Cuboids, a class of clutters

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Abstract

The τ=2 Conjecture, the Replication Conjecture and the f-Flowing Conjecture, and the classification of binary matroids with the sums of circuits property are foundational to Clutter Theory and have far-reaching consequences in Combinatorial Optimization, Matroid Theory and Graph Theory. We prove that these conjectures and result can equivalently be formulated in terms of cuboids, which form a special class of clutters. Cuboids are used as means to (a) manifest the geometry behind primal integrality and dual integrality of set covering linear programs, and (b) reveal a geometric rift between these two properties, in turn explaining why primal integrality does not imply dual integrality for set covering linear programs. Along the way, we see that the geometry supports the τ=2 Conjecture. Studying the geometry also leads to over 700 new ideal minimally non-packing clutters over at most 14 elements, a surprising revelation as there was once thought to be only one such clutter.

Introduction

Let E be a finite set of elements, and let C be a family of subsets of E, called members. We say that C 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 C 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 C is defined asQ(C):={xR+E:x(C)1CC} while the set covering polytope of C refers toP(C):={x[0,1]E:x(C)1CC}. Here, x(C) is shorthand notation for eCxe.

Proposition 1.1 folklore

Let C be a clutter. Then the integral extreme points of Q(C) are precisely the incidence vectors of the minimal covers of C and the integral extreme points of P(C) are precisely the incidence vectors of the covers of C. Moreover, Q(C) is an integral polyhedron if, and only if, P(C) is an integral polytope.

We say that C is ideal if the corresponding set covering polyhedron (or polytope) is integral [11]. Consider the primal-dual pair of linear programs(P)minwxs.t.x(C)1CCx0(D)max1ys.t.(yC:eCC)weeEy0. It is well-known that C is an ideal clutter if, and only if, the primal linear program (P) has an integral optimal solution for all wZ+E (see [8], Theorem 4.1). We say that (P) is totally dual integral if for all wZ+E, the dual linear program (D) has an integral optimal solution. It is also well-known that if (P) is totally dual integral, then C 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 τ(C) as the minimum cardinality of a cover, and the packing number ν(C) as the maximum number of pairwise disjoint members. As every member of a packing picks a distinct element of a cover, it follows that τ(C)ν(C). If equality holds here, then C packs, otherwise it is non-packing. Observe that τ(C) and ν(C) are the integral optimal values of (P) and (D), respectively, for w=1. Thus, if (P) is totally dual integral, then C must pack.

Consider the clutter over ground set {1,,6} whose members areQ6:={{2,4,6},{1,3,6},{1,4,5},{2,3,5}}. Notice that Q6 is isomorphic to the clutter of triangles (or claws) of the complete graph on four vertices. This clutter does not pack as τ(Q6)=2>1=ν(Q6). This clutter was found by Lovász [26], but Seymour [36] was the person who established the significant role of Q6 among non-packing clutters in his seminal paper on the matroids with the max-flow min-cut property. Even though Q6 does not pack, it is an ideal clutter [36]. In fact, as we will see in §4,

Proposition 1.2

Q6 is the only ideal non-packing clutter over at most 6 elements, up to isomorphism.

Given disjoint sets I,JE, the minor of C obtained after deleting I and contracting J is the clutterCI/J:= the minimal sets of {CJ:CC,CI=}. We say that the minor is proper if IJ. 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 Q6 is in fact an ideal minimally non-packing clutter. Despite what Seymour [36] conjectured, Q6 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 {Qr,t:r1,t1} 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 τ=2 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 τ=2 Conjecture. Take an integer n1. We will be working over {0,1}n, the vertices of the unit n-dimensional hypercube, represented for convenience as 0,1 strings of length n. Take a set S{0,1}n. The cuboid of S, denoted cuboid(S), is the clutter over ground set [2n] whose members have incidence vectors1(x1,1x1,,xn,1xn)xS. Observe that every member of cuboid(S) has cardinality n, and that for each i[n], {2i1,2i} is a cover. For example, the cuboid of R1,1:={000,110,101,011}{0,1}3 is {{2,4,6},{1,3,6},{1,4,5},{2,3,5}} which is Q6. 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 {Qr,t,r1,t1} – 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 τ=2 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 n1 be an integer and S{0,1}n an arbitrary set of vertices of the unit n-dimensional hypercube. Take a coordinate i[n]. To twist coordinate i is to replace S bySei:={xei:xS}; 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 S is obtained from S after twisting and relabeling some coordinates, then we say that S is isomorphic to S and write it as SS. Notice that if S,S are isomorphic, then so are their cuboids.

The set obtained from S{x:xi=0} after dropping coordinate i is called the 0-restriction of S over coordinate i, and the set obtained from S{x:xi=1} after dropping coordinate i is called the 1-restriction of S over coordinate i. If S is obtained from S after 0- and 1-restricting some coordinates, then we say that S is a restriction of S. The set obtained from S after dropping coordinate i is called the projection of S over coordinate i. If S is obtained from S after projecting away some coordinates, then we say that S is a projection of S. If S is obtained from S after a series of restrictions and projections, then we say that S is a minor of S; we say that S 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 n1 and a set S{0,1}n. Then, for each i[n], the following statements hold:

  • If S is the 0-restriction of S over i, then cuboid(S)=cuboid(S)(2i1)/2i.

  • If S is the 1-restriction of S over i, then cuboid(S)=cuboid(S)/(2i1)2i.

  • If S is the projection of S over i, then cuboid(S)=cuboid(S)/{2i1,2i}.

If S is a minor of S, we will say that cuboid(S) is a cuboid minor of cuboid(S).

Inequalities of the form 1xi0,i[n] are called hypercube inequalities, and the ones of the formiIxi+jJ(1xj)1I,J[n],IJ= are called generalized set covering inequalities. Observe that these two classes of inequalities are closed under twistings, i.e. the change of variables xi1xi,i[n].

We say that S is cube-ideal if its convex hull conv(S) can be described using hypercube and generalized set covering inequalities. For instance, the set R1,1={000,110,101,011} is cube-ideal as its convex hull isconv(R1,1)={x[0,1]3:(1x1)+x2+x31x1+(1x2)+x31x1+x2+(1x3)1(1x1)+(1x2)+(1x3)1}, as illustrated in Fig. 1.

Remark 1.5

Take an integer n1 and a cube-ideal set S{0,1}n. If S is isomorphic to a minor of S, then S is cube-ideal.

Proof

Since the hypercube and generalized set covering inequalities are closed under relabelings and the transformation xi1xi,i[n], being cube-ideal is closed under relabelings and twistings. It therefore suffices to prove the remark for the case when S is obtained from S after a single minor operation. Suppose that conv(S)={x[0,1]n:iIxi+jJ(1xj)1,(I,J)V} for an appropriate V. If S is obtained from S after 0-restricting coordinate 1, then conv(S)={x[0,1]n1:iI{1}xi+jJ(1xj)1,(I,J)V,1J}. If S is obtained from S after 1-restricting coordinate 1, then conv(S)={x[0,1]n1:iIxi+jJ{1}(1xj)1,(I,J)V,1I}. If S is obtained from S after projecting away coordinate 1, then conv(S)={x[0,1]n1:iIxi+jJ(1xj)1,(I,J)V,1IJ}. In each case, we see that S 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 n1 and a set S{0,1}n. Then S is cube-ideal if, and only if, cuboid(S) 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 GF(2), or simply a binary space, if abS for all (possibly equal) points a,bS. A binary space is by definition the cycle space of a binary matroid (see [30]). For instance, R1,1 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 x{0,1}n. The induced clutter of S with respect to x, denoted ind(Sx), is the clutter over ground set [n] whose members areind(Sx)= the minimal sets of {C[n]:χCSx}. In words, ind(Sx) is the clutter corresponding to the points of Sx of minimal support. Notice that if S= then every induced clutter is {}, and in general, if xS then ind(Sx)={}. Observe thatind(Sx)=cuboid(S)/{2i:i[n],xi=0}/{2i1:i[n],xi=1}. Hence,

Remark 1.7

Take an integer n1 and a set S{0,1}n. Then the 2n induced clutters ind(Sx),x{0,1}n are in correspondence with the 2n minors of cuboid(S) obtained after contracting, for each i[n], exactly one of 2i1,2i.

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 n1 and a set S{0,1}n. Then S is cube-ideal if, and only if, every induced clutter of S is ideal.

Let P be a minor-closed property defined on clutters. Motivated by Theorem 1.8, we say that P is a local property if for all integers n1 and sets S{0,1}n, the following statements are equivalent:

  • cuboid(S) has property P,

  • the induced clutters of S have property P.

Otherwise, we say that P is a non-local property. Notice that Theorem 1.6, Theorem 1.8 imply that idealness is a local property. Using the locality of idealness, we will be able to use the famous result of Edmonds and Johnson on T-join polytopes [14] to prove Seymour's result that graphic matroids have the sums of circuits property [35], as well as find a new link between the binary matroids with the sums of circuits property and the f-flowing binary matroids, and formulate the famous f-Flowing Conjecture in terms of cube-idealness of binary spaces (§2).

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 R1,1 again. The induced clutters of this set are isomorphic to either {} or {{1},{2},{3}}, so they all have the packing property, whereas cuboid(R1,1)=Q6 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 n1 be an integer. A pair of points a,b{0,1}n are antipodal if a+b=1. Take a set S{0,1}n. We will refer to the points in S as feasible points and to the points in {0,1}nS 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:{x,1x}S for some x{0,1}norS{x:xi=a} for some i[n] and a{0,1}. For instance, the set R1,1 is non-polar. Notice that if a set is polar, then so is every twisting of it. Moreover,

Remark 1.9

Take an integer n1 and a set S{0,1}n. Then S is polar if, and only if, cuboid(S) packs.

We say that S is strictly polar if every restriction, including S itself, is polar.

Remark 1.10

Take an integer n1 and a strictly polar set S{0,1}n. If S is isomorphic to a minor of S, then S 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 S be a minor of S. Then there are disjoint sets I,J,K[n] such that S is obtained after 0-restricting I, 1-restricting J and projecting away K; among all possible I,J,K 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 S 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 S. Otherwise, the points in R agree on a coordinate, so by the minimality of K, the points in the projection S also agree on the same coordinate. In either cases, we see that S 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 cuboid(S) 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 cuboid(S) 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 n1 and a set S{0,1}n. Then the following statements are equivalent:

  • (i)

    S is not strictly polar,

  • (ii)

    there are distinct points a,b,cS such that the restriction of S containing them of smallest dimension is not polar.

As a result, in time O(n|S|4) 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 Q6. 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 cuboid(S) is ideal if, and only if, cuboid(S) 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 τ=2 Conjecture.

Take an integer n3 and a set S{0,1}n. We say that S is critically non-polar if it is strictly non-polar and, for each i[n], 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 n1,n21 and sets S1{0,1}n1 and S2{0,1}n2. Define the productS1×S2:={(x,y){0,1}n1×{0,1}n2:xS1 and yS2} and the coproductS1S2:={(x,y){0,1}n1×{0,1}n2:xS1 or yS2}. Observe that S1S2=S1×S2, 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 productS1S2:=(S1×S2)(S1×S2). In words, the reflective product S1S2 is obtained from S1 after replacing each feasible point by a copy of S2 and each infeasible point by a copy of S2. Observe that S1S2=S1S2 and S1S2=S1S2=S1S2. We will see in §5 that,

Theorem 1.16

Take integers n1,n21 and sets S1{0,1}n1 and S2{0,1}n2. If S1,S1,S2,S2 are cube-ideal, then so are S1S2,S1S2.

That is, by Theorem 1.6, if cuboid(S1),cuboid(S1),cuboid(S2),cuboid(S2) are ideal, then so are cuboid(S1S2),cuboid(S1S2). In contrast, the analogue of this for the packing property does not hold. For instance, let S1:={00,11} and S2:={0}. Then cuboid(S1),cuboid(S1),cuboid(S2),cuboid(S2) all have the packing property, while cuboid(S1S2),cuboid(S1S2) are isomorphic to Q6 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 n1,n21 and sets S1{0,1}n1 and S2{0,1}n2, where cuboid(S1), cuboid(S1),cuboid(S2),cuboid(S2) have the packing property. If S1S2 is strictly polar, then cuboid(S1S2) 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 n1,n21 and sets S1{0,1}n1 and S2{0,1}n2, where S1S2 is strictly non-polar. Then the following statements hold:

  • (1)

    S1,S1,S2,S2 are nonempty.

  • (2)

    Either n1=1 and S2 is antipodally symmetric, or n2=1 and S1 is antipodally symmetric. In particular, S1S2S1S2.

  • (3)

    S1S2 is critically non-polar.

  • (4)

    If cuboid(S1),cuboid(S1),cuboid(S2),cuboid(S2) have the packing property, then cuboid(S1S2) is an ideal minimally non-packing clutter.

For an integer k1, letRk,1:={0k+1,1k+1}{0}{0,1}k+2. (Hereinafter, 0m,1m are the m-dimensional vectors all of whose entries are 0,1, respectively.) See Fig. 2 for an illustration of R1,1 and R2,1. The reader can readily check that {Rk,1:k1} are strictly non-polar sets, and that their cuboids are isomorphic to the ideal minimally non-packing clutters {Qk,1:k1}.

Take an integer n1. Denote by Gn the skeleton graph of {0,1}n whose vertices are the points in {0,1}n and two points u,v are adjacent if they differ in exactly one coordinate. A set R{0,1}n is connected if Gn[R] is connected. We say that R{0,1}n 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 n1 and an antipodally symmetric set S{0,1}n such that S{0} is strictly non-polar. If S{0} is not isomorphic to any of {Rk,1:k1}, then both S and S are strictly connected.

For an integer k5, letCk1:={i=1dei,1k1i=1dei:d[k1]}{0,1}k1Rk:=Ck1{0}{0,1}k. See Fig. 2 for an illustration of R5. Notice that Gk1[Ck1] is a cycle of length 2(k1). The reader can readily check that {Rk:k5} are strictly non-polar sets, and that Ck1,Ck1 are strictly connected, verifying Theorem 1.19. The cuboid of R5 is the ideal minimally non-packing clutter Q10 found in [4], and as we will see in §3, the cuboids of {Rk:k6} are not ideal and not minimally non-packing.

Take an integer n1 and a set S{0,1}n. For an integer k{0,1,,n}, we say that S has degree at most k if every infeasible point has at most k infeasible neighbors in Gn. We say that S has degree k if it has degree at most k and not k1. 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 k1, it is known that every strictly non-polar set of degree at most k must have dimension at most 4k+1 ([4], Theorem 1.10 (i)). It was also shown that, up to isomorphism, the strictly non-polar sets of degree at most 2 are R1,1,R2,1,R5, as displayed in Fig. 2 ([4], Theorem 1.9). We will improve in §6 the upper bound of 4k+1 as follows:

Theorem 1.20

Take an integer k2 and a strictly non-polar set S of degree k, whose dimension is n. Then the following statements hold:

  • (1)

    n{k,k+1,,2k+1}.

  • (2)

    If n=k+1, then either S is minimally non-polar, or after a possible relabeling,S{x{0,1}k+1:xk=xk+1} and the projection of S over coordinate k+1 is a critically non-polar set that is the reflective product of two other sets.

  • (3)

    If nk+2, then S is critically non-polar.

  • (4)

    If n=2k+1, then |S|=2n1, every infeasible point has exactly k infeasible neighbors, and cuboid(S) 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 R5, which is of degree 2 and dimension 5, has 16 points, every infeasible point has exactly 2 infeasible neighbors, and cuboid(R5)=Q10 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 τ=2 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

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