1, …, x n 〉 is not a unique factorization ring, we note that variable-disjoint factorization in F 〈 x 1, …, x n 〉 has the uniqueness property. Furthermore, we prove that computing the variable-disjoint factorization is polynomial-time equivalent to Polynomial Identity Testing (both when the input polynomial is given by an arithmetic circuit or an algebraic branching program). We also show that variable-disjoint factorization in the black-box setting can be efficiently computed (where the computed factors will be also given by black-boxes). • As a consequence of the previous result we show that homogeneous noncommutative polynomials and multilinear noncommutative polynomials have unique factorizations in the usual sense, which can be efficiently computed. • Finally, we discuss a polynomial decomposition problem in F 〈 x 1, …, x n 〉 which is a natural generalization of homogeneous polynomial factorization and prove some complexity bounds for it.

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MFCS Conference 2016 Conference Paper

The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

Vikraman Arvind
Frank Fuhlbrück
Johannes Köbler
Sebastian Kuhnert
Gaurav Rattan

In this paper we study the complexity of the following problems: 1. Given a colored graph X=(V, E, c), compute a minimum cardinality set of vertices S (subset of V) such that no nontrivial automorphism of X fixes all vertices in S. A closely related problem is computing a minimum base S for a permutation group G <= S_n given by generators, i. e. , a minimum cardinality subset S of [n] such that no nontrivial permutation in G fixes all elements of S. Our focus is mainly on the parameterized complexity of these problems. We show that when k=|S| is treated as parameter, then both problems are MINI[1]-hard. For the dual problems, where k=n-|S| is the parameter, we give FPT~algorithms. 2. A notion closely related to fixing is called individualization. Individualization combined with the Weisfeiler-Leman procedure is a fundamental technique in algorithms for Graph Isomorphism. Motivated by the power of individualization, in the present paper we explore the complexity of individualization: what is the minimum number of vertices we need to individualize in a given graph such that color refinement "succeeds" on it. Here "succeeds" could have different interpretations, and we consider the following: It could mean the individualized graph becomes: (a) discrete, (b) amenable, (c)compact, or (d) refinable. In particular, we study the parameterized versions of these problems where the parameter is the number of vertices individualized. We show a dichotomy: For graphs with color classes of size at most 3 these problems can be solved in polynomial time, while starting from color class size 4 they become W[P]-hard.

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TCS Journal 2015 Journal Article

On the isomorphism problem for decision trees and decision lists

V. Arvind
Johannes Köbler
Sebastian Kuhnert
Gaurav Rattan
Yadu Vasudev

We study the complexity of isomorphism testing for boolean functions that are represented by decision trees or decision lists. Our results are the following: • Isomorphism testing of rank 1 decision trees is complete for logspace. • For any constant r ≥ 2, isomorphism testing for rank r decision trees is polynomial-time equivalent to Graph Isomorphism. As a consequence of our reduction, we obtain our main result for decision trees: A 2 n ( log ⁡ s ) O ( 1 ) time algorithm for isomorphism testing of decision trees of size s over n variables. • The isomorphism problem for decision lists admits a Schaefer-type trichotomy: depending on the class of base functions, the isomorphism problem is either in L, or polynomial-time equivalent to Graph Isomorphism, or coNP -hard.

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