Constraint Solving via Fractional Edge Covers (1711.04506v1)

Abstract: Many important combinatorial problems can be modeled as constraint satisfaction problems. Hence identifying polynomial-time solvable classes of constraint satisfaction problems has received a lot of attention. In this paper, we are interested in structural properties that can make the problem tractable. So far, the largest structural class that is known to be polynomial-time solvable is the class of bounded hypertree width instances introduced by Gottlob et al. Here we identify a new class of polynomial-time solvable instances: those having bounded fractional edge cover number. Combining hypertree width and fractional edge cover number, we then introduce the notion of fractional hypertree width. We prove that constraint satisfaction problems with bounded fractional hypertree width can be solved in polynomial time (provided that a the tree decomposition is given in the input). Together with a recent approximation algorithm for finding such decompositions by Marx, it follows that bounded fractional hypertree width is now the most general known structural property that guarantees polynomial-time solvability.

• The paper presents fractional edge cover number as a novel structural measure that ensures polynomial-time solvability for certain CSP instances.
• It introduces fractional hypertree width by combining hypertree width with fractional edge covers, generalizing tractability conditions for CSPs.
• The authors propose polynomial-time algorithms supported by an approximation scheme to efficiently compute fractional hypertree decompositions.

An Analysis of "Constraint Solving via Fractional Edge Covers"

This essay provides a comprehensive analysis of the paper titled "Constraint Solving via Fractional Edge Covers" by Martin Grohe and Daniel Marx. The paper focuses on the tractability of constraint satisfaction problems (CSPs) through structural properties of their associated hypergraphs, advancing the understanding of polynomial-time solvable classes in CSPs.

Summary of the Paper

The core of the paper is the identification of a new polynomial-time solvable class of CSP instances characterized by bounded fractional edge cover number. This establishes a novel structural property for CSPs that guarantees tractability, potentially expanding the class beyond hypertree width. The authors proceed to introduce a refined measure called fractional hypertree width, combining hypertree width and fractional edge cover number. They demonstrate that CSP instances with bounded fractional hypertree width can be solved in polynomial time if a fractional hypertree decomposition is provided. This combines with an approximation algorithm for finding such decompositions, positioning bounded fractional hypertree width as the most general known tractability condition for CSPs.

Key Contributions

1. Fractional Edge Cover: The paper introduces fractional edge cover number as a metric for CSP tractability. A CSP instance's hypergraph with a bounded fractional edge cover number can be solved in polynomial time due to its strategic structure.
2. Fractional Hypertree Width: By integrating hypertree width with fractional edge covers, the authors develop an invariant known as fractional hypertree width. This invariant generalizes both hypertree width and fractional edge cover number, facilitating a broader set of tractable CSP cases.
3. Algorithmic Implications: The authors propose polynomial-time algorithms for CSPs with bounded fractional hypertree width, underlining the feasibility of their approach. Furthermore, these algorithms are supported by a theoretical approximation mechanism that guarantees bounded width fractional hypertree decompositions.
4. Demonstration of Tractability: The paper proves that CSPs can be efficiently solved with a bounded fractional hypertree width. This is achieved by ensuring that, if a decomposition is available, CSP satisfiability can be checked in polynomial time.

Technical Results

• The paper employs Shearer's Lemma to establish bounds on the number of solutions. If the hypergraph has fractional edge cover number $\rho^*(H)$, the CSP has at most $\|I\|^{\rho^*(H)}$ solutions.
• It explores the fractional hypertree width in detail, demonstrating that it offers a broader applicability than hypertree width without increasing computational complexity unnecessarily.

Future Directions

The paper opens several avenues for future research. While it successfully benchmarks fractional hypertree width as a tool for CSP solvability, it remains unclear whether there exists a polynomial-time algorithm for finding exact fractional hypertree decompositions for every fixed width. Achieving a sharper understanding of this could substantially improve algorithmic efficiency.

Another open question lies in identifying tractable classes beyond fractional hypertree width. The submodular width has been presented as a possible extension that generalizes fractional hypertree width, indicating room for further exploration.

Conclusion

Martin Grohe and Daniel Marx's paper represents a significant addition to the field of CSP tractability. By introducing fractional edge covers and fractional hypertree width, the authors not only broaden the scope of tractable CSP instances but also deliver powerful algorithmic tools for addressing these instances. The work inherently suggests further theoretical and practical investigations, promising advancements in both complexity theory and real-world applications of constraints solving methodologies.