arXiv subjects

Timo Camillo Merkl

Publications and source records attributed to Timo Camillo Merkl.

8 recordsLinked to original sources

arXiv2026

Convolution Sum-Product Queries

We study query evaluation for an extension of sum-product queries (SPQ) that allows atoms with linear combinations of variables (e.g. $A(2X+3Y-Z, Y+Z)$), which we call convolution sum-product queries (CSPQs). These queries arise in both practical settings (e.g. image processing workloads) and theoretical ones (e.g. the $(\min,+)$-convolution and $k$-SUM conjectures), and capture SPQs with added linear equality constraints. While prior work has considered the impact of linear in- and dis-equality constraints on query evaluation, the techniques developed in that setting are asymptotically sub-optimal for CSPQs. To address this, we describe several evaluation algorithms for CSPQs that leverage linear algebra techniques like rank analysis, quotient spaces, and variable substitution. First, we adapt Worst-Case Optimal Joins to CSPQs, achieving a runtime similar in spirit to that for conjunctive queries. Then, we extend the definition of tree decompositions (TDs) to CSPQs, and describe a factorized execution. In this extension, linear combinations are first-class citizens and play the same role as variables in traditional TDs. We define three width measures that bound the complexity of this execution with respect to the size of the domain values, of the active domain, and of the relation's support. Lastly, we show how these methods can be applied to arbitrary fields beyond the rationals.

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arXiv2026

From FPT Decision to FPT Enumeration

Fixed-parameter tractable (FPT) algorithms have been successfully applied to many intractable problems -- with a focus on decision and optimization problems. Their aim is to confine the exponential explosion to some parameter, while the time complexity only depends polynomially on the instance size. In contrast, intractable enumeration problems have received comparatively little attention so far. The goal of this work is to study how FPT decision algorithms could be turned into FPT enumeration algorithms. We thus inspect several fundamental approaches for designing FPT decision or optimization algorithms and we present ideas how they can be extended to FPT enumeration algorithms.

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arXiv2025

The Space-Time Complexity of Sum-Product Queries

While extensive research on query evaluation has achieved consistent improvements in the time complexity of algorithms, the space complexity of query evaluation has been largely ignored. This is a particular challenge in settings with strict pre-defined space constraints. In this paper, we examine the combined space-time complexity of conjunctive queries (CQs) and, more generally, of sum-product queries (SPQs). We propose several classes of space-efficient algorithms for evaluating SPQs, and we show that the optimal time complexity is almost always achievable with asymptotically lower space complexity than traditional approaches.

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arXiv2025

Query Answering under Volume-Based Diversity Functions

When query evaluation produces too many tuples, a new approach in query answering is to retrieve a diverse subset of them. The standard approach for measuring the diversity of a set of tuples is to use a distance function between tuples, which measures the dissimilarity between them, to then aggregate the pairwise distances of the set into a score (e.g., by using sum or min aggregation). However, as we will point out in this work, the resulting diversity measures may display some unintuitive behavior. Moreover, even in very simple settings, finding a maximally diverse subset of the answers of fixed size is, in general, intractable and little is known about approximations apart from some hand-picked distance-aggregator pairs. In this work, we introduce a novel approach for computing the diversity of tuples based on volume instead of distance. We present a framework for defining volume-based diversity functions and provide several examples of these measures applied to relational data. Although query answering of conjunctive queries (CQ) under this setting is intractable in general, we show that one can always compute a (1-1/e)-approximation for any volume-based diversity function. Furthermore, in terms of combined complexity, we connect the evaluation of CQs under volume-based diversity functions with the ranked enumeration of solutions, finding general conditions under which a (1-1/e)-approximation can be computed in polynomial time.

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arXiv2025

Diversity of Answers to Conjunctive Queries

Enumeration problems aim at outputting, without repetition, the set of solutions to a given problem instance. However, outputting the entire solution set may be prohibitively expensive if it is too big. In this case, outputting a small, sufficiently diverse subset of the solutions would be preferable. This leads to the Diverse-version of the original enumeration problem, where the goal is to achieve a certain level d of diversity by selecting k solutions. In this paper, we look at the Diverse-version of the query answering problem for Conjunctive Queries and extensions thereof. That is, we study the problem if it is possible to achieve a certain level d of diversity by selecting k answers to the given query and, in the positive case, to actually compute such k answers.

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arXiv2025

Partition Constraints for Conjunctive Queries: Bounds and Worst-Case Optimal Joins

In the last decade, various works have used statistics on relations to improve both the theory and practice of conjunctive query execution. Starting with the AGM bound which took advantage of relation sizes, later works incorporated statistics like functional dependencies and degree constraints. Each new statistic prompted work along two lines; bounding the size of conjunctive query outputs and worst-case optimal join algorithms. In this work, we continue in this vein by introducing a new statistic called a \emph{partition constraint}. This statistic captures latent structure within relations by partitioning them into sub-relations which each have much tighter degree constraints. We show that this approach can both refine existing cardinality bounds and improve existing worst-case optimal join algorithms.

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arXiv2024

Towards Tractability of the Diversity of Query Answers: Ultrametrics to the Rescue

The set of answers to a query may be very large, potentially overwhelming users when presented with the entire set. In such cases, presenting only a small subset of the answers to the user may be preferable. A natural requirement for this subset is that it should be as diverse as possible to reflect the variety of the entire population. To achieve this, the diversity of a subset is measured using a metric that determines how different two solutions are and a diversity function that extends this metric from pairs to sets. In the past, several studies have shown that finding a diverse subset from an explicitly given set is intractable even for simple metrics (like Hamming distance) and simple diversity functions (like summing all pairwise distances). This complexity barrier becomes even more challenging when trying to output a diverse subset from a set that is only implicitly given such as the query answers of a query and a database. Until now, tractable cases have been found only for restricted problems and particular diversity functions. To overcome these limitations, we focus on the notion of ultrametrics, which have been widely studied and used in many applications. Starting from any ultrametric $d$ and a diversity function $δ$ extending $d$, we provide sufficient conditions over $δ$ for having polynomial-time algorithms to construct diverse answers. To the best of our knowledge, these conditions are satisfied by all diversity functions considered in the literature. Moreover, we complement these results with lower bounds that show specific cases when these conditions are not satisfied and finding diverse subsets becomes intractable. We conclude by applying these results to the evaluation of conjunctive queries, demonstrating efficient algorithms for finding a diverse subset of solutions for acyclic conjunctive queries when the attribute order is used to measure diversity.

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arXiv2024

Consistent Query Answering over SHACL Constraints

The Shapes Constraint Language (SHACL) was standardized by the World Wide Web as a constraint language to describe and validate RDF data graphs. SHACL uses the notion of shapes graph to describe a set of shape constraints paired with targets, that specify which nodes of the RDF graph should satisfy which shapes. An important question in practice is how to handle data graphs that do not validate the shapes graph. A solution is to tolerate the non-validation and find ways to obtain meaningful and correct answers to queries despite the non-validation. This is known as consistent query answering (CQA) and there is extensive literature on CQA in both the database and the KR setting. We study CQA in the context of SHACL for a fundamental fragment of the Semantic Web query language SPARQL. The goal of our work is a detailed complexity analysis of CQA for various semantics and possible restrictions on the acceptable repairs. It turns out that all considered variants of the problem are intractable, with complexities ranging between the first and third level of the polynomial hierarchy.

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