Cuboids, a class of clutters
Introduction
Let E be a finite set of elements, and let be a family of subsets of E, called members. We say that is a clutter over ground set E if no member is contained in another one [12]. Two clutters are isomorphic if one is obtained from the other after relabeling its ground set. A cover of is a subset of E that intersects every member, and a cover is minimal if it does not properly contain another cover. The set covering polyhedron of is defined as while the set covering polytope of refers to Here, is shorthand notation for .
Proposition 1.1 folklore Let be a clutter. Then the integral extreme points of are precisely the incidence vectors of the minimal covers of and the integral extreme points of are precisely the incidence vectors of the covers of . Moreover, is an integral polyhedron if, and only if, is an integral polytope.
We say that is ideal if the corresponding set covering polyhedron (or polytope) is integral [11]. Consider the primal-dual pair of linear programs It is well-known that is an ideal clutter if, and only if, the primal linear program (P) has an integral optimal solution for all (see [8], Theorem 4.1). We say that (P) is totally dual integral if for all , the dual linear program (D) has an integral optimal solution. It is also well-known that if (P) is totally dual integral, then is an ideal clutter ([20], [13], see also [8], Theorem 4.26). The converse however does not hold, as we will explain shortly.
Define the covering number as the minimum cardinality of a cover, and the packing number as the maximum number of pairwise disjoint members. As every member of a packing picks a distinct element of a cover, it follows that . If equality holds here, then packs, otherwise it is non-packing. Observe that and are the integral optimal values of (P) and (D), respectively, for . Thus, if (P) is totally dual integral, then must pack.
Consider the clutter over ground set whose members are Notice that is isomorphic to the clutter of triangles (or claws) of the complete graph on four vertices. This clutter does not pack as . This clutter was found by Lovász [26], but Seymour [36] was the person who established the significant role of among non-packing clutters in his seminal paper on the matroids with the max-flow min-cut property. Even though does not pack, it is an ideal clutter [36]. In fact, as we will see in §4,
Proposition 1.2 is the only ideal non-packing clutter over at most 6 elements, up to isomorphism.
Given disjoint sets , the minor of obtained after deleting I and contracting J is the clutter We say that the minor is proper if . In terms of the set covering polyhedron, contractions correspond to restricting the corresponding coordinates to 0, while deletions correspond to projecting away the corresponding coordinates; in terms of the set covering polytope, deletions can also be thought of as restricting the corresponding coordinates to 1, which is sometimes convenient. Due to these geometric interpretations, if a clutter is ideal then so is every minor of it [36]. A clutter is minimally non-ideal if it is not ideal but every proper minor is ideal. In the same vein, a clutter is minimally non-packing if it does not pack but every proper minor packs. A minimally non-packing clutter is either ideal or minimally non-ideal – this is a fascinating consequence of Lehman's seminal theorem on minimally non-ideal clutters [25] and was first noticed in [10].
Proposition 1.2 implies that is in fact an ideal minimally non-packing clutter. Despite what Seymour [36] conjectured, is not the only ideal minimally non-packing clutter. Schrijver [31] found an ideal minimally non-packing clutter over 9 elements, which was a minor of the clutter of dijoins of a directed graph, as a counterexample to a conjecture of Edmonds and Giles [13]. Two decades later, Cornuéjols, Guenin and Margot grew the known list to a dozen sporadic instances as well as an infinite class of ideal minimally non-packing clutters [10]. All their examples of ideal minimally non-packing clutters, however, have covering number two, so they conjecture the following:
The Conjecture [10] Every ideal minimally non-packing clutter has covering number two.
We will prove this conjecture for clutters over at most 8 elements (§4). For the most part, however, we take a different perspective towards the Conjecture. Take an integer . We will be working over , the vertices of the unit n-dimensional hypercube, represented for convenience as strings of length n. Take a set . The cuboid of S, denoted , is the clutter over ground set whose members have incidence vectors1 Observe that every member of has cardinality n, and that for each , is a cover. For example, the cuboid of is which is . Thus, the smallest ideal minimally non-packing clutter is a cuboid. Abdi, Cornuéjols and Pashkovich showed that cuboids play a central role among all ideal minimally non-packing clutters [4]. They found two new ideal minimally non-packing cuboids, and observed that each clutter of – the only known infinite class of ideal minimally non-packing clutters – is a cuboid. This was also observed by Flores, Gitler and Reyes, who referred to cuboids as 2-partitionable clutters [16]. However, to emphasize the fact that these clutters come from subsets of a hypercube, we refrain from this terminology. The following theorem further stresses the importance of cuboids among ideal minimally non-packing clutters:
Theorem 1.3 Every minimally non-packing cuboid is ideal.
This theorem is proved in §2. In this paper, we will see that the Conjecture is equivalent to a conjecture on cuboids (§4), and furthermore, we will show how Seymour's classification of binary matroids with the sums of circuits property [33], his characterization of binary matroids with the max-flow min-cut property [36], as well as his f-Flowing Conjecture [36], [33] translate into the world of cuboids (§2 and §3). We will also reduce the Replication Conjecture of Conforti and Cornuéjols [7] to cuboids (§4). After reading this paper, we hope to have convinced the reader that cuboids are an important class of clutters.
Let be an integer and an arbitrary set of vertices of the unit n-dimensional hypercube. Take a coordinate . To twist coordinate i is to replace S by this terminology is due to Bouchet [6]. (The symmetric difference operator △ performs coordinatewise addition modulo 2. Novick and Sebő [29] refer to twisting as switching.) Observe that the cuboid of S encodes all of its twistings. If is obtained from S after twisting and relabeling some coordinates, then we say that is isomorphic to S and write it as . Notice that if are isomorphic, then so are their cuboids.
The set obtained from after dropping coordinate i is called the 0-restriction of S over coordinate i, and the set obtained from after dropping coordinate i is called the 1-restriction of S over coordinate i. If is obtained from S after 0- and 1-restricting some coordinates, then we say that is a restriction of S. The set obtained from S after dropping coordinate i is called the projection of S over coordinate i. If is obtained from S after projecting away some coordinates, then we say that is a projection of S. If is obtained from S after a series of restrictions and projections, then we say that is a minor of S; we say that is a proper minor if at least one minor operation is applied. These minor operations can be defined directly on cuboids:
Remark 1.4 [4] Take an integer and a set . Then, for each , the following statements hold: If is the 0-restriction of S over i, then . If is the 1-restriction of S over i, then . If is the projection of S over i, then .
If is a minor of S, we will say that is a cuboid minor of .
Inequalities of the form are called hypercube inequalities, and the ones of the form are called generalized set covering inequalities. Observe that these two classes of inequalities are closed under twistings, i.e. the change of variables .
We say that S is cube-ideal if its convex hull can be described using hypercube and generalized set covering inequalities. For instance, the set is cube-ideal as its convex hull is as illustrated in Fig. 1.
Remark 1.5 Take an integer and a cube-ideal set . If is isomorphic to a minor of S, then is cube-ideal. Proof Since the hypercube and generalized set covering inequalities are closed under relabelings and the transformation , being cube-ideal is closed under relabelings and twistings. It therefore suffices to prove the remark for the case when is obtained from S after a single minor operation. Suppose that for an appropriate . If is obtained from S after 0-restricting coordinate 1, then . If is obtained from S after 1-restricting coordinate 1, then . If is obtained from S after projecting away coordinate 1, then . In each case, we see that is still cube-ideal, thereby finishing the proof. □
Cube-idealness of subsets of a hypercube can be defined solely in terms of cuboids:
Theorem 1.6 Take an integer and a set . Then S is cube-ideal if, and only if, is an ideal clutter.
Using this theorem, which is proved in §2, we can use cube-idealness to link idealness to another deep property. We say that S is a vector space over , or simply a binary space, if for all (possibly equal) points . A binary space is by definition the cycle space of a binary matroid (see [30]). For instance, is a binary space, and it corresponds to the cycle space of the graph on two vertices and three parallel edges. We will see in §2 that a binary space is cube-ideal if, and only if, the associated binary matroid has the sums of circuits property. Paul Seymour introduced this rich property in [35], and after developing his splitter theorems and decomposition of regular matroids [32], he classified the binary matroids with the sums of circuits property. (In that paper, he also posed the cycle double cover conjecture [33], [37].)
Theorem 1.6 reduces cube-idealness of subsets of a hypercube to clutter idealness; Theorem 4.3 gives a converse reduction (though with an exponential blow-up). As such, cube-idealness provides a framework to interpret clutter idealness geometrically, rather than combinatorially, as foreseen by Jon Lee [22]. To this end, take a point . The induced clutter of S with respect to x, denoted , is the clutter over ground set whose members are In words, is the clutter corresponding to the points of of minimal support. Notice that if then every induced clutter is {}, and in general, if then . Observe that Hence,
Remark 1.7 Take an integer and a set . Then the induced clutters are in correspondence with the minors of obtained after contracting, for each , exactly one of .
It therefore follows from Theorem 1.6 that if S is cube-ideal, then all of its induced clutters are ideal. The converse of this statement, proved in §2, is also true:
Theorem 1.8 Take an integer and a set . Then S is cube-ideal if, and only if, every induced clutter of S is ideal.
Let be a minor-closed property defined on clutters. Motivated by Theorem 1.8, we say that is a local property if for all integers and sets , the following statements are equivalent:
- •
has property ,
- •
the induced clutters of S have property .
We say that a clutter has the packing property if every minor, including the clutter itself, packs. Notice that a clutter has the packing property if, and only if, it has no minimally non-packing minor. Let us consider again. The induced clutters of this set are isomorphic to either or , so they all have the packing property, whereas does not pack. Therefore, in contrast to idealness, the packing property is non-local. We will now see what causes the packing property to become local.
Let be an integer. A pair of points are antipodal if . Take a set . We will refer to the points in S as feasible points and to the points in as infeasible points. We say that S is polar if either there are antipodal feasible points or all the feasible points agree on a coordinate: For instance, the set is non-polar. Notice that if a set is polar, then so is every twisting of it. Moreover,
Remark 1.9 Take an integer and a set . Then S is polar if, and only if, packs.
We say that S is strictly polar if every restriction, including S itself, is polar.
Remark 1.10 Take an integer and a strictly polar set . If is isomorphic to a minor of S, then is polar. Proof Being polar is closed under twistings and relabelings, so it suffices to prove that every minor of S is polar. To this end, let be a minor of S. Then there are disjoint sets such that is obtained after 0-restricting I, 1-restricting J and projecting away K; among all possible we may assume that K is minimal, so that no single projection can be replaced by a single restriction. Let R be the restriction of S obtained after 0-restricting I and 1-restricting J; notice that is obtained from R after projecting away K. Since S is strictly polar, it follows from the definition that R is polar. If R contains antipodal points, then the same points give antipodal points in the projection . Otherwise, the points in R agree on a coordinate, so by the minimality of K, the points in the projection also agree on the same coordinate. In either cases, we see that is polar, as required. □
As a result, a set is strictly polar if, and only if, every cuboid minor of the corresponding cuboid packs. In particular, if has the packing property, then S is strictly polar. We will see that once strict polarity is extracted, the non-local packing property becomes a local property:
Theorem 1.11 Let S be a strictly polar set. Then has the packing property if, and only if, all of the induced clutters of S have the packing property.
This theorem is proved in §3. In that section, we will also see that strict polarity is a tractable property:
Theorem 1.12 Take an integer and a set . Then the following statements are equivalent: S is not strictly polar, there are distinct points such that the restriction of S containing them of smallest dimension is not polar.
As a result, in time one can certify whether or not S is strictly polar.
A set is strictly non-polar if it is not polar and every proper restriction is polar. Theorem 1.12 equivalently states that every strictly non-polar set has three distinct feasible points that do not all agree on a coordinate. A set is minimally non-polar if it is not polar and every proper minor is polar. A minimally non-polar set is strictly non-polar, and as we will see in §3, there are strictly non-polar sets that are not minimally non-polar. Observe that a set is strictly polar if, and only if, it has no strictly non-polar restriction if, and only if, it has no minimally non-polar minor.
A fascinating consequence of Lehman's theorem on minimally non-ideal clutters [25] is the following:
Theorem 1.13 [10] If a clutter has the packing property, then it is ideal.
The converse however is not true, as there are ideal non-packing cuboids such as . And after all, we should not expect the two properties to be the same, because idealness is a local property but the packing property is not. However, as Theorem 1.11 shows, strict polarity makes the packing property local. We conjecture that strict polarity does far more than that:
The Polarity Conjecture Let S be a strictly polar set. Then is ideal if, and only if, has the packing property.
Justified by Theorem 1.6, Theorem 1.13, we may rephrase this conjecture as follows:
The Polarity Conjecture rephrased If a set is cube-ideal and strictly polar, then its cuboid has the packing property.
As we will see in §4,
Theorem 1.14 The Polarity Conjecture is equivalent to the Conjecture.
Take an integer and a set . We say that S is critically non-polar if it is strictly non-polar and, for each , both the 0- and 1-restrictions of S over coordinate i have antipodal points. We will see in §3 that critical non-polarity implies minimal non-polarity. In §4, we will see that if the Polarity Conjecture is true, then so is the following conjecture:
Conjecture 1.15 If a set is cube-ideal and critically non-polar, then its cuboid is minimally non-packing.
We will show in §4 that the Polarity Conjecture and Conjecture 1.15 are true for sets of degree at most 8 – this notion is defined later in the introduction.
In §5 we study three basic binary operations on pairs of sets. Take integers and sets and . Define the product and the coproduct Observe that , thereby justifying our terminology. We will observe that if the cuboids of two sets are ideal (resp. have the packing property), then so is (resp. does) the cuboid of their product. Moreover, by exploiting the locality of idealness, and the locality of the packing property once strict polarity is enforced, we show that if the cuboids of two sets are ideal (resp. have the packing property), then so is (resp. does) the cuboid of their coproduct. Define the reflective product In words, the reflective product is obtained from after replacing each feasible point by a copy of and each infeasible point by a copy of . Observe that and . We will see in §5 that,
Theorem 1.16 Take integers and sets and . If are cube-ideal, then so are .
That is, by Theorem 1.6, if are ideal, then so are . In contrast, the analogue of this for the packing property does not hold. For instance, let and . Then all have the packing property, while are isomorphic to and therefore do not pack. This phenomenon raises an intriguing question: can we build a counterexample to the Polarity Conjecture by taking the reflective product of two sets that are not counterexamples? As we will prove in §5, the answer is no:
Theorem 1.17 Take integers and sets and , where , have the packing property. If is strictly polar, then has the packing property.
A set is antipodally symmetric if a point is feasible if and only if its antipodal point is feasible. We will prove the following in §5:
Theorem 1.18 Take integers and sets and , where is strictly non-polar. Then the following statements hold: are nonempty. Either and is antipodally symmetric, or and is antipodally symmetric. In particular, . is critically non-polar. If have the packing property, then is an ideal minimally non-packing clutter.
For an integer , let (Hereinafter, are the m-dimensional vectors all of whose entries are , respectively.) See Fig. 2 for an illustration of and . The reader can readily check that are strictly non-polar sets, and that their cuboids are isomorphic to the ideal minimally non-packing clutters .
Take an integer . Denote by the skeleton graph of whose vertices are the points in and two points are adjacent if they differ in exactly one coordinate. A set is connected if is connected. We say that is strictly connected if every restriction of R is connected.
The following result, proved in §5, is the second half of Theorem 1.18:
Theorem 1.19 Take an integer and an antipodally symmetric set such that is strictly non-polar. If is not isomorphic to any of , then both S and are strictly connected.
For an integer , let See Fig. 2 for an illustration of . Notice that is a cycle of length . The reader can readily check that are strictly non-polar sets, and that are strictly connected, verifying Theorem 1.19. The cuboid of is the ideal minimally non-packing clutter found in [4], and as we will see in §3, the cuboids of are not ideal and not minimally non-packing.
Take an integer and a set . For an integer , we say that S has degree at most k if every infeasible point has at most k infeasible neighbors in . We say that S has degree k if it has degree at most k and not . As a result, given a set of degree k, every infeasible point has at most k infeasible neighbors, and there is an infeasible point achieving this bound. For each , it is known that every strictly non-polar set of degree at most k must have dimension at most ([4], Theorem 1.10 (i)). It was also shown that, up to isomorphism, the strictly non-polar sets of degree at most 2 are , as displayed in Fig. 2 ([4], Theorem 1.9). We will improve in §6 the upper bound of as follows:
Theorem 1.20 Take an integer and a strictly non-polar set S of degree k, whose dimension is n. Then the following statements hold: . If , then either S is minimally non-polar, or after a possible relabeling, and the projection of S over coordinate is a critically non-polar set that is the reflective product of two other sets. If , then S is critically non-polar. If , then , every infeasible point has exactly k infeasible neighbors, and is an ideal minimally non-packing clutter.
Part (4) is done by using Mantel's Theorem [27] as well as the local structure of delta free clutters. Notice that , which is of degree 2 and dimension 5, has 16 points, every infeasible point has exactly 2 infeasible neighbors, and is an ideal minimally non-packing clutter. In §6 we will describe a computer code, whose correctness relies on Theorem 1.20 (1), that generates all the non-isomorphic strictly non-polar sets of degree at most 3, as well as all the non-isomorphic strictly non-polar sets of degree 4 and dimension at most 7. As we will see, there are exactly 745 non-isomorphic strictly non-polar sets of degree at most 4 and dimension at most 7, explicitly described in Appendix A and summarized in Fig. 3, 716 sets of which have ideal minimally non-packing cuboids.
Section snippets
Cube-ideal sets
In this section we prove Theorem 1.3, Theorem 1.6, Theorem 1.8. We will also characterize the cube-ideal binary spaces, discuss the sums of circuits property, the theorem of Edmonds and Johnson on T-join polytopes [14] and the f-Flowing Conjecture.
Strictly polar sets
In this section we prove Theorem 1.11, Theorem 1.12, discuss strict, minimal and critical non-polarity, characterize when the cuboid of a strictly non-polar set is minimally non-packing, characterize the strictly polar binary spaces, and discuss Seymour's characterization of the binary matroids with the max-flow min-cut property.
The Polarity Conjecture
In this section, we prove Proposition 1.2 and Theorem 1.14, and show that the Polarity Conjecture implies Conjecture 1.15. We will also prove the Conjecture for clutters over at most 8 elements, and the Polarity Conjecture and Conjecture 1.15 for sets of degree at most 8. Moreover, we will see how the Replication Conjecture of Conforti and Cornuéjols [7] can be reduced to cuboids.
Basic binary operations
In this section, we prove Theorem 1.16, Theorem 1.17, Theorem 1.18, Theorem 1.19. We need a few basic facts on the products and coproducts of clutters and sets.
The spectrum of strictly non-polar sets of constant degree
Here we prove Theorem 1.20, and describe a code that generates the strictly non-polar sets of degree at most 3.
Concluding remarks and open questions
Cuboids, a natural home to ideal minimally non-packing clutters with covering number two, were comprehesively studied in this paper. Ideal minimally non-packing cuboids of bounded degree were studied, and more than seven hundred non-isomorphic ones over at most 14 elements were generated. Cuboids were also used as a tool to manifest the geometry behind idealness and the packing property. We saw that idealness is a local property while the packing property is not, resulting in a geometric rift
Acknowledgements
We would like to thank Kanstantsin Pashkovich for helpful discussions about parts of this paper. We would also like to thank two anonymous referees for their diligent review of our work; it has led to us to publish our code on GitHub [1]. This work was supported in parts by ONR grant 00014-18-12129, NSF grant CMMI-1560828, and an NSERC PDF grant 516584-2018. Our code is written in DrRacket, Version 7.3 [15]. Finally, many thanks to Christopher Calzonetti as well as the Math Faculty Computing
References (37)
- et al.
On the cycle polytope of a binary matroid
J. Combin. Theory Ser. B
(1986) - et al.
Ideal 0, 1 matrices
J. Combin. Theory Ser. B
(1994) - et al.
Bottleneck extrema
J. Combin. Theory Ser. B
(1970) - et al.
A min-max relation for submodular functions on graphs
A short proof of Seymour's characterization of the matroids with the max-flow min-cut property
J. Combin. Theory Ser. B
(2002)A counterexample to a conjecture of Edmonds and Giles
Discrete Math.
(1980)Decomposition of regular matroids
J. Combin. Theory Ser. B
(1980)Matroids and multicommodity flows
European J. Combin.
(1981)The matroids with the max-flow min-cut property
J. Combin. Theory Ser. B
(1977)- A. Abdi, A code for generating all non-isomorphic strictly non-polar sets of fixed dimension and bounded degree, GitHub...