# Maximum Quadratic Assignment Problem: Reduction from Maximum Label Cover and LP-based Approximation Algorithm

TL;DR: In this article, it was shown that unless NP ⊂ BPQP, it is impossible to approximate the maximum quadratic assignment problem within a factor better than 2 log 1-en by a reduction from the maximum label cover problem.

Abstract: We show that for every positive e > 0, unless NP ⊂ BPQP, it is impossible to approximate the maximum quadratic assignment problem within a factor better than 2log1-en by a reduction from the maximum label cover problem. Our result also implies that Approximate Graph Isomorphism is not robust and is, in fact, 1 - e versus e hard assuming the Unique Games Conjecture.Then, we present an O(√n)-approximation algorithm for the problem based on rounding of the linear programming relaxation often used in state-of-the-art exact algorithms.

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14 Jun 2020TL;DR: This paper proposes two theoretical approaches that it sees as central for understanding the foundations of vector embeddings and draws connections between the various approaches and suggests directions for future research.

Abstract: Vector representations of graphs and relational structures, whether hand-crafted feature vectors or learned representations, enable us to apply standard data analysis and machine learning techniques to the structures. A wide range of methods for generating such embeddings have been studied in the machine learning and knowledge representation literature. However, vector embeddings have received relatively little attention from a theoretical point of view. Starting with a survey of embedding techniques that have been used in practice, in this paper we propose two theoretical approaches that we see as central for understanding the foundations of vector embeddings. We draw connections between the various approaches and suggest directions for future research.

88 citations

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11 Sep 2022

TL;DR: It is argued that neural network loss landscapes contain (nearly) a single basin after accounting for all possible permutation symmetries of hidden units a la Entezari et al. (2021), and intriguing phenomena relating model width and training time to mode connectivity are identified.

Abstract: The success of deep learning is due in large part to our ability to solve certain massive non-convex optimization problems with relative ease. Though non-convex optimization is NP-hard, simple algorithms -- often variants of stochastic gradient descent -- exhibit surprising effectiveness in fitting large neural networks in practice. We argue that neural network loss landscapes often contain (nearly) a single basin after accounting for all possible permutation symmetries of hidden units a la Entezari et al. 2021. We introduce three algorithms to permute the units of one model to bring them into alignment with a reference model in order to merge the two models in weight space. This transformation produces a functionally equivalent set of weights that lie in an approximately convex basin near the reference model. Experimentally, we demonstrate the single basin phenomenon across a variety of model architectures and datasets, including the first (to our knowledge) demonstration of zero-barrier linear mode connectivity between independently trained ResNet models on CIFAR-10. Additionally, we identify intriguing phenomena relating model width and training time to mode connectivity. Finally, we discuss shortcomings of the linear mode connectivity hypothesis, including a counterexample to the single basin theory.

64 citations

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TL;DR: A generalized graph alignment formulation that considers both matches and mismatches in a standard QAP formulation is proposed that significantly outperforms other methods in the alignment of regular graph structures, which is one of the most difficult graph alignment cases.

Abstract: Graph alignment refers to the problem of finding a bijective mapping across vertices of two graphs such that, if two nodes are connected in the first graph, their images are connected in the second graph. This problem arises in many fields such as computational biology, social sciences, and computer vision and is often cast as a quadratic assignment problem (QAP). Most standard graph alignment methods consider an optimization that maximizes the number of matches between the two graphs, ignoring the effect of mismatches. We propose a generalized graph alignment formulation that considers both matches and mismatches in a standard QAP formulation. This modification can have a major impact in aligning graphs with different sizes and heterogenous edge densities. Moreover, we propose two methods for solving the generalized graph alignment problem based on spectral decomposition of matrices. We compare the performance of proposed methods with some existing graph alignment algorithms including Natalie2, GHOST, IsoRank, NetAlign, Klau's approach as well as a semidefinite programming-based method over various synthetic and real graph models. Our proposed method based on simultaneous alignment of multiple eigenvectors leads to consistently good performance in different graph models. In particular, in the alignment of regular graph structures which is one of the most difficult graph alignment cases, our proposed method significantly outperforms other methods.

43 citations

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TL;DR: It is shown that it is possible to achieve the information‐theoretic limit of graph sparsity in time polynomial in the number of vertices n, and theNumber of seeds needed for perfect recovery in polynometric‐time can be as low as nϵ in the sparse graph regime and Ω(logn) in the dense graph regime.

Abstract: We study a well known noisy model of the graph isomorphism problem. In this model, the goal is to perfectly recover the vertex correspondence between two edge-correlated Erdős-Renyi random graphs, with an initial seed set of correctly matched vertex pairs revealed as side information. For seeded problems, our result provides a significant improvement over previously known results. We show that it is possible to achieve the information-theoretic limit of graph sparsity in time polynomial in the number of vertices $n$. Moreover, we show the number of seeds needed for exact recovery in polynomial-time can be as low as $n^{3\epsilon}$ in the sparse graph regime (with the average degree smaller than $n^{\epsilon}$) and $\Omega(\log n)$ in the dense graph regime.
Our results also shed light on the unseeded problem. In particular, we give sub-exponential time algorithms for sparse models and an $n^{O(\log n)}$ algorithm for dense models for some parameters, including some that are not covered by recent results of Barak et al.

34 citations

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06 Jan 2019TL;DR: It is shown that it is possible to achieve the information-theoretic limit of graph sparsity in time polynomial in the number of vertices, which can be as low as $n^{3\epsilon}$ in the sparse graph regime (with the average degree smaller than $n^{\epsil on}$) and $\Omega(\log n) $ in the dense graph regime.

Abstract: We study a well known noisy model of the graph isomorphism problem. In this model, the goal is to perfectly recover the vertex correspondence between two edge-correlated graphs, with an initial seed set of correctly matched vertex pairs revealed as side information. Specifically, the model first generates a parent graph G0 from Erdos-Renyi random graph G(n,p) and then obtains two children graphs G1 and G2 by subsampling the edge set of G0 twice independently with probability s = Θ(1). The vertex correspondence between G1 and G2 is obscured by randomly permuting the vertex labels of G1 according to a latent permutation π*. Finally, for each i, π* (i) is revealed independently with probability α as seeds.In the sparse graph regime where np ≤ ne for any e

33 citations

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Princeton University

^{1}, Bell Labs^{2}, Stanford University^{3}, Massachusetts Institute of Technology^{4}TL;DR: It is proved that no MAX SNP-hard problem has a polynomial time approximation scheme, unless NP = P, and there exists a positive ε such that approximating the maximum clique size in an N-vertex graph to within a factor of Nε is NP-hard.

Abstract: We show that every language in NP has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof. If a string is in the language, then there exists a proof such that the verifier accepts with probability 1 (i.e., for every choice of its random string). For strings not in the language, the verifier rejects every provided “proof” with probability at least 1/2. Our result builds upon and improves a recent result of Arora and Safra [1998] whose verifiers examine a nonconstant number of bits in the proof (though this number is a very slowly growing function of the input length).As a consequence, we prove that no MAX SNP-hard problem has a polynomial time approximation scheme, unless NP = P. The class MAX SNP was defined by Papadimitriou and Yannakakis [1991] and hard problems for this class include vertex cover, maximum satisfiability, maximum cut, metric TSP, Steiner trees and shortest superstring. We also improve upon the clique hardness results of Feige et al. [1996] and Arora and Safra [1998] and show that there exists a positive e such that approximating the maximum clique size in an N-vertex graph to within a factor of Ne is NP-hard.

1,501 citations

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24 Oct 1992TL;DR: Agarwal et al. as discussed by the authors showed that the MAXSNP-hard problem does not have polynomial-time approximation schemes unless P=NP, and for some epsilon > 0 the size of the maximal clique in a graph cannot be approximated within a factor of n/sup 1/ε / unless P = NP.

Abstract: The class PCP(f(n),g(n)) consists of all languages L for which there exists a polynomial-time probabilistic oracle machine that used O(f(n)) random bits, queries O(g(n)) bits of its oracle and behaves as follows: If x in L then there exists an oracle y such that the machine accepts for all random choices but if x not in L then for every oracle y the machine rejects with high probability. Arora and Safra (1992) characterized NP as PCP(log n, (loglogn)/sup O(1)/). The authors improve on their result by showing that NP=PCP(logn, 1). The result has the following consequences: (1) MAXSNP-hard problems (e.g. metric TSP, MAX-SAT, MAX-CUT) do not have polynomial time approximation schemes unless P=NP; and (2) for some epsilon >0 the size of the maximal clique in a graph cannot be approximated within a factor of n/sup epsilon / unless P=NP. >

1,277 citations

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TL;DR: It is shown that approximating Clique and Independent Set, even in a very weak sense, is NP-hard, and the class NP contains exactly those languages for which membership proofs can be verified probabilistically in polynomial time.

Abstract: We give a new characterization of NP: the class NP contains exactly those languages L for which membership proofs (a proof that an input x is in L) can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof.We discuss implications of this characterization; specifically, we show that approximating Clique and Independent Set, even in a very weak sense, is NP-hard.

1,261 citations