A homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. In the graph homomorphism problem, denoted by $Hom(H)$, the graph $H$ is fixed and we need to determine if there exists a homomorphism from an instance graph $G$ to $H$. We study the complexity of the problem parameterized by the cutwidth of $G$. We aim, for each $H$, for algorithms for $Hom(H)$ running in time $c_H^k n^{\mathcal{O}(1)}$ and matching lower bounds that exclude $c_H^{k \cdot o(1)}n^{\mathcal{O}(1)}$ or $c_H^{k(1-\Omega(1))}n^{\mathcal{O}(1)}$ time algorithms under the (Strong) Exponential Time Hypothesis. In the paper we introduce a new parameter that we call $\mathrm{mimsup}(H)$. Our main contribution is strong evidence of a close connection between $c_H$ and $\mathrm{mimsup}(H)$: * an information-theoretic argument that the number of states needed in a natural dynamic programming algorithm is at most $\mathrm{mimsup}(H)^k$, * lower bounds that show that for almost all graphs $H$ indeed we have $c_H \geq \mathrm{mimsup}(H)$, assuming the (Strong) Exponential-Time Hypothesis, and * an algorithm with running time $\exp ( {\mathcal{O}( \mathrm{mimsup}(H) \cdot k \log k)}) n^{\mathcal{O}(1)}$. The parameter $\mathrm{mimsup}(H)$ can be thought of as the $p$-th root of the maximum induced matching number in the graph obtained by multiplying $p$ copies of $H$ via certain graph product, where $p$ tends to infinity. It can also be defined as an asymptotic rank parameter of the adjacency matrix of $H$. Our results tightly link the parameterized complexity of a problem to such an asymptotic rank parameter for the first time.
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Vertex splitting is a graph operation that replaces a vertex $v$ with two nonadjacent new vertices and makes each neighbor of $v$ adjacent with one or both of the introduced vertices. Vertex splitting has been used in contexts from circuit design to statistical analysis. In this work, we explore the computational complexity of achieving a given graph property $\Pi$ by a limited number of vertex splits, formalized as the problem $\Pi$ Vertex Splitting ($\Pi$-VS). We focus on hereditary graph properties and contribute four groups of results: First, we classify the classical complexity of $\Pi$-VS for graph properties characterized by forbidden subgraphs of size at most 3. Second, we provide a framework that allows to show NP-completeness whenever one can construct a combination of a forbidden subgraph and prescribed vertex splits that satisfy certain conditions. Leveraging this framework we show NP-completeness when $\Pi$ is characterized by forbidden subgraphs that are sufficiently well connected. In particular, we show that $F$-Free-VS is NP-complete for each biconnected graph $F$. Third, we study infinite families of forbidden subgraphs, obtaining NP-hardness for Bipartite-VS and Perfect-VS. Finally, we touch upon the parameterized complexity of $\Pi$-VS with respect to the number of allowed splits, showing para-NP-hardness for $K_3$-Free-VS and deriving an XP-algorithm when each vertex is only allowed to be split at most once.
Techniques based on $k$-th order Hodge Laplacian operators $L_k$ are widely used to describe the topology as well as the governing dynamics of high-order systems modeled as simplicial complexes. In all of them, it is required to solve a number of least square problems with $L_k$ as coefficient matrix, for example in order to compute some portions of the spectrum or integrate the dynamical system. In this work, we introduce the notion of optimal collapsible subcomplex and we present a fast combinatorial algorithm for the computation of a sparse Cholesky-like preconditioner for $L_k$ that exploits the topological structure of the simplicial complex. The performance of the preconditioner is tested for conjugate gradient method for least square problems (CGLS) on a variety of simplicial complexes with different dimensions and edge densities. We show that, for sparse simplicial complexes, the new preconditioner reduces significantly the condition number of $L_k$ and performs better than the standard incomplete Cholesky factorization.
We study the problem of allocating indivisible resources under the connectivity constraints of a graph $G$. This model, initially introduced by Bouveret et al. (published in IJCAI, 2017), effectively encompasses a diverse array of scenarios characterized by spatial or temporal limitations, including the division of land plots and the allocation of time plots. In this paper, we introduce a novel fairness concept that integrates local comparisons within the social network formed by a connected allocation of the item graph. Our particular focus is to achieve pairwise-maximin fair share (PMMS) among the "neighbors" within this network. For any underlying graph structure, we show that a connected allocation that maximizes Nash welfare guarantees a $(1/2)$-PMMS fairness. Moreover, for two agents, we establish that a $(3/4)$-PMMS allocation can be efficiently computed. Additionally, we demonstrate that for three agents and the items aligned on a path, a PMMS allocation is always attainable and can be computed in polynomial time. Lastly, when agents have identical additive utilities, we present a pseudo-polynomial-time algorithm for a $(3/4)$-PMMS allocation, irrespective of the underlying graph $G$. Furthermore, we provide a polynomial-time algorithm for obtaining a PMMS allocation when $G$ is a tree.
We use Markov categories to develop generalizations of the theory of Markov chains and hidden Markov models in an abstract setting. This comprises characterizations of hidden Markov models in terms of local and global conditional independences as well as existing algorithms for Bayesian filtering and smoothing applicable in all Markov categories with conditionals. We show that these algorithms specialize to existing ones such as the Kalman filter, forward-backward algorithm, and the Rauch-Tung-Striebel smoother when instantiated in appropriate Markov categories. Under slightly stronger assumptions, we also prove that the sequence of outputs of the Bayes filter is itself a Markov chain with a concrete formula for its transition maps. There are two main features of this categorical framework. The first is its generality, as it can be used in any Markov category with conditionals. In particular, it provides a systematic unified account of hidden Markov models and algorithms for filtering and smoothing in discrete probability, Gaussian probability, measure-theoretic probability, possibilistic nondeterminism and others at the same time. The second feature is the intuitive visual representation of information flow in these algorithms in terms of string diagrams.
Calibrating simulation models that take large quantities of multi-dimensional data as input is a hard simulation optimization problem. Existing adaptive sampling strategies offer a methodological solution. However, they may not sufficiently reduce the computational cost for estimation and solution algorithm's progress within a limited budget due to extreme noise levels and heteroskedasticity of system responses. We propose integrating stratification with adaptive sampling for the purpose of efficiency in optimization. Stratification can exploit local dependence in the simulation inputs and outputs. Yet, the state-of-the-art does not provide a full capability to adaptively stratify the data as different solution alternatives are evaluated. We devise two procedures for data-driven calibration problems that involve a large dataset with multiple covariates to calibrate models within a fixed overall simulation budget. The first approach dynamically stratifies the input data using binary trees, while the second approach uses closed-form solutions based on linearity assumptions between the objective function and concomitant variables. We find that dynamical adjustment of stratification structure accelerates optimization and reduces run-to-run variability in generated solutions. Our case study for calibrating a wind power simulation model, widely used in the wind industry, using the proposed stratified adaptive sampling, shows better-calibrated parameters under a limited budget.
Most formal methods see the correctness of a software system as a binary decision. However, proving the correctness of complex systems completely is difficult because they are composed of multiple components, usage scenarios, and environments. We present QuAC, a modular approach for quantifying the correctness of service-oriented software systems by combining software architecture modeling with deductive verification. Our approach is based on a model of the service-oriented architecture and the probabilistic usage scenarios of the system. The correctness of a single service is approximated by a coverage region, which is a formula describing which inputs for that service are proven to not lead to an erroneous execution. The coverage regions can be determined by a combination of various analyses, e.g., formal verification, expert estimations, or testing. The coverage regions and the software model are then combined into a probabilistic program. From this, we can compute the probability that under a given usage profile no service is called outside its coverage region. If the coverage region is large enough, then instead of attempting to get 100% coverage, which may be prohibitively expensive, run-time verification or testing approaches may be used to deal with inputs outside the coverage region. We also present an implementation of QuAC for Java using the modeling tool Palladio and the deductive verification tool KeY. We demonstrate its usability by applying it to a software simulation of an energy system.
Join-preserving maps on the discrete time scale $\omega^+$, referred to as time warps, have been proposed as graded modalities that can be used to quantify the growth of information in the course of program execution. The set of time warps forms a simple distributive involutive residuated lattice -- called the time warp algebra -- that is equipped with residual operations relevant to potential applications. In this paper, we show that although the time warp algebra generates a variety that lacks the finite model property, it nevertheless has a decidable equational theory. We also describe an implementation of a procedure for deciding equations in this algebra, written in the OCaml programming language, that makes use of the Z3 theorem prover.
Knowledge graph completion aims to predict missing relations between entities in a knowledge graph. While many different methods have been proposed, there is a lack of a unifying framework that would lead to state-of-the-art results. Here we develop PathCon, a knowledge graph completion method that harnesses four novel insights to outperform existing methods. PathCon predicts relations between a pair of entities by: (1) Considering the Relational Context of each entity by capturing the relation types adjacent to the entity and modeled through a novel edge-based message passing scheme; (2) Considering the Relational Paths capturing all paths between the two entities; And, (3) adaptively integrating the Relational Context and Relational Path through a learnable attention mechanism. Importantly, (4) in contrast to conventional node-based representations, PathCon represents context and path only using the relation types, which makes it applicable in an inductive setting. Experimental results on knowledge graph benchmarks as well as our newly proposed dataset show that PathCon outperforms state-of-the-art knowledge graph completion methods by a large margin. Finally, PathCon is able to provide interpretable explanations by identifying relations that provide the context and paths that are important for a given predicted relation.
Incompleteness is a common problem for existing knowledge graphs (KGs), and the completion of KG which aims to predict links between entities is challenging. Most existing KG completion methods only consider the direct relation between nodes and ignore the relation paths which contain useful information for link prediction. Recently, a few methods take relation paths into consideration but pay less attention to the order of relations in paths which is important for reasoning. In addition, these path-based models always ignore nonlinear contributions of path features for link prediction. To solve these problems, we propose a novel KG completion method named OPTransE. Instead of embedding both entities of a relation into the same latent space as in previous methods, we project the head entity and the tail entity of each relation into different spaces to guarantee the order of relations in the path. Meanwhile, we adopt a pooling strategy to extract nonlinear and complex features of different paths to further improve the performance of link prediction. Experimental results on two benchmark datasets show that the proposed model OPTransE performs better than state-of-the-art methods.
Neural machine translation (NMT) is a deep learning based approach for machine translation, which yields the state-of-the-art translation performance in scenarios where large-scale parallel corpora are available. Although the high-quality and domain-specific translation is crucial in the real world, domain-specific corpora are usually scarce or nonexistent, and thus vanilla NMT performs poorly in such scenarios. Domain adaptation that leverages both out-of-domain parallel corpora as well as monolingual corpora for in-domain translation, is very important for domain-specific translation. In this paper, we give a comprehensive survey of the state-of-the-art domain adaptation techniques for NMT.
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