On the Number of Circuits in Regular Matroids (with Connections to Lattices and Codes)

07/13/2018
by   Rohit Gurjar, et al.
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We show that for any regular matroid on m elements and any α≥ 1, the number of α-minimum circuits, or circuits whose size is at most an α-multiple of the minimum size of a circuit in the matroid is bounded by m^O(α^2). This generalizes a result of Karger for the number of α-minimum cuts in a graph. As a consequence, we obtain similar bounds on the number of α-shortest vectors in "totally unimodular" lattices and on the number of α-minimum weight codewords in "regular" codes.

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1 Introduction

We study a general question about the number of certain structures, with respect to their sizes, arising in three different settings: matroids, codes, and lattices. More precisely, we are interested in the growth of the number of circuits in a matroid, the number codewords in a code, and the number of vectors in an integral lattice with respect to their size, weight, and length, respectively. These questions have been extensively studied in various forms in areas such as combinatorial optimization, information and coding theory, and discrete geometry (e.g., see

[18, 16, 23, 5]).

In all of these cases, a trivial (and rough) upper bound on the number of these objects of size is , where is the underlying ground set size in the case of matroids, length for codes, and dimension in the case of lattices. There are also elementary constructions of matroids/codes/lattices where these upper bounds are tight (e.g., graphic matroid of the complete graph, the trivial code with distance , the lattice ). However, consider the setting where the shortest size of such an object is . The question of interest is, when the size is close to the shortest size , whether the number of objects still grows like . Since circuits are well-studied objects in matroids, this is a natural question in the context of matroids. In coding theory, the motivation to study this question comes from “list decodability” (see [29]), and in lattices, this relates to the widely studied question of the “kissing number” of a lattice packing (see [5]). As we subsequently explain, these three questions turn out to be intimately connected in certain cases.

Circuits in matroids.

A circuit of a matroid is a minimal dependent set of its ground elements. A motivation for the above question is a seminal result of Karger [19] who showed that for a cographic matroid, the number of near-minimum circuits – circuits whose sizes are at most a constant multiple of the minimum size of a circuit– is bounded by , that is, independent of the minimum size. The circuits of a cographic matroid are the simple cut-sets of the associated graph, and Karger’s result was actually presented in terms of the number of near-minimum cuts in a graph.

An analogous result is also known in the “dual” setting of graphic matroids. The circuits of a graphic matroid are simple cycles in a graph. Subramanian [28] (building on [30]) showed a bound on the number of near-minimum cycles. Quantitatively, the results of Karger and Subramanian show that in a graphic/cographic matroid, if the shortest circuit has size , then the number of circuits of size at most , or -minimum circuits, is bounded by . Subramanian raised the question of identifying other matroids that have only polynomially many near-minimum circuits.

Do all matroids have such a property? The answer is no: the uniform matroid can have exponentially many shortest circuits.111Consider the uniform matroid , a matroid of ground set size where every subset of size at most is independent. A circuit of is any subset of size . Thus, the number of shortest circuits is , i.e., exponential in . Since a uniform matroid is representable (by a family of vectors over some field), one can also rule out the possibility of an affirmative answer for all representable matroids.

The next natural candidate to consider would be the class of binary matroids – matroids representable over – that also contains graphic and cographic matroids. Circuits of binary matroids are closely connected to codewords in binary linear codes and have received considerable attention from this perspective.

Binary linear codes.

Let be a matrix over representing a binary matroid on the ground set , i.e., has columns, and a set is independent in if and only if the corresponding set of columns in is linearly independent. Consider the linear code whose parity check matrix is . The codewords of are the vectors in and thus, are precisely the disjoint unions of circuits of . More importantly, the minimum weight of a codeword in and the minimum size of a circuit in are same. And thus, any -minimum weight codeword of comes from a union of -minimum circuits of . Thus, the question arises: do all binary linear codes have a small number of minimum or near-minimum weight codewords?

This question derives interest from the perspective of list decoding. Alon [2] gave a construction of a binary linear code where there are codewords of minimum weight. Kalai and Linial [16] studied distance distributions of codes and conjectured that the above number should be for all binary linear codes. However, Ashikhmin, Barg and Vlăduţ [3] disproved this conjecture by giving an explicit binary code with minimum weight codewords. The question is actually much easier to answer when we consider near-minimum weight codewords. Most binary linear codes have exponentially many near-minimum weight codewords222For a binary code with a random parity check matrix of dimensions , all its codewords have weights in the range

with a high probability, when

(see [4])..

In short, we cannot get the desired polynomial bound for all binary matroids or binary linear codes. Can we identify a subclass of binary matroids where this is true? Let us first briefly see the history of the analogous question in lattices.

Lattices.

The question of the number of shortest vectors in a lattice has attracted a lot of attention in mathematics. This number is also referred to as the “kissing number” of a lattice packing of spheres. Consider, for example, the lattice . The number of shortest vectors in this lattice is simply . Moreover, the number of near-shortest vectors – whose length is at most a constant multiple of the shortest length – is bounded by .

Such a bound does not hold for general lattices. It is widely conjectured that there exists a lattice packing with an exponentially large kissing number. However, the best known lower bound on the kissing number of an -dimensional lattice is only ([22], also see [6]). On the other hand, if we consider the number of near-shortest vectors in lattices, much higher bounds are known. Ajtai [1] showed that for some constants and infinitely many integers , there exists an -dimensional lattice that has at least vectors of length at most times the length of the shortest vector. A polynomial bound on the number of near-shortest vectors could still hold for some special class of lattices. It is an interesting question to characterize such lattices.

1.1 Our results

We make progress on the above questions about matroids, lattices, and codes by showing that, for a large class of each, the number of -minimum circuits, vectors, or codewords grow as Our main result concerns matroids and the others are derived via connections between matroids and lattices and matroids and codes.

Near-minimum circuits in matroids.

We answer the above question about -minimum circuits in affirmative for an extensively studied subclass of binary matroids – called regular matroids. These are matroids that can be represented by a family of vectors over every field.

Theorem 1.1 (Number of near-minimum circuits in a regular matroid).

Let be a regular matroid with ground set size . Suppose that has no circuits of size at most . Then for any , the number of circuits in of size at most is bounded by .

Since graphic and cographic matroids are the two simplest cases of regular matroids, our result significantly generalizes the results of Karger [19] and Subramanian [28]. Moreover, our result also holds for a class, more general than regular matroids, namely max-flow min-cut matroids (see Section 7). This is the most general class of binary matroids, where a natural generalization of max-flow min-cut theorem continues to hold.

A recent work [14] took a step towards answering this question for regular matroids. They showed a polynomial upper bound on the number of circuits whose size is less than times the minimum size of a circuit. However, a serious shortcoming of their work is that the proof breaks down for (see Section 2).

List decodability of codes.

Since binary matroids have close connections with binary linear codes, our Theorem 1.1 implies a list decodability result for certain special binary linear codes, called regular codes. A regular code – defined in [20] – is a binary linear code such that the columns of its parity check matrix represent a regular matroid. We get that for a regular code with distance and for any constant , the number of codewords with Hamming weight at most is polynomially bounded. This means that is -list decodable, for any constant . This is in contrast to general binary linear codes, which are not list-decodable beyond the minimum distance.

Corollary 1.2 (List decodability of regular codes).

For a regular code with distance , and any , the number of codewords with Hamming weight at most is bounded by .

To see this, observe that any codeword of weight at most comes from a combination of at most circuits of , since each circuit has size at least . Since we have a bound of on the number of circuits from Theorem 1.1, the bound on the number of codewords follows.

Near-shortest vectors in lattices.

In general, matroids are not related to lattices. However, since regular matroids are also representable over the real field, they happen to be connected to certain lattices. A result in matroid theory (see [25]) states that a matroid is regular if and only if it can be represented by the set of columns of a totally unimodular matrix (TUM). A matrix (over reals) is a TUM if each of its minors is either , , or . TUM are of fundamental importance in discrete optimization, as they are related to the integrality of polyhedra.

We define a lattice corresponding to a TUM, called totally unimodular lattice. For a TUM , the lattice of the set of integral vectors in is said to be a totally unimodular lattice.

It turns out that the near-shortest vectors in a totally unimodular lattice can be related to the near-minimum circuits of the associated regular matroid. And thus, our Theorem 1.1 implies a polynomial upper bound on the number of near-shortest vectors in totally unimodular lattices.

Theorem 1.3 (Number of near-shortest vectors in TU lattices).

Let be an totally unimodular matrix. Suppose any nonzero vector has a length more than . Then for any , the number of vectors in of length at most is .

Here by the length of a vector we mean its -norm.333Our proof works for any norm (), with appropriate dependence on .

1.2 Techniques

A landmark result in the study of regular matroids was Seymour’s decomposition theorem [26], that also implied the first polynomial time algorithm for testing total unimodularity of a matrix. The theorem states that every regular matroid can be decomposed into simpler matroids, each of which is graphic, cographic, or a special -element matroid . The theorem has found many uses in discrete optimization and also used to prove some structural results about regular matroids. To prove any result about regular matroids, a generic approach can be to first prove the corresponding results for the above simpler matroids and then “piece” them together using Seymour’s theorem to “lift” the result to regular matroids. The following results about regular matroids are some examples where this approach has been successful: extended formulations for independent set polytope [15], finding minimum cycle basis [13], and deciding first order logic properties [10].

Although Seymour’s theorem gives a meta-strategy, it does not automatically imply a result for regular matroids, given the result for graphic/cographic/ matroids. Each setting, where one wants to apply Seymour’s theorem to prove something about regular matroids, requires some new ideas. In fact, in many settings, Seymour’s theorem does not work as it is and a strengthening of its statement is required. Indeed, the recent work [14] on near-minimum circuits uses a refined version of Seymour’s theorem (given by [32]). There are a few other results on regular matroids that have used a more refined version (see [32]): faster algorithm to test total unimodularity [31], upper bounding the cycle cover ratio [21], and approximating the partition function of the ferromagnetic Ising model [12].

Some recent works, solving discrete optimization problems for regular matroids, have used a further stronger version of Seymour’s theorem. The strongest form was presented recently by Dinitz and Kortsarz [7], which gives the flexibility to decompose a regular matroid in many different possible sequences. They used it to solve the matroid secretary problem for regular matroids [7]. Later, Fomin, Golovach, Lokshtanov, and Saurabh utilized it in designing parameterized algorithms for the space cover problem [8] and the spanning circuits problem [9].

Our main result (Theorem 1.1) also takes advantage of this strongest version of Seymour’s theorem. One of our novel ideas is to use different sequences of decompositions for the same matroid to upper bound different classes of circuits. In contrast, the result of [14], which only works for a multiplicative factor smaller than , uses a fixed decomposition tree. In the next section, we give an overview of our proof and explain why the techniques of [14] fail to generalize to an arbitrary multiplicative factor.

1.3 Future directions

One natural question motivated by our work is to find out which other matroids have only polynomially many near-minimum circuits. Analogous to Seymour’s decomposition theorem, Geelen, Gerards, and Whittle [11] have proposed a structure theorem for any proper minor-closed class of matroids representable over a finite field. Can we use this structure theorem to upper bound the number of near-minimum circuits in these matroids.

Similarly, for what all lattices can we prove a polynomial bound on the number of near-shortest vectors. An interesting candidate to examine would be the lattice for any matrix whose entries are .

2 Overview of our proof and comparison with previous work

As mentioned earlier, Theorem 1.1 was already known in the special cases of graphic and cographic matroids.

Theorem 2.1 (Number of near-minimum circuits in a graphic or cographic matroid [19, 28]).

Let be a graphic or cographic matroid with ground set size . If every circuit in has size more than , then for any , the number of circuits in of size at most is bounded by .

A key component of our proof of Theorem 1.1 is a deep result of Seymour [26] about decomposition of regular matroids. Seymour’s Theorem states that every regular matroid can be built from piecing together some simpler matroids, each of which is graphic, cographic or a special matroid with 10 elements, . These building blocks are composed together via a binary operation on matroids, called “-sum” for . One can visualize this in terms of a “decomposition tree” – a binary tree where each node represents a matroid such that each internal node is a -sum of its two children, each leaf node is a graphic, cographic or the matroid, and the root node is the desired regular matroid. It is important to note that this decomposition tree is not unique, and one can perform the decomposition in different ways, optimizing various parameters, e.g., the tree depth.

The -sum of two matroids and is a matroid where and interact through a common set of elements (of size , or ) and each circuit of is obtained by picking a circuit each from and , and taking their sum via elements in . Thus, one can hope to upper bound the number of circuits in using such bounds for and . We know a polynomial upper bound on the number of near-minimum circuits for the matroids that are building blocks of Seymour’s decomposition theorem: Theorem 2.1 shows it for graphic and cographic matroids; and for the matroid , the bound holds trivially since it has only constantly many circuits. The challenge is to show that the polynomial bound still holds when we compose these matroids together via a sequence of an arbitrary number of -sum operations.

A natural attempt to get this bound would be to use an induction based on the decomposition tree of a regular matroid. This was precisely the approach [14] took and showed a polynomial upper bound on the number of circuits in a regular matroid whose size is at most times the shortest circuit size. However, their proof technique does not generalize to an arbitrary multiplicative factor . The use of such an induction severely restricts the power of the argument and it cannot be made to work for arbitrary .

Theorem 2.2 ([14]).

Let be a regular matroid with ground set size . Suppose that has no circuits of size less than . Then the number of circuits in of size less than is bounded by .

Limitations of the arguments in [14].

To see why the arguments in [14] fail to generalize to an arbitrary multiplicative factor, we need to take a closer look at the -sum operations. If is a -sum of two matroids and with the set of common elements, then any circuit of

  • is a circuit in or that avoids any element from , or

  • is the same as , where and are circuits of and , respectively that both contain a single element from .

Let us say, the shortest circuit size in is more than and we want to upper bound the number of circuits in it of size at most . The starting point would be to get such bounds for and , using induction. However, we do not know the shortest circuit sizes of and . We only know something weaker: that any circuit in or that avoids elements from has size more than (since it is also a circuit of ).

The idea of [14] is to divide the circuits in into two classes according to their decomposition into circuits of and :

  1. when the size of is small, that is, less than , and

  2. when the size of is at least .

It turns out that if we assume that the size of is less than , then there is a unique possibility of (this is the reason for such a classification). In this case, they just absorb into by assigning appropriate weights to the common elements. And thus, the question just reduces to bounding the number of circuits of weight at most in under the assumption that the shortest circuit weight in is more than . This is done by induction.

In case (ii), such a trick does not work. Here, one has to work only with a weaker assumption that any circuit in that avoids the common element has size more than . But on the positive side, the size of can be at most and thus, we have to bound a smaller set of circuits.

The problem arises with this weaker assumption. We can try to prove the desired bound with the weaker assumption, by using the same inductive proof methodology. However, as we go deeper into the case (ii) induction, the set of elements (say ) that needs be avoided in the assumption grows in size. We can no longer prove the uniqueness of (case (i)), as it might or might not contain elements from . The problem can be fixed in a limited case when has size , by ensuring that is completely contained in and thereby, is disjoint from . This follows from a slightly stronger decomposition theorem [32], which works only when . This restricts the case (ii) induction depth to a single level. Recall that for every level we go down in the case (ii) induction, we get a decrement of in . Moreover, we do not need to enter case (ii) when circuits have a size less than since here either or has a size less than . Hence, we have

This is the reason why the proof of [14] breaks down beyond .

Proof overview of Theorem 1.1

As we have seen, the induction based proof of [14] does not extend for a multiplicative factor larger than . On the other hand, such an inductive approach seems unavoidable if we work with a given decomposition tree, since it gives a fixed sequence of -sum operations. To overcome this issue, we need several new ideas. As noted earlier, the decomposition tree of a regular matroid is not unique. Our first main idea is to use different decomposition trees to upper bound different kinds of circuits. For instance, we can use a decomposition for one kind of circuits and use another decomposition for the rest.

However, to switch between different decompositions, we need an “associativity property” for the -sum operations. As defined, the -sum operations do not seem to be associative. However, a closer inspection tells us that in fact, they can be made associative. Using this associativity, Seymour’s Theorem can be adapted to give, what we call, an unordered decomposition tree (UDT) of a regular matroid , which allows us to construct many different possible decomposition trees for . This form of Seymour’s Theorem seems to be a folklore knowledge, but is first formally given in [7] (also see [17]). We believe that using this more structured version of Seymour’s theorem is necessary to obtain a result corresponding to an arbitrary multiplicative factor .

For a regular matroid , its UDT is an undirected tree such that each of its nodes corresponds to a graphic/cographic/ matroid. Each subtree of the UDT corresponds to a unique regular matroid . Moreover, if and are the two subtrees obtained by deleting an arbitrary edge from a subtree , then

Thus, a UDT gives us many possible ways of decomposing , depending on the order we choose for deleting edges from the UDT.

In the proof of Theorem 1.1, instead of an inductive argument, we directly use the structure of the unordered decomposition tree. We first give a brief outline of the steps in our proof. Assume that the shortest size of a circuit in is more than , and we would like to bound the number of circuits of size at most .

  • Circuit decomposition. We first argue that any circuit in can be written as a sum of its projections on the nodes of the UDT, which themselves are circuits in the corresponding matroids (Observation 6.1).

  • Balanced division of UDT. For any given circuit of size in , we divide the UDT into a balanced set of subtrees, i.e., we find a set of center nodes in the UDT such that each subtree obtained by deleting the center nodes has a projection of of size at most (Claim 6.3).

  • Classifying the circuits. We then classify these circuits based on the division of the UDT they produce and argue that the number of different classes is polynomially bounded. To bound the number of circuits in a given class, we work with the corresponding division of the UDT (Claim 6.4).

  • Uniqueness of small projections. The number of circuits in a class is bounded by the product of the numbers of possible projections on the center nodes and the remaining subtrees. We first show that if a subtree has a projection of size at most , then the projection is unique for all the circuits in the class (Lemma 6.7).

  • Number of projections on the center nodes. The main technical step is to bound the number of possible projections on the center nodes. Since there are only constantly many center nodes, it suffices to do it for a given center node. This task is reduced to bounding the number of weighted circuits in a graphic/cographic/ matroid, through various technical ideas (Lemma 6.9) such as

    • absorbing small projections via common elements,

    • avoiding common elements that connect to large projections.

Some of our lower level techniques are inspired from [14]: uniqueness of small projections, assigning weights, and avoiding a set of elements. We elaborate on each of our proof steps.

Recall that -sum operations are defined in a way such that any circuit of a matroid is either completely contained in or , or it can be written as a sum of two circuits, one coming from each and , which we refer as projections of the given circuit. When a circuit of is completely contained in , we say it has an empty projection on . A crucial property of the UDT is that any division of the UDT into subtrees gives us a valid decomposition of the matroid into smaller matroids corresponding to the subtrees. Thus, we can also obtain the projection of a circuit on any subtree of the UDT.

A balanced division of the UDT.

Our next idea is based on the observation that any weighted tree can be broken down into a balanced set of subtrees, i.e., it has a node such that its removal produces subtrees that have weights at most half of the total weight of the tree. For a given circuit of , we consider its projection size on a node of the UDT as the weight of node and use this balanced division recursively on the UDT. By this, we obtain certain center nodes of the UDT such that if we delete these center nodes, then each of the obtained subtrees has only a small projection of . Here, by a small projection we mean that its size should be at most . We argue that if the size of is at most , then the number of centers would be at most .

Next, we classify the circuits (of size ) according to the set of centers they produce. Since is a constant, the number of possible classes is polynomially bounded. Hence, we just need to upper bound the number of circuits in a given class.

Bounding the number of circuits in a class.

To upper bound the number of circuits in a class, let us fix a set of centers. We divide the UDT into the center nodes, and the subtrees we obtain if we delete centers, and write as a sum of the matroids corresponding to these centers and subtrees. Further, we write any circuit as a sum of its projections on the matroids associated with the centers and the subtrees. If we upper bound the number of distinct possible projections on each of these smaller matroids, then their product bounds the number of all the desired circuits. Since there can be subtrees, this product can easily become exponentially large even if we have just two possible projections for each subtree. We sidestep this exponential blow-up as follows.

If a projection is small, it is unique.

The first important step is to show that there is only a unique possibility of the projection on a subtree (to a certain extent), besides the empty projection, if we assume that the size of the projection is at most . It is true because, if there are more than one such projections then we can show that any two of them combine to give a circuit of size at most in , which contradicts the initial assumption. This takes care of the projections on the subtrees in the above division of the UDT.

Upper bounding the number of projections on a center node.

The next step is the most technically involved part of the proof: bounding the number of projections on the center nodes. As there are only a constant number of center nodes, it suffices to polynomially bound the number of distinct projections on a given center node. Recall that each projection is a circuit in the respective matroid. One might think that since the matroid associated with a single node is graphic, cographic or , it would be easy to bound the number of circuits in it, but it is not that straightforward. The main problem is that we do not know the size of the shortest circuit in this component matroid; in fact, it could be arbitrarily small. And thus, we cannot just get a polynomial bound on the number of circuits whose sizes can be up to . Here, we need to use a more sophisticated argument. There are two main technical ingredients involved.

Technique I: Absorbing small projections on subtrees via common elements. To see the first ingredient, let us consider a relatively simpler case when there is only one center node. In this case, all the subtrees attached to the center node have projections of sizes at most . Let be the matroid associated with the center node. The idea is to give some weights to the elements of and define a map from circuits of to circuits of in a way that a circuit of with weight is mapped to a circuit in with size . If we can design such a weight assignment then we can assume that the smallest weight of a circuit in is more than . Then we would just need to bound the number of near-smallest weight circuits in a graphic/cographic matroid, which can be done by a weighted version of Theorem 2.1.

The next question is: how to design such weights? Let be any subtree attached to the center node and be the matroid associated with it. Let be one of the common elements between and . As discussed above, there is a unique projection on that has size at most and contains . Let that projection be . We put a weight on the element of , which is supposed to be a representative of the projection on . That is, if a circuit of takes the element then it means that will be summed up with to form a circuit of , otherwise it will not be. We put such weights for every subtree attached to the center node, and every element common between and . It turns out that this gives us a weighting scheme with exactly the desired property: a circuit of with weight is mapped to a circuit in with size .

We need to consider the case when there are more than one center nodes. The problem with the above argument would be that if we pick any one of the center nodes and delete it from UDT, some of the obtained subtrees can have projection sizes more than , and thus, there is no uniqueness of the projection (recall that we get sizes less than only when we delete all center nodes). This is where we require the second technical ingredient.

Technique II: Avoiding common elements that connect to large projections. Let us pick one of the center nodes and say, the associated matroid is . Observe that there can be many subtrees attached to the center node that have projection sizes larger than , but the crucial fact is that their number can be at most (since the total size of the circuit is at most ).

Let be one such subtree and be one of the common elements between and . Unlike what we did in the first technique, there is no unique way of assigning a weight to . Instead, we just consider circuits in that avoid the element . In short, here again, we can have a weight-preserving map from circuits in to circuits in , but the domain is restricted to those circuits in which avoid common elements , for any subtree with a large projection.

Using this map, we can argue that the weight of any circuit in , that avoids elements , is more than . This is a weaker assumption than what we require, in the sense that a circuit in that takes some elements can have arbitrary small weight. It turns out that such a weaker assumption is sufficient to give a polynomial upper bound on the number of circuits of weight in a graphic/cographic matroid, as long as the number of elements that we need to avoid remains a constant (Lemma 6.8). We claim that the number of elements are at most , which is a constant. This is true because and can have at most elements in common, and as we saw above, the number of subtrees with a large projection can be at most .

Finishing the proof.

In essence, through various technical components, we reduce the problem to a single component matroid, which is graphic/cographic/. However, we have to consider weights on the elements and also have a weaker assumption about the smallest weight of a circuit – that is – we assume a lower bound on the weight of only those circuits that avoid a fixed set of elements. It turns out that even with this weaker assumption, we can get an upper bound on the number of desired circuits in a graphic/cographic matroid by cleverly modifying the proofs of [19, 28]. This was partially done in [14], where they only considered the case of . Here, we generalize their proof to an arbitrary set (Lemma 6.8).

Organization of the rest of the paper.

In Section 3, we introduce well-known concepts from matroid theory and describe Seymour’s Theorem for regular matroids. In Section 4, we relate circuits of a regular matroid with vectors of a totally unimodular lattice and prove Theorem 1.3 using Theorem 1.1. Section 5 talks about a refinement of Seymour’s Theorem, where we have more structure in the decomposition of a regular matroid. In Section 6 we use this structured decomposition to upper bound the number of near-minimum circuits in regular matroids. Section 7 describes max-flow min-cut matroids. We argue that the same proof technique gives a polynomial bound on the number of near-minimum circuits in these matroids. For proving Theorem 1.1, we first need that result for graphic and cographic matroids, but in a stronger form. Appendix A is devoted to prove this.

3 Matroid preliminaries

3.1 Matroids and circuits

Definition 3.1 (Matroid).

For a finite set and a nonempty collection of its subsets, the pair is called a matroid if

  1. for every , implies ,

  2. if with then there exists an element such that .

Every set in is said to be an independent set of .

Every independent set of maximum size is called a base of . A subset of that is not independent is said to be dependent. Note that since is nonempty, the empty set must be an independent set.

Definition 3.2 (Circuit).

For a matroid , any inclusion-wise minimal dependent subset is called a circuit.

We define some special classes of matroids that are useful for us. For a matrix , let be the set of its columns and be the collection of all linearly independent sets of columns. It is known that is a matroid.

Definition 3.3 (Linear, binary, and regular matroids).

A matroid is called representable over a field if there exists a matrix over such that .

  • A matroid representable over some field is called linear.

  • A matroid representable over is called binary.

  • A matroid representable over every field is called regular.

Regular matroids are known to be characterized by totally unimodular matrix.

Definition 3.4 (Totally unimodular matrix).

A matrix over real numbers is said to be totally unimodular if every square submatrix of has determinant , , or .

Note that by definition, each entry in a totally unimodular matrix is , or .

Theorem 3.5 (Characterization of regular matroids, see [25]).

A matroid is regular if and only if there is a totally unimodular matrix such that .

Two well-known special cases of regular matroids are graphic and cographic matroids.

Definition 3.6 (Graphic matroids).

The graphic matroid for a graph is defined as , where is the set of edges in and is the collection of all sets of edges without cycles.

For a graph , its cographic matroid is the duals the graphic matroid . For a matroid , its dual is where,

Observe that a base set of a graphic matroid is a spanning tree in (or spanning forest, if is not connected). Thus, a set of edges is independent in if and only if its removal keeps connected.

The circuits of graphic and cographic matroids are easy to characterize in terms of cycles and cut-sets. For a graph , and any partition of its vertices, the set of edges connecting a vertex in to another in is called a cut-set.

Fact 3.7 (Circuits in graphic and cographic matroids).

For a graph ,

  • a circuit of the graphic matroid is any simple cycle of and

  • a circuit of the cographic matroid is any inclusion-wise minimal cut-set of .

Recall that the symmetric difference of two sets and is given by . Note that the symmetric difference of two cycles in a graph can be expressed as a disjoint union of cycles in the graph. Same holds true for two cut-sets. These two statements are special cases a more general fact about binary matroids. Recall that graphic and cographic matroids are regular and thus, binary.

Fact 3.8.

For two circuits and of a binary matroid , their symmetric difference is a disjoint union of circuits of .

It is known that for a matroid , one can delete one of its ground set elements to obtain another matroid defined as follows.

Definition 3.9 (Deletion).

For a matroid and an element , is defined to be matroid on the ground set such that any independent set of not containing is an independent set of .

For a graphic matroid, deletion of an element corresponds to the deletion of the edge from the graph, while for a cographic matroid, it corresponds to the contraction of the edge. It is easy to characterize the circuits of .

Fact 3.10.

The circuits of are those circuits of that do not contain .

Another operation one can do on a matroid is to add a new element parallel to an existing element. This new element is essentially a copy of the existing element. Formally, let be a matroid with a given element . One can define a new matroid on the ground set with a new element such that is a circuit. This also implies that for any circuit of that contains , the set is a circuit of . In the case of a graphic matroid, adding a parallel element means adding a parallel edge in the graph, while in the cographic case, it means splitting an edge into two by adding a new vertex.

Fact 3.11.

The classes of regular matroids, graphic matroids, and cographic matroids are closed under the deletion operation and under the addition of a parallel element.

3.2 -sums and Seymour’s Theorem

To prove Theorem 1.1, a crucial ingredient is the remarkable decomposition theorem for regular matroids by Seymour. Seymour [26] showed that every regular matroid can be constructed by piecing together three special kinds of matroids – graphic matroids, cographic matroids and a certain matroid of size 10. The operation involved in this composition is called a -sum, for , , or . The -sum operation is defined for arbitrary binary matroids.

Definition 3.12 (Sum of two matroids, [26, 25]).

Let and be two binary matroids with . The matroid is defined over the ground set such that the circuits of are the minimal non-empty subsets of that are of the form , where is a (possibly empty) disjoint union of circuits of for .

The fact that the above definition indeed gives a matroid can be verified from the circuit characterization of a matroid [25, Theorem 1.1.4]. We are only interested in three special cases of this sum, called -sum, -sum, and -sum.

Definition 3.13 (-sums).

The sum of two binary matroids and with is called

  1. a -sum if ,

  2. a -sum if , is not a circuit of or their duals, and , and

  3. a -sum if , is a circuit of and , does not contain a circuit of the duals of and , and .

The conditions on the ground set sizes are there to avoid degenerate cases. The following facts follows from the definition and Fact 3.8.

Fact 3.14.

For , if is a circuit of that does not contain any elements from then is a circuit of .

Fact 3.15.

Let be a disjoint union of circuits of for . If is a subset of then it is a disjoint union of circuits in .

The operation of -sum is the easiest sum operation.

Fact 3.16 (Circuits in a -sum).

If is a -sum of and then any circuit of is either a circuit of or a circuit of .

Characterizing the circuits of a -sum or a -sum is a bit more non-trivial. The following lemma [26, Lemma 2.7] gives a way to represent circuits of in terms of circuits of and .

Proposition 3.17 (Circuits in a -sum or a -sum, [26]).

Let and be the sets of circuits of and , respectively. Let be a -sum or a -sum of and and let ( or ). Then for any circuit of , exactly one of the following holds:

  1. with

  2. with

  3. there exist unique , , and such that

With all the required definitions, we can finally present Seymour’s theorem for regular matroids [26, Theorem 14.3].

Theorem 3.18 (Seymour’s Theorem).

Every regular matroid can be obtained by means of -sums, -sums and -sums, starting from matroids which are graphic, cographic or .

The matroid , which forms one of the building blocks for Seymour’s Theorem, is represented by the following matrix over .

The following fact about is useful.

Fact 3.19.

The matroid obtained by deleting any element from is a graphic matroid.

4 Number of -shortest vectors in a totally unimodular lattice

In this section, we show how Theorem 1.3 follows from Theorem 1.1. Recall that for an matrix , the lattice is defined as:

We first define circuits of a matrix , which are vectors in and show a correspondence between the circuits of a TU matrix and the circuits of the associated regular matroid . Thus, an upper bound on number of near-minimum circuits in (Theorem 1.1) implies an upper bound on number of near-minimum circuits of the matrix . Finally, we argue that any -minimum vector in comes from a combination of at most -many -minimum circuits of . A specialized version of this statement was shown in [14] – any -minimum vector in comes from a -minimum circuit of .

Definition 4.1 (Circuits of a matrix).

For a matrix , a vector is a circuit of if for any vector with , it must be that for some integer .

Note that the circuits of come in pairs, in the sense that if is a circuit of , then so is . The following is well-known for the circuits of a TU matrix (see [24, Lemma 3.18]).

Fact 4.2.

Every circuit of a TU matrix has its coordinates in .

We show a correspondence between circuits of and circuits of when is TU.

Lemma 4.3 (Circuits of a TU matrix and a regular matroid).

Let be a TU matrix and be the regular matroid represented by it. Then the circuits of have a one to one correspondence with the circuits of  (up to change of sign).

Proof.

By definition, a circuit of has an inclusion-wise minimal support. Thus for a circuit of , the columns in  corresponding to the set  are minimally dependent. We know that a minimal dependent set is a circuit in the associated matroid. Hence, the set  is a circuit of matroid .

In the other direction, if is a circuit of matroid , then the set of columns of  corresponding to  is minimally linear dependent. Hence, there is a unique linear dependence (up to a multiplicative factor) among the set of columns corresponding to . This means that there are precisely two circuits with their support being . ∎

Lemma 4.3 together with Theorem 1.1 gives the following corollary. Let denote the -norm of a vector.

Corollary 4.4.

Let be an TU matrix. If for every circuit of , we have then the number of its circuits with is .

Proof.

If is a circuit of with then (from Fact 4.2). Thus, any circuit of with corresponds to a circuit of the regular matroid of size at most . The desired bound follows from Theorem 1.1. ∎

We now show that we can get an upper bound on the number of all short vectors in from Corollary 4.4. We define a notion of conformality among two vectors and show that every vector in is a conformal combination of circuits of .

Definition 4.5 (Conformal vectors [24]).

Let . We say that  is conformal to , denoted by , if and , for each .

The following lemma which says that each vector in is a conformal sum of circuits, follows from [24, Lemma 3.2 and 3.19].

Lemma 4.6.

Let be a TU matrix. Then for any nonzero vector , we have

where each is a circuit of  and is conformal to .

We have the following easy observation for a conformal sum.

Observation 4.7.

If are such that and each is conformal to then

We are ready to prove Theorem 1.3.

Proof of Theorem 1.3.

From the assumption in the theorem, for any circuit of , we have . Consider a vector with . From Lemma 4.6, we can write as a conformal sum of circuits

We know that for each . This together with Observation 4.7 implies that

Thus, we get that .

Note that each is smaller than and hence, . From Corollary 4.4, the number of circuits of with is . Thus, there can be at most vectors of the form . This gives a bound of on the number of vectors with . ∎

5 A strengthening of Seymour’s Theorem

In this section, we look at a stronger version of Seymour’s Theorem, which gives a more structured decomposition of a regular matroid. One way to present Seymour’s Theorem (Theorem 3.18) can be in terms of a decomposition tree.

Theorem 5.1 (Seymour’s Theorem).

For every regular matroid , there exists a decomposition tree – a rooted binary tree whose every vertex is regular matroid such that

  • every internal vertex is a -sum of its two children for , or ,

  • every leaf vertex is a graphic matroid, a cographic matroid or the matroid,

  • the root vertex is the matroid .

A few observations can be made about such a decomposition tree. Recall from Section 3 that for two matroids with grounds sets and , their -sum is a matroid on the ground set .

-sum

-sum
Figure 1: The decomposition tree of a matroid given by .
Observation 5.2.

For any vertex of the decomposition tree of a binary matroid ,

  1. Each element in belongs to exactly one of its children matroids. Arguing recursively, each element in belongs to a unique leaf in the subtree rooted at .

  2. The ground set of is the symmetric difference of all the ground sets of the leaf vertices in the subtree rooted at .

For example, Figure 1 shows the decomposition tree for a matroid . Note that the decomposition tree specifies an order of the decomposition or composition, that is, can be obtained by first taking a -sum of and and then taking a -sum of the resulting matroid with . It is not clear if the -sum operations are associative. It turns out that one can strengthen the decomposition theorem such that the -sum operations involved in the composition are associative up to a certain extent.

5.1 Associativity of the -sum

The operations of -sums are trivially associative. It can be shown that the -sum operations are always associative and so are -sum operations in some special cases. The following lemma gives a criterion when the associativity holds.

Lemma 5.3 (Associativity of -sums).

Let be binary matroids with ground sets , respectively. Let be a -sum of and for , or with . Let be a -sum of and for , or with . Further, we assume that the set , which is contained in , is entirely contained in (or in , which is a similar case). Then

where is defined via the common set and is defined via the common set .

Proof.

We will show that the sets of circuits of the two matroids and are the same. This would imply that the two matroids are the same. Consider a circuit of . From Proposition 3.17, there are two possibilities for the circuit . First is when is a circuit of or that avoids the common elements . We skip this easy case and only consider the other possibility which is non-trivial. In the other possibility, must be of the form , where and are circuits in and , respectively and . Similarly for , there exist circuits and of and respectively such that .

Since is contained entirely in we get that

Thus, is a subset of and hence, is a disjoint union of circuits of , from Fact 3.15. Since is a circuit of , it follows that is a disjoint union of circuits in , again from Fact 3.15. But,

Thus, is a disjoint union of circuits in .

The other direction is similar. Consider a circuit of . In the non-trivial case, the circuit must be of the form , where and are circuits of and , respectively, with (Proposition 3.17). Similarly, , where and are circuits in and , respectively. Since is disjoint from , it must be that . Thus, is a subset of and hence, is a disjoint union of circuits of (Fact 3.15). Similarly, since is a circuit in , it follows that is a disjoint union of circuits in . But,

Thus, is a disjoint union of circuits in .

We have shown that a circuit of one matroid is a disjoint union of circuits in the other matroid and vice-versa. Consequently, by the minimality of circuits, it follows that their sets of circuits must be the same. ∎

To summarize the above lemma, the sequence of two -sums is associative, when the common sets involved in the -sum operations are completely contained in the starting matroids. Note that this is always true for a -sum operation since the common set has a single element in this case. However, this need not be always true in the case of a -sum. It is possible that the common set between and has elements in both and . Dinitz and Kortsarz [7] call such a set as a bad sum-set. More generally, they define a notion of good or bad for a decomposition tree of a regular matroid obtained from Seymour’s Theorem.

Definition 5.4 (Good decomposition tree [7]).

The decomposition tree of a regular matroid , as in Theorem 5.1, is said to be good if for every internal vertex of the tree , which is a -sum of its two children and , the common set between and is completely contained in one of the leaf vertices of the subtree rooted at and also in one of the leaf vertices of the subtree rooted at .

As we see later, if we have a good decomposition tree of a binary matroid, then the -sum operations involved in the decomposition have a suitably defined associative property. Dinitz and Kortsarz [7] showed how to modify a given decomposition tree of a regular matroid to get a good decomposition tree. To do this, their basic step involves ‘moving’ an element from one matroid to another matroid. Consider the above discussed example of a matroid given by and assume that the common set of elements between and , i.e., , has elements in both and . In this case, they delete an element from one of the matroids, say , and add an element in parallel444Two elements and of a matroid are in parallel if is a circuit. to an existing element, to create new matroids and such that

Doing this repeatedly with the decomposition tree, starting from the leaves and going up to the root vertex, they obtain the desired decomposition tree. Formally, [7, Lemma 3.1] implies the following stronger version of Seymour’s Theorem.

Lemma 5.5.

For any regular matroid, there is a good decomposition tree such that each of the leaf vertices is a graphic, cographic or the matroid.

Proof.

[7, Lemma 3.1] says that for any regular matroid and a given decomposition tree of , one can construct a good decomposition tree with the same tree structure, but each leaf vertex is possibly replaced with another matroid obtained from by deleting some elements in it and/or adding elements parallel to some elements in it. Recall that the classes of graphic and cographic matroids are closed under deletion of an element or addition of an element in parallel (Fact 3.11). Since the matroid does not have any circuit with elements, the procedure of Dinitz and Kortsarz [7] can only delete an element from and not add one. The matroid obtained by deleting some elements from is a graphic matroid (Fact 3.19). Thus, all the leaf vertices remain graphic, cographic or . ∎

5.2 Unordered decomposition tree

Next, we define an unordered decomposition tree (UDT), which allows us to decompose a matroid in many different ways. We show that we can obtain an unordered decomposition tree of a matroid from its good decomposition tree.

Let be a tree with its vertex set such that each vertex has a corresponding binary matroid with the ground set . Further, for any two vertices and of the tree we have the following.

(1)

In particular, this means that any element is a part of at most two ground sets. For any subtree of this tree , we define a binary matroid with its ground being . It is defined recursively as follows:

  • if is a vertex, say , then .

  • Otherwise, let be an edge in the tree . Let and be the two subtrees obtained by removing the edge from . From (1), it follows that . Define

Here, the sum is a -sum, -sum or a -sum depending on the size of . At first, it is not clear if (and ) is uniquely defined since one can pick any edge from and get different subtrees (see fig. 2). In the following, we argue that is indeed uniquely defined. For this we need the associativity of the -sum operations proved in Lemma 5.3.

Figure 2: The unordered decomposition tree (UDT) representing the matroid given by .
Claim 5.6.

For a tree as above, the matroid is uniquely defined.

Proof.

We want to show that is the same matroid whatever be our choice of edge to decompose it into two subtrees. We argue inductively. We assume that for any subtree of tree , the matroid is uniquely defined. We first consider two neighboring edges and in , and the subtrees obtained after removing them. Let and and be the two subtrees obtained by removing