Tractable hypergraph properties for constraint satisfaction and conjunctive queries (0911.0801v3)

Abstract: An important question in the study of constraint satisfaction problems (CSP) is understanding how the graph or hypergraph describing the incidence structure of the constraints influences the complexity of the problem. For binary CSP instances (i.e., where each constraint involves only two variables), the situation is well understood: the complexity of the problem essentially depends on the treewidth of the graph of the constraints. However, this is not the correct answer if constraints with unbounded number of variables are allowed, and in particular, for CSP instances arising from query evaluation problems in database theory. Formally, if H is a class of hypergraphs, then let CSP(H) be CSP restricted to instances whose hypergraph is in H. Our goal is to characterize those classes of hypergraphs for which CSP(H) is polynomial-time solvable or fixed-parameter tractable, parameterized by the number of variables. Note that in the applications related to database query evaluation, we usually assume that the number of variables is much smaller than the size of the instance, thus parameterization by the number of variables is a meaningful question. The most general known property of H that makes CSP(H) polynomial-time solvable is bounded fractional hypertree width. Here we introduce a new hypergraph measure called submodular width, and show that bounded submodular width of H implies that CSP(H) is fixed-parameter tractable. In a matching hardness result, we show that if H has unbounded submodular width, then CSP(H) is not fixed-parameter tractable, unless the Exponential Time Hypothesis fails.

• The paper introduces submodular width, a novel hypergraph measure, to classify tractable constraint satisfaction problems and conjunctive queries.
• It presents algorithmic strategies for decomposing CSP instances and matching hardness results under the Exponential Time Hypothesis (ETH).
• Bounded submodular width precisely characterizes the tractable cases for CSPs, establishing a clear boundary for computational feasibility.

Tractable Hypergraph Properties for Constraint Satisfaction and Conjunctive Queries

The paper "Tractable Hypergraph Properties for Constraint Satisfaction and Conjunctive Queries" by Dániel Marx presents a rigorous investigation into the complexity of constraint satisfaction problems (CSP) and conjunctive queries from a hypergraph perspective. It aims to characterize hypergraph classes that guarantee tractable CSP solutions, focusing on the relationship between hypergraph measures and the computational feasibility of these problems.

Key Contributions

The research introduces submodular width, a novel hypergraph measure, and establishes its significance in determining CSP tractability. The concept of submodular width extends existing hypergraph measures such as fractional hypertree width and adaptive width, providing a broader framework for understanding CSP complexity.

Algorithmic Insights

The algorithmic contributions are centered on partitioning CSP instances into uniform instances that feature bounded submodular width. The paper devises a strategy where a CSP instance is decomposed into smaller problem instances. Each of these instances allows dynamic programming techniques to efficiently evaluate the number of potential solutions, leveraging tree decompositions rooted in submodular width.

Hardness Results

The paper presents matching hardness results asserting that CSPs become intractable for hypergraph classes with unbounded submodular width, substantiated under the Exponential Time Hypothesis (ETH). Notably, if hypergraphs possess large submodular width, simulating 3SAT efficiently becomes feasible, which affirms that CSP(H) cannot achieve fixed-parameter tractability.

Theoretical Implications

The theoretical implications are profound, offering a complete classification of hypergraph properties that dictate CSP evaluability in polynomial or fixed-parameter time. Bounded submodular width precisely encapsulates the tractable cases, providing an explicit boundary past which CSPs are intractable, barring ETH failure.

Future Potential

While successfully demarcating tractable domains in CSPs, further exploration into the fixed-parameter tractability versus polynomial-time solvability dichotomy is necessary. The paper inspires a conjecture on the relationship between hypergraph parameters and potential polynomial-time algorithms, urging researchers to explore beyond the incremental extensions of current hypergraph measures.

Conclusion

Marx's work establishes a robust framework for understanding constraint satisfaction complexity vis-à-vis hypergraph properties, paving the way for deeper insights into CSP structural boundaries. The introduction of submodular width is pivotal, not only extending known results but also providing a landscape for future computational explorations in CSPs and database queries.