Fractional Hypertree Width Research Articles

Fractional covers of hypergraphs with bounded multi-intersection

Fractional (hyper-)graph theory is concerned with the specific problems that arise when fractional analogues of otherwise integer-valued (hyper-)graph invariants are considered. The focus of this paper is on fractional edge covers of hypergraphs. Our main technical result generalizes and unifies previous conditions under which the size of the support of fractional edge covers is bounded independently of the size of the hypergraph itself. We show how this combinatorial result can be used to extend previous tractability results for checking if the fractional hypertree width of a given hypergraph is ≤k for some constant k. Moreover, we show a dual version of our main result for fractional hitting sets.

Finding Optimal Triangulations Parameterized by Edge Clique Cover

We consider problems that can be formulated as a task of finding an optimal triangulation of a graph w.r.t. some notion of optimality. We present algorithms parameterized by the size of a minimum edge clique cover (texttt {cc}) to such problems. This parameterization occurs naturally in many problems in this setting, e.g., in the perfect phylogeny problem texttt {cc} is at most the number of taxa, in fractional hypertreewidth texttt {cc} is at most the number of hyperedges, and in treewidth of Bayesian networks texttt {cc} is at most the number of non-root nodes. We show that the number of minimal separators of graphs is at most 2^texttt {cc}, the number of potential maximal cliques is at most 3^texttt {cc}, and these objects can be listed in times O^*(2^texttt {cc}) and O^*(3^texttt {cc}), respectively, even when no edge clique cover is given as input; the O^*(cdot ) notation omits factors polynomial in the input size. These enumeration algorithms imply O^*(3^texttt {cc}) time algorithms for problems such as treewidth, weighted minimum fill-in, and feedback vertex set. For generalized and fractional hypertreewidth we give O^*(4^m) time and O^*(3^m) time algorithms, respectively, where m is the number of hyperedges. When an edge clique cover of size texttt {cc}' is given as a part of the input we give O^*(2^{texttt {cc}'}) time algorithms for treewidth, minimum fill-in, and chordal sandwich. This implies an O^*(2^n) time algorithm for perfect phylogeny, where n is the number of taxa. We also give polynomial space algorithms with time complexities O^*(9^{texttt {cc}'}) and O^*(9^{texttt {cc}+ O(log ^2 texttt {cc})}) for problems in this framework.

Fractional Covers of Hypergraphs with Bounded Multi-Intersection

Fractional (hyper-)graph theory is concerned with the specific problems that arise when fractional analogues of otherwise integer-valued (hyper-)graph invariants are considered. The focus of this paper is on fractional edge covers of hypergraphs. Our main technical result generalizes and unifies previous conditions under which the size of the support of fractional edge covers is bounded independently of the size of the hypergraph itself. This allows us to extend previous tractability results for checking if the fractional hypertree width of a given hypergraph is $\leq k$ for some constant $k$. We also show how our results translate to fractional vertex covers.

HyperBench

To cope with the intractability of answering Conjunctive Queries (CQs) and solving Constraint Satisfaction Problems (CSPs), several notions of hypergraph decompositions have been proposed—giving rise to different notions of width, noticeably, plain, generalized, and fractional hypertree width (hw, ghw, and fhw). Given the increasing interest in using such decomposition methods in practice, a publicly accessible repository of decomposition software, as well as a large set of benchmarks, and a web-accessible workbench for inserting, analyzing, and retrieving hypergraphs are called for. We address this need by providing (i) concrete implementations of hypergraph decompositions (including new practical algorithms), (ii) a new, comprehensive benchmark of hypergraphs stemming from disparate CQ and CSP collections, and (iii) HyperBench, our new web-interface for accessing the benchmark and the results of our analyses. In addition, we describe a number of actual experiments we carried out with this new infrastructure.

Functional Aggregate Queries with Additive Inequalities

Motivated by fundamental applications in databases and relational machine learning, we formulate and study the problem of answering functional aggregate queries (FAQ) in which some of the input factors are defined by a collection of additive inequalities between variables. We refer to these queries as FAQ-AI for short. To answer FAQ-AI in the Boolean semiring, we define relaxed tree decompositions and relaxed submodular and fractional hypertree width parameters. We show that an extension of the InsideOut algorithm using Chazelle’s geometric data structure for solving the semigroup range search problem can answer Boolean FAQ-AI in time given by these new width parameters. This new algorithm achieves lower complexity than known solutions for FAQ-AI. It also recovers some known results in database query answering. Our second contribution is a relaxation of the set of polymatroids that gives rise to the counting version of the submodular width, denoted by #subw. This new width is sandwiched between the submodular and the fractional hypertree widths. Any FAQ and FAQ-AI over one semiring can be answered in time proportional to #subw and respectively to the relaxed version of #subw. We present three applications of our FAQ-AI framework to relational machine learning: k -means clustering, training linear support vector machines, and training models using non-polynomial loss. These optimization problems can be solved over a database asymptotically faster than computing the join of the database relations.

Solving Graph Problems via Potential Maximal Cliques

The BT algorithm of Bouchitté and Todinca based on enumerating potential maximal cliques, originally proposed for the treewidth and minimum fill-in problems, yields improved exact exponential-time algorithms for various graph optimization problems related to optimal triangulations. While the BT algorithm has received significant attention in terms of theoretical analysis, less attention has been paid on engineering efficient implementations of the algorithm for different problems and thereby on empirical studies on its effectiveness in practice. In this work, we provide an experimental evaluation of an implementation of the BT algorithm, based on our second-place winning entry in the 2nd Parameterized Algorithms and Computational Experiments Challenge (PACE 2017), extended to several related graph problems: treewidth, minimum fill-in, generalized and fractional hypertreewidth, and the total table size problem. Instead of focusing on problem-specific optimization of BT for a particular problem, our focus in this work is on studying the applicability of BT more generally to a range of problems. Based on the results, we conclude that an efficient implementation of the BT algorithm yields an empirically competitive approach to each of the considered problems when compared to available implementations of alternative problem-specific algorithmic approaches.

It’s All a Matter of Degree

We optimize multiway equijoins on relational tables using degree information. We give a new bound that uses degree information to more tightly bound the maximum output size of a query. On real data, our bound on the number of triangles in a social network can be up to 95 times tighter than existing worst case bounds. We show that using only a constant amount of degree information, we are able to obtain join algorithms with a running time that has a smaller exponent than existing algorithms–for any database instance. We also show that this degree information can be obtained in nearly linear time, which yields asymptotically faster algorithms in the serial setting and lower communication cost algorithms in the MapReduce setting. In the serial setting, the data complexity of join processing can be expressed as a function O(IN x + OUT) in terms of input size IN and output size OUT in which x depends on the query. An upper bound for x is given by the fractional hypertreewidth. We are interested in situations in which we can get algorithms for which x is strictly smaller than the fractional hypertreewidth. We say that a join can be processed in subquadratic time if x < 2. Building on the AYZ algorithm for processing cycle joins in quadratic time, for a restricted class of joins which we call 1-series-parallel graphs, we obtain a complete decision procedure for identifying subquadratic solvability (subject to the 3-SUM problem requiring quadratic time). Our 3-SUM based quadratic lower bound is tight, making it the only known tight bound for joins that does not require any assumption about the matrix multiplication exponent ω. We also give a MapReduce algorithm that meets our improved communication bound and handles essentially optimal parallelism.

Constraint Solving via Fractional Edge Covers

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 article, 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. [2002]. 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 the tree decomposition is given in the input). Together with a recent approximation algorithm for finding such decompositions [Marx 2010], it follows that bounded fractional hypertree width is now the most generally known structural property that guarantees polynomial-time solvability.

Tractable Hypergraph Properties for Constraint Satisfaction and Conjunctive Queries

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 (that is, 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 [Grohe 2007; Marx 2010b]. 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 (which is a strictly more general property than bounded fractional hypertree width) 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 (and hence not polynomial-time solvable), unless the Exponential Time Hypothesis (ETH) fails. The algorithmic result uses tree decompositions in a novel way: instead of using a single decomposition depending on the hypergraph, the instance is split into a set of instances (all on the same set of variables as the original instance), and then the new instances are solved by choosing a different tree decomposition for each of them. The reason why this strategy works is that the splitting can be done in such a way that the new instances are “uniform” with respect to the number extensions of partial solutions, and therefore the number of partial solutions can be described by a submodular function. For the hardness result, we prove via a series of combinatorial results that if a hypergraph H has large submodular width, then a 3SAT instance can be efficiently simulated by a CSP instance whose hypergraph is H . To prove these combinatorial results, we need to develop a theory of (multicommodity) flows on hypergraphs and vertex separators in the case when the function b ( S ) defining the cost of separator S is submodular, which can be of independent interest.

Computing hypergraph width measures exactly

Hypergraph width measures are important in studying the complexity of constraint satisfaction problems (CSPs). We present a general exact exponential algorithm for a large variety of these measures. As a consequence, we obtain algorithms which, for a hypergraph H on n vertices and m hyperedges, compute its generalized hypertree-width in time O ⁎ ( 2 n ) and its fractional hypertree-width in time O ( 1.734601 n ⋅ m ) . 3 3 We omit factors polynomial in n whenever the base of the exponent is rounded. This is justified as c n ⋅ n O ( 1 ) = O ( ( c + ϵ ) n ) for every ϵ > 0 . We also use the notation O ⁎ to suppress polynomial factors.

Approximating Width Parameters of Hypergraphs with Excluded Minors

The notions of hypertree width and generalized hypertree width were introduced by Gottlob, Leone, and Scarcello in order to extend the concept of hypergraph acyclicity. These notions were further generalized by Grohe and Marx, who introduced the fractional hypertree width of a hypergraph. All these width parameters on hypergraphs are useful for extending the tractability of many problems in database theory and artificial intelligence. In this paper, we study the approximability of (generalized, fractional) hypertree width of sparse hypergraphs where the criterion of sparsity reflects the sparsity of their incidence graphs. Our first step is to prove that the (generalized, fractional) hypertree width of a hypergraph is constant factor sandwiched by the treewidth of its incidence graph when the incidence graph belongs to some apex-minor-free graph class (the family of apex-minor-free graph classes includes planar graphs and graphs of bounded genus). This determines the combinatorial borderline above in which the notion of (generalized, fractional) hypertree width becomes essentially more general than treewidth, justifying that way its functionality as a hypergraph acyclicity measure. While for more general sparse families of hypergraphs treewidth of incidence graphs and all hypertree width parameters may differ arbitrarily, there are sparse families where a constant factor approximation algorithm is possible. In particular, we give a constant factor approximation polynomial time algorithm for (generalized, fractional) hypertree width on hypergraphs whose incidence graphs belong to some -minor-free graph class.

Approximating fractional hypertree width

Fractional hypertree width is a hypergraph measure similar to tree width and hypertree width. Its algorithmic importance comes from the fact that, as shown in previous work, Constraint Satisfaction Problems (CSP) and various problems in database theory are polynomial-time solvable if the input contains a bounded-width fractional hypertree decomposition of the hypergraph of the constraints. In this article, we show that for every fixed w ≥ 1, there is a polynomial-time algorithm that, given a hypergraph H with fractional hypertree width at most w , computes a fractional hypertree decomposition of width O ( w 3 ) for H . This means that polynomial-time algorithms relying on bounded-width fractional hypertree decompositions no longer need to be given a decomposition explicitly in the input, since an appropriate decomposition can be computed in polynomial time. Therefore, if H is a class of hypergraphs with bounded fractional hypertree width, then a CSP restricted to instances whose structure is in H is polynomial-time solvable. This makes bounded fractional hypertree width the most general known hypergraph property that makes CSP, Boolean conjunctive queries, and conjunctive query containment polynomial-time solvable.

Tractable Structures for Constraint Satisfaction with Truth Tables

The way the graph structure of the constraints influences the complexity of constraint satisfaction problems (CSP) is well understood for bounded-arity constraints. The situation is less clear if there is no bound on the arities. In this case the answer depends also on how the constraints are represented in the input. We study this question for the truth table representation of constraints. We introduce a new hypergraph measure adaptive width and show that CSP with truth tables is polynomial-time solvable if restricted to a class of hypergraphs with bounded adaptive width. Conversely, assuming a conjecture on the complexity of binary CSP, there is no other polynomial-time solvable case. Finally, we present a class of hypergraphs with bounded adaptive width and unbounded fractional hypertree width.