Fractional Covers of Hypergraphs with Bounded Multi-Intersection

07/03/2020 ∙ by Georg Gottlob, et al. ∙ TU Wien Birkbeck, University of London 0

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 ≤ k for some constant k. We also show how our results translate to fractional vertex covers.

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1 Introduction

Fractional (hyper-)graph theory [10] has evolved into a mature discipline in graph theory – building upon early research efforts that date back to the 1970s [2]

. The crucial observation in this field is that many integer-valued (hyper-)graph invariants have a meaningful fractional analogue. Frequently, the integer-valued invariants are defined in terms of some integer linear program (ILP) and the fractional analogue is obtained by the fractional relaxation. Examples of problems which have been studied in fractional (hyper-)graph theory comprise matching problems, coloring problems, covering problems, and many more.

Covering problems come in two principal flavors, namely vertex covers and edge covers. We shall concentrate on edge covers in the first place, but we will later also mention how our results translate to vertex covers. Fractional edge covers have attracted a lot of attention in recent times. On the one hand, this is due to a deep connection between information theory and database theory. Indeed, the famous “AGM bound” – named after the authors of [1] – establishes a tight upper bound on the number of result tuples of relational joins in terms of fractional edge covers. On the other hand, fractional hypertree width () is to date the most general width-notion that allows one to define tractable fragments of solving Constraint Satisfaction Problems (CSPs), answering Conjunctive Queries (CQs), and solving the Homomorphism Problem [8]. The fractional hypertree width of a hypergraph is defined in terms of the size of fractional edge covers of the bags in a tree decomposition.

Fractional (hyper-)graph invariants give rise to new challenges that do not exist in the integral case. Intuitively, if a fractional (hyper-)graph invariant is obtained by the relaxation of a linear program (LP), one would expect things to become easier, since we move from the intractable problem of ILPs to the tractable problem of LPs. However, also the opposite may happen, namely that the fractional relaxation introduces complications not present in the integral case. To illustrate such an effect, we first recall some basic definitions.

A hypergraph is a tuple , consisting of a set of vertices and a set of hyperedges (or simply “edges”), which are non-empty subsets of .

Let be a function of the form . Then the set of vertices “covered” by is defined as . Intuitively, assigns weights to the edges and a vertex is covered if the total weight of the edges containing is at least 1.

A fractional edge cover of is a function with . An integral edge cover is obtained by restricting the function values of to . The support of is defined as . The weight of is defined as . The minimum weight of a fractional (resp. integral) edge cover of a hypergraph is referred to as the fractional (resp. integral) edge cover number of . The following example adapted from [5] illustrates which complications may arise if we move from the integral to the fractional case.

Consider the family of hypergraphs with defined as

with and for .

The integral edge cover number of each is 2 and an optimal integral edge cover can be obtained, e.g., by setting and for all other edges. In contrast, the fractional edge cover number is and the unique optimal fractional edge cover is with and for each . For the support of these two covers, we have and . Hence, the support of the optimal edge covers is bounded in the integral case but unbounded in the fractional case.

As mentioned above, fractional hypertree width () is to date the most general width-notion that allows one to define tractable fragments of classical NP-complete problems, such as CSP solving and CQ answering. However, recognizing if a given hypergraph has for fixed is itself an NP-complete problem [5]. It has recently been shown that the problem of checking becomes tractable if we can efficiently enumerate the fractional edge covers of size [7]. This fact can be exploited to show that, for classes of hypergraphs with bounded rank (i.e., max. size of edges), bounded degree (i.e., max. number of edges containing a particular vertex), or bounded intersection (i.e., max. number of vertices in the intersection of two edges), checking becomes tractable. The size of the support has been recently [7] identified as a crucial parameter for the efficient enumeration of fractional edge covers of weight for given .

The overarching goal of this work is to further extend and provide a uniform view of previously known structural properties of hypergraphs that guarantee a bound on the size of the support of fractional edge covers of a given weight. In particular, when looking at Example 1, we want to avoid the situation that the support of fractional edge covers increases with the size of the hypergraph. Our main combinatorial result (Theorem 3) will be that the size of the support of a fractional edge cover does not depend on the number of vertices or edges of a hypergraph but instead only on the weight of the cover as well as the structure of its edge intersections.

Formally, the structure of the edge intersections is captured by the so-called Bounded-Multi-Intersection-Propery (BMIP) [5]: a class of hypergraphs has this property, if in every hypergraph , the intersection of edges of has at most elements, for constants and . The BMIP thus generalizes all of the above mentioned hypergraph properties that ensure bounded support of fractional edge covers of given weight and, hence, also guarantee tractability of checking , namely bounded rank, bounded degree, and bounded intersection. Moreover, when considering the incidence graph of , the BMIP corresponds to not having large complete bipartite graphs. A notable result in the area of parameterized complexity [9] is the polynomial kernelizability of the Dominating Set Problem for graphs without . A minor tweaking of the results yields a polynomial kernelization for the Set Cover Problem if the corresponding incidence graph does not contain . Our result thus makes an interesting connection: it shows that a condition that enables efficient solving of the Set Cover problem also enables efficient checking of bounded fractional hypertree width.

In summary, the main results of this paper are as follows:

  • First of all, we show that the size of the support of a fractional edge cover only depends on the weight of the cover and of the structure of its edge intersections (Theorem 3). More specifically, if the intersection of edges of a hypergraph has at most elements, and has a fractional edge cover of weight , then also has a fractional edge cover of weight with a support whose size only depends on , and .

  • As an important consequence of this result, we show that the problem of checking if a given hypergraph has is tractable for hypergraph classes satisfying the BMIP (Theorem 5.1). In particular, BMIP generalizes all previously known hypergraph classes with tractable -checking, namely bounded rank, bounded degree, and bounded intersection.

  • We transfer our results on fractional edge covers to fractional vertex covers, where we again vastly generalize previously known hypergraph classes (such as hypergraphs of bounded rank [6]) that guarantee a bound on the size of the support of fractional vertex covers (Theorem 5.2).

The paper is organized as follows: after recalling some basic notions and results in Section 2, we will present our main technical result on fractional edge covers in Section 3. The detailed proof of a crucial lemma is separated in Section 4. In Section 5, we apply our result on the bounded support of fractional edge covers to fractional hypertree width and fractional vertex covers. Finally, in Section 6, we summarize our results and discuss some directions for future research. Due to space limitations, some proofs are given in the appendix.

2 Preliminaries

Some general notation. It is convenient to use the following short-hand notation for various kinds of sets: we write for the set of natural numbers. Let be a set of sets. Then we write and for the intersection and union, respectively, of the sets in , i.e., for all and for some .

Hypergraphs. We recall some basic notions on hypergraphs next. We have already introduced in Section 1 hypergraphs as pairs consisting of a set of vertices and a set of edges. Without loss of generality, we assume throughout this paper that a hypergraph neither contains isolated vertices (i.e., vertices that do not occur in any edge) nor empty edges. We call a hypergraph reduced if, in addition to these restrictions, it contains no two vertices of the same type, i.e., there do not exist in such that . Note that, for computing (edge or vertex) covers, we may always assume that a hypergraph is reduced. It is sometimes convenient to identify a hypergraph with its set of edges with the understanding that . A subhypergraph of a hypergraph is obtained by taking a subset of the edges of . By slight abuse of notation, we may thus write for a subhypergraph of .

Given a hypergraph , the dual hypergraph is defined as and . If is reduced, then we have , i.e., the dual of the dual of is itself. The incidence graph of a hypergraph is a bipartite graph with , such that, for every and , there is an edge in iff . Note that a hypergraph and its dual hypergraph have the same incidence graph.

In this work, we are particularly interested in the structure of the edge intersections of a hypergraph. To this end, recall the notion of -hypergraphs for integers and from [7]: is a -hypergraph if the intersection of any distinct edges in has at most elements, i.e., for every subset with , we have . A class of hypergraphs is said to satisfy the bounded multi-intersection property (BMIP) [5], if there exist and , such that every in is a -hypergraph. As a special case studied in [4, 5], a class of hypergraphs is said to satisfy the bounded intersection property (BIP), if there exists , such that every in is a -hypergraph.

We now recall tree decompositions, which form the basis of various notions of width. A tuple is a tree decomposition (TD) of a hypergraph , if is a tree, every is a subset of and the following two conditions are satisfied:

  1. For every edge there is a node in , such that , and

  2. for every vertex , is connected in .

The vertex sets are usually referred to as the bags of the TD. Note that, by slight abuse of notation, we write to express that is a node in .

For a function , the -width of a TD is defined as and the -width of a hypergraph is the minimal -width over all its TDs.

An edge weight function is a function . We call a fractional edge cover of a set by edges in , if for every , we have . The weight of a fractional edge cover is defined as . For a set , we define , i.e., the total weight of the edges in . For , we write to denote the minimal weight over all fractional edge covers of . The fractional hypertree width (fhw) of a hypergraph , denoted , is then defined as the -width for . Likewise, the of a TD of is its -width.

We state an important technical lemma for weight-functions of -hypergraphs.

There is a function with the following property: let be a -hypergraph and let be an edge weight function with . Moreover, let and assume that, for each , . Let be the set of all vertices of weight at least . Then holds.

The above lemma is essentially an extract of Lemma 7.3 in [7]. For convenience, we have included a proof in the appendix.

Linear Programs. We assume some familiarity with Linear Programs (LPs). Formally, we are dealing here with minimization problems of the form subject to and , where

is a vector of

variables, is a vector of constants, is an matrix, is a vector of constants, and stands for the -dimensional zero-vector. More specifically, for a hypergraph and vertices , the fractional edge cover number of is obtained as the optimal value of the following LP: let and , then is the -dimensional vector , is the -dimensional vector , and , such that if and otherwise. In the sequel, we will refer to such LPs with , and as unary linear programs.

For given number of edges, there are at most possible different inequalities of the form . We thus get the following property of unary LPs, which intuitively states that if the optimum is bigger than some threshold , then it exceeds by some distance.

For every positive integers and , there is an integer such that for any unary LP of at most variables if then , where denotes the minimum of the LP.

3 Bounding the Support of Fractional Edge Covers

In this section we establish our main combinatorial result, Theorem 3. Every set of vertices in a -hypergraph can be covered in a way such that the size of the support depends only on , , and the set’s fractional edge cover number. Due to space constraints proofs of some statements have to be omitted and we refer to the appendix for additional details.

There is a function such that the following is true. Let be constants. Let be a -hypergraph and let Assume that . Then there exists an assignment such that , and .

The first step of our reasoning is to consider the situation where is bounded. In this case it is easy to transform into the desired . Partition all the hyperedges of into equivalence classes corresponding to non-empty subsets of such that two edges and are equivalent if and only if . Then let be the total weight (under ) of all the edges from the equivalence class where . Identify one representative of each (non-empty) equivalence class and let be the representative of the equivalence class corresponding to . Then define as follows. For each corresponding to a non-empty equivalence class, set . For each edge whose weight has not been assigned in this way, set . It is clear that and that the support of is at most , which is bounded by assumption.

Of course, in general we cannot assume that is bounded. Therefore, as the next step of our reasoning, we consider a more general situation where we have a bounded set where each is a set of at most hyperedges such that the following conditions hold regarding : (i) for each , and (ii) the set is of bounded size. Then the assignment as in Theorem 3 can be defined as follows. For each , set . Next, we observe that for the subhypergraph , is bounded, where subscript means that we consider for hypergraph and is restricted accordingly. Therefore, we define on the remaining edges as in the paragraph above. It is not hard to see that the support of the resulting is of size at most . We are going to show that such a family of sets of edges can always be found for hypergraphs (after a possible modification of ).

[Well-formed pair] Let be a hypergraph and let be an edge weight function. We say is a well-formed pair (with regard to ) if it satisfies the following conditions:

  1. where each is a set of at most hyperedges of .

  2. .

We denote by and refer to it as the size of .

[Perfect well-formed pairs] A well-formed pair is perfect if there is an assignment with and such that .

Our aim now is to demonstrate the existence of a perfect pair of size bounded by a function depending on , , and . Clearly, this will imply Theorem 3.

In particular, we will define the initial pair which is a well-formed pair but not necessarily perfect. Then we will define two transformations from one well-formed pair into another and prove existence of a function so that if is transformed into , then . We will then prove that if we form a sequence of well-formed pairs starting from the initial pair and obtain every next element by a transformation of the last one then, after a bounded number of steps we obtain a perfect well-formed pair. We start by defining the initial pair.

[Initial pair] The initial pair is where and .

There is a function such that .

Proof.

where is as in Lemma 2 (for ) and by construction. ∎

We now introduce two kinds of transformations, folding and extension. A folding removes a set of edges from and adds to the vertices in the intersection of the edges of

. In the resulting well-ordered pair

, has one less element than and , compared to , has a bounded size increase of at most vertices. Thus the action of folding gets the resulting well-formed pair closer to one with empty first component, which is a perfect pair according to the paragraph immediately after the statement of Theorem 3.

[Folding] Let be a well-formed pair such that contains elements of size . Let such that . Let and . We call a folding of .

The folding, however, is possible only if has an element of size . Otherwise, we need a more complicated transformation called an extension. The extension takes an element of size and expands it by replacing with several subsets of each containing all the edges of plus one extra edge. This replacement may miss some of the elements of simply because is not contained in any of these extra edges. This excess of missed elements is added to and thus all the conditions of a well-formed pair are satisfied.

[Extension] Let be a well-formed pair with such that every element of is of size at most . For the extension, we identify be an element called the extended element and a set of hyperedges called the extending set. We refer to as the set of light vertices. An extension of is where and .

With data as in Definition 3, is a well-formed pair.

At the first glance the transformation performed by the extension is radically opposite to the one done by the folding: the first component grows rather than shrinks. Note, however, that the new sets replacing the removed one contain a larger number of edges and thus they are closer to being of size at which stage the folding can be applied to them. Our claim is that after a sufficiently large number of foldings and extensions, a well-formed pair with empty first component is eventually obtained.

For our overall goal we then need to show that the size of the resulting perfect pair is indeed bounded by a function of , , and . To that end, the following lemma first establishes that a single step in this process increases the size of the well-formed pair in a controlled manner. To streamline our path to the main result, the proof of the lemma is deferred to Section 4.

There is a function such that the following holds. Let be a well-formed pair with such that every element of is of size at most . Then one of the following two statements is true.

  1. is a perfect pair.

  2. There is an extension of such that . We refer to as a bounded extension of .

For the sake of syntactical convenience, we unify the notions of folding and bounded extension into a single notion of transformation and prove the related statement following from Lemma 3 and the definition of folding.

[Transformation] Let and be well-formed pairs. We say that is a transformation of if it is either a folding or a bounded extension of .

There is a monotone function with for any natural number such that the following holds. If be a well-formed pair, then one of the following two statements is true.

  1. is a perfect pair.

  2. There is a transformation of such that .

Proof.

Assume that is not a perfect pair. Then is not empty (see the discussion at the beginning of this section). Suppose that an element of is of size . Then we set to be a folding of . By definition of the folding and of -hypergraphs, is obtained from by removal of an element from and adding at most vertices to . Hence the size of is clearly bounded in the size of . If all elements of are of size at most then by Lemma 3, there is a bounded extension of .

Clearly, we can specify a function so that in both cases . To satisfy the requirement for , set for each natural number . ∎

Now that we know that each individual step on our path to a perfect pair increases the size only in a bounded fashion, we need to establish that the number of steps is also bounded by a function of , , and . The following auxiliary theorem states that such a bound exists. A full proof of Theorem 3 is available in the appendix.

A sequence of is a sequence of transformations if for each the following two statements hold

  1. is not a perfect pair.

  2. is a transformation of .

There is a monotone function such that the following is true. Let be a sequence of transformations. Then .

In summary, we have shown that we can reach a perfect pair in a bounded number of transformations. Moreover, each transformation increases the size of a pair in a controlled manner. We are now ready to prove our main result.

Proof of Theorem 3.

Consider the following algorithm.

  1. Let be the initial pair (see Definition 3).

  2. While is not a perfect pair

    1. Let be a transformation of existing by Lemma 3

By Theorem 3, the above algorithm stops with the final being at most . It follows from the description of the algorithm that is a perfect pair. It remains to show that its size is bounded by a function of .

(1)

the second inequality follows from Lemma 3 and the monotonicity of . Next, by the properties of , an inductive application of Lemma 3 and Lemma 3 yields

(2)

where superscript means that the function is applied times.

Let . It follows from combination of (1) and (2) that . ∎

4 Proof of Lemma 3

The first step of the proof is to define a unary linear program of bounded size associated with . Then we will demonstrate that if the optimal value of this linear program is at most , then is perfect. Otherwise, we show that a bounded extension can be constructed.

In order to define the linear program, we first formally define equivalence classes of edges covering (see the informal discussion at the beginning of Section 3).

[Working subset, witnessing edge] A set of vertices is called  working set (for ) if there is such that . This is called a witnessing edge of and the set of all witnessing edges of is denoted by .

Continuing on the previous definition, it is not hard to see that the sets partition the set of edges of having a non-empty intersection with . Choose an arbitrary but fixed representative of each and let be the set of these representatives which we also refer to as the set of witnessing representatives. Now, we are ready to define the linear program.

[] The linear program of has the set of variables . The objective function is the minimization of . The constraints are of the following two kinds.

  1. where is .

  2. where is where is the subset of consisting of all the edges containing .

Assume that the optimal solution of is at most . Then is a perfect pair.

Proof.

Each variable of corresponds to an edge and this correspondence is injective. For each , let be the value of in the optimal solution. For each edge not having a corresponding edge, set . It follows from a direct inspection that and the size of support of is at most . ∎

As stated above, in case the optimal value of is greater than we are going to demonstrate existence of a bounded extension of . The first step towards identifying such an extension is to identify the extending element of . Combining Lemma 2 from Section 2 with Lemma 4 below, we observe that has an element such that is much smaller than . This will be the extended element.

Let be a well-formed pair. Let be the subset of consisting of all such that . Let be an optimal solution for . Then .

Proof.

Let be an arbitrary assignment of weights to the hyperedges of . We say that satisfies a constraint for if and that satisfies the constraint for if .

We are going to demonstrate an assignment of weights whose total weight exceeds that by at most and that satisfies all the constraints and . Clearly, this will imply correctness of this theorem.

For each choose an arbitrary edge and let be the set of all such edges. For each , let .

Let be obtained from as follows. If then . Otherwise, . It is not hard to see that satisfies the constraints for each , that , and that, since does not decrease the weight of any edge, .

Let be all the working subsets of and let be the respective witnessing representatives. Then the assignment of weights is defined as follows.

  1. If there is such that then if and otherwise.

  2. Otherwise, .

Let . Note that, by construction, and the weights of edges outside are the same under and and thus, . Moreover since does not intersect with , satisfies the constraints for all .

It remains to show that satisfies the constraints for each . Let be the edges of containing , let be the working subsets of containing , and let be the respective witnessing representatives. As , it follows that . By construction, for each and for each . Consequently, . We conclude that satisfies . ∎

Lemma 2 and Lemma 4 imply the following corollary.

Let be a well-formed pair. Assume that while . Let . Then there is an with . In particular this means that is not empty where is as in Lemma 4.

Proof.

Note that the number of variables of is at most . It follows from the combination of Lemma 2 and Lemma 4 that and, since , and hence there is with . ∎

Proof of Lemma 3.

If the value of the optimal solution of is at most , we are done by Proposition 4.

Otherwise, let be as in Corollary 4. Let . It follows from Corollary 4 that vertices of need weight contribution of at least from hyperedges of other than . We define the extending set to be the set of all hyperedges of other than whose weight is at least and therefore . Accordingly, we define the set of light vertices to be the subset of consisting of all vertices that, besides are contained only in hyperedges of weight smaller than . By Lemma 2, and the size of is clearly bounded by a function on and . It is not hard to see that the size of the resulting extension is bounded as well. ∎

5 Applications and Extensions

5.1 Checking Fractional Hypertree Width

Now that our main combinatorial result has been established we move our attention to an algorithmic application of the support bound. In particular, we are interested in the problem of deciding whether for an input hypergraph and constant we have . The problem is known to be NP-hard even for  [5]. However, as noted in the introduction, in recently published research we were able to show that for hypergraph classes which enjoy bounded intersection or bounded degree, it is indeed tractable to check for constant  [7].

Due to limited space we will recall the main components of the framework for tractable width checking developed in [7] and use them in a black-box fashion.

[-limited fractional hypertree width] Let be the minimal weight of an assignment such that and . We define the -limited fractional hypertree width of a hypergraph as its -width.

[Theorem 4.5 & Lemma 6.2 in [7]] Fix , , and as constant integers. There is a polynomial time algorithm testing whether a given -hypergraph has a -limited fractional hypertree width at most .

The underlying intuition of -limited is that the bounded support allows for a polynomial time enumeration of all the (inclusion) maximal covers of sufficient weight. For -hypergraphs it is then possible to compute a set of candidate bags such that a fitting tree decomposition, if one exists, uses bags only from this set. Deciding whether a tree decomposition can be created from a given set of candidate bags is tractable under some minor restrictions to the structure of the resulting decomposition (not of any concern to the case discussed here).

We now apply our main result and show that, under BMIP, there exists a constant such that the -limited fractional hypertree width always equals fractional hypertree width. From the previous lemma is then straightforward to arrive at the desired tractability result.

There is a polynomial time algorithm for testing whether the of the given -hypergraph is at most (the degree of the polynomial is upper bounded by a fixed function depending on ).

Proof.

It follows from Theorem 3 that if for a -hypergraph then the -limited of is also at most .

Indeed, let be a tree decomposition with at most . Then, according to Theorem 3, for each node in there is an edge weight function with such that . In other words, it follows that has -width at most where is . Thus, also has -limited fractional hypertree width at most .

Thus to test whether , it is enough to test whether the -limited of is at most . This can be done in polynomial time according to Lemma 5.1. ∎

5.2 Extension to Fractional Vertex Cover

There are two natural dual concepts of fractional edge cover. One is the fractional vertex cover problem which is the dual in the sense that it is equivalent to the fractional edge cover on the dual hypergraph. The other, the fractional independent set problem, corresponds to the dual linear program of a linear programming formulation of finding an optimal fractional cover. Here we discuss how our results extend to vertex covers and discuss how the resulting statement generalizes, in a particular sense, a well-known statement of Füredi [6]. Some notes on connections to fractional independent sets are given in our discussion of future work in Section 6.

We start by giving a formal definition of the fractional vertex cover problem. Let be a hypergraph and be an assignment of weights to the vertices of . Analogous to the definition of fractional edge covers we define

  • ,

  • ,

  • and .

A fractional vertex cover is also called a transversal in some contexts (cf. [10]). For a set of edges we denote the weight of the minimal fractional vertex cover such that as . For hypergraph , we say . Recall, that we assume reduced hypergraphs and therefore there is a one-to-one correspondence of vertices in and edges in . We will make use of the following well-known fact about the connection of what we will call dual weight assignments. Let be a (reduced) hypergraph and its dual. We write to identify the edge in that corresponds to the vertex in . The following two statements hold:

  • For every and the function with it holds that .

  • For every and the function with it holds that .

In the following we extend Theorem 3 to an analogous statement for fractional vertex covers thereby generalizing the previous proposition significantly. To derive the result we need a final observation about -hypergraphs. In a sense, we show that bounded multi-intersection is its own dual property.

Let be a -hypergraph. Then the dual hypergraph is a -hypergraph.111Note that the superscript of only signifies that it is the dual of . It is not connected to the integer constant used for the multi-intersection size of .

Proof.

Let be distinct arbitrary vertices of a -hypergraph . We write for the set of edges incident to a vertex . Since is a -hypergraph, it must hold that has no more than elements. Otherwise, there would be at least edges in that share vertices, i.e., a contradiction to the assumption that is a -hypergraph.

Now, consider the edges in that correspond to the vertices in . It follows from the definition of the dual hypergraph that since any two edges in share exactly one vertex for each edge in that they are both incident to. We know from above that . As this applies to any choice of vertices in , and thus also to any choice of edges in , we see that any intersection of edges in has cardinality less or equal . ∎

There is a function such that the following is true. Let be constants. Let be a -hypergraph and be an assignment of weights to . Assume that . Then there is an assignment of weights to such that , and .

Proof.

Let be the dual weight assignment of as in Proposition 5.2. That is, is an edge weight assignment in the dual hypergraph with and .

From Lemma 5.2 we have that is a -hypergraph and thus by Theorem 3 there is an edge weight function with and . Let now be the dual weight assignment of . By Proposition 5.2 we then see that also and . ∎

To conclude this section we wish to highlight the connection of Theorem 5.2 to a classical result on fractional edge covers. The following result is due to Füredi [6], who extended earlier results by Chung et al. [3].

[[6], page 152,  Proposition 5.11.(iii)] For every hypergraph of rank (i.e., maximal edge size) , and every fractional vertex cover for satisfying , the property holds.

Recall that a hypergraph with rank is also a -hypergraph. Hence, the above proposition means that, for a -hypergraph , there is a fractional vertex cover of optimal weight whose support is bounded by a function of the weight and . Theorem 5.2 generalizes Proposition 5.2 in two aspects. First, Theorem 5.2 considers -hypergraphs with and second, it applies to assignments of weights to vertices in general not just to those that establish an optimal fractional vertex cover. An important aspect of Proposition 5.2 not reflected in Theorem 5.2 is a concrete upper bound on the size of the support. Optimizing the upper bound following from Theorem 3 is left for future research.

6 Conclusion and Outlook

We have proved novel upper bounds on the size of the support of fractional edge covers and vertex covers. These bounds have then been fruitfully applied to the problem of checking for given hypergraph . Recall that, without imposing any restrictions on the hypergraph , this problem is NP-complete even for  [5], thus ruling out XP-membership. In contrast, for hypergraph classes that exhibit bounded multi-intersection, we have actually managed to establish XP-membership, that is, checking for hypergraphs in such a class is feasible in polynomial time for any constant .

However, there is still room for improvement: first, our tractability result depends on a big constant . Hence, an important next step for future research will be a deeper investigation of algorithms for checking in case of the BMIP and either further improve the runtime or prove a matching lower bound. Moreover, XP-membership is only “second prize” in terms of a parameterized complexity result. It will be interesting to search for further restrictions on the hypergraphs to achieve fixed-parameter tractability (FPT).

Another major challenge for future research is the computation of . Note that our tractability result refers to the decision problem of checking . However, at its heart, dealing with fractional hypertree width is an optimization problem, namely computing the minimum possible width of all fractional hypertree decompositions of . The difficulty here is that our bound tends to infinity as approaches the actual value of . Substantial new ideas are required to overcome this problem.

Our bound on the support of fractional vertex covers generalizes a classical result by Füredi in two aspects. In future work, we plan to explore how this generalization can be applied to known consequences (cf. [6]) of Füredi’s result. Finally, we have left open the extension to fractional independent sets. By use of the complementary slackness of linear programs our main result also implies structural restrictions for optimal fractional independent sets since there can only be a bounded number of constraints that have slack. We believe that an in-depth study of the connections to independent sets is merited.

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Appendix

Appendix A Full Proofs

a.1 Proof of Lemma 2

Proof.

A constraint of a unary linear program can be associated with a subset of the set of variables and a unary linear program is just a set of constraints. Hence there are at most non-isomorphic unary linear programs with variables. Clearly, the number of optimal values of these programs that are greater than are at most that many and (by assumption) at least . Therefore, these optimal values have a minimum. Denote this minimum by . Set

a.2 Proof of Lemma 2

Recall our claim from Lemma 2. There is a function with the following property: let be a -hypergraph and let be an edge weight function with . Moreover, let and assume that, for each , . Let be the set of all vertices of weight at least . Then holds.

Proof of Lemma 2.

The proof is based on the following claim (which is Lemma 7.2 in [