Isolating a Vertex via Lattices: Polytopes with Totally Unimodular Faces

We deterministically construct quasi-polynomial weights in quasi-polynomial time, such that in a given polytope with totally unimodular constraints, one vertex is {\em isolated}, i.e., there is a unique minimum weight vertex. More precisely, the property that we need is that every face of the polytope lies in an affine space defined by a totally unimodular matrix. This derandomizes the famous \emph{Isolation Lemma} by Mulmuley, Vazirani, and Vazirani for this setting, generalizes the recent derandomization results for {bipartite perfect matching} and {matroid intersection}, and resolves the problem of derandomizing the isolation lemma for totally unimodular polytopes raised in a recent work. We prove our result by associating a {\em lattice} to each face of the polytope and showing that if there is a totally unimodular kernel matrix for this lattice, then the number of near-shortest vectors in it is polynomially bounded. The proof of this latter geometric fact is combinatorial and follows from a polynomial bound on the number of near-shortest circuits in a regular matroid. This is the technical core of the paper and relies on Seymour's decomposition theorem for regular matroids. It generalizes an influential result by Karger on the number of minimum cuts in a graph to regular matroids; answering another open problem. Both of our results, on lattices and matroids, should be of independent interest.


Introduction
The Isolation Lemma by Mulmuley, Vazirani, and Vazirani [MVV87] states that for any given family of subsets of a ground set E, if we assign random weights (bounded in magnitude by poly(|E|)) to the elements of E then, with high probability, the minimum weight set in the family is unique. Such a weight assignment is called an isolating weight assignment. The lemma was introduced in the context of parallel algorithms for the matching problem. Since then it has found numerous other applications: a reduction from CLIQUE to UNIQUE-CLIQUE [MVV87], NL/poly ⊆ ⊕L/poly [Wig94], NL/poly = UL/poly [RA00], an RNC-algorithm for linear matroid intersection [NSV94], and an RP-algorithm for disjoint paths [BH14]. In all these results, the Isolation Lemma is the only place where they need randomness (or advice string). Thus, if the Isolation Lemma can be derandomized, i.e., a polynomially bounded isolating weight assignment can be deterministically constructed, then so can the aforementioned results that rely on it. Unfortunately, it is easy to see that it is impossible to design a polynomially bounded isolating weight assignment for all possible families of subsets of E. This is because there are exponentially many subsets and the weights are polynomially bounded. For any polynomially bounded weight assignment, there will be two subsets of E with the same weights. These two subsets form a family where the weight assignment fails. Even a relaxed task -of constructing a poly-size list of poly-bounded weight functions such that for each family B ⊆ 2 E , one of the weight functions in the list is isolatingis impossible. This can be shown via arguments involving the polynomial identity testing (PIT) problem. The PIT problem, that is another important consequence of derandomizing the Isolation Lemma, asks if an implicitly given multivariate polynomial is identically zero. Here, the family of sets one needs to consider comes from the family of monomials that have nonzero coefficients in the polynomial. In essence, construction of such a list would imply that there exists a small set of points in R E such that any |E|-variate polynomial of small degree is nonzero on at least one of the points in the set. One can rule this out by designing a small degree polynomial that vanishes on all the points in the set. Hence, a natural question is to solve the isolation question for families B, that have a succinct representation.
In this work, we derandomize the Isolation Lemma for a large class of families via a geometric approach. For a family of sets B ⊆ 2 E , define the polytope P (B) ⊆ R E to be the convex hull of the characteristic vectors of the sets in B. We show that there exists an m O(log m) -sized family of weights, with weights bounded by m O(log m) that isolates any family B whose corresponding polytope P (B) satisfies the following property: the affine space spanned by any face of P (B) is parallel to the null space of some totally unimodular (TU) matrix; see Theorem 2.3. Our weight construction is black-box in the sense that it does not need the description of the family or the polytope.
A large variety of polytopes satisfy this property and, as a consequence, have been extensively studied in combinatorial optimization. The simplest class of polytopes that satisfy this property is when the polytope P (B) has a description Ax ≤ b with A being a TU matrix. Thus, a simple consequence of our main result is a resolution to the problem of derandomizing the isolation lemma for TU polytopes raised in a recent work [ST17]. Further, our results significantly generalize recent work [FGT16] for the family of perfect matchings in a bipartite graph and for the family of common bases of two matroids [GT17]. In the case of perfect matchings in bipartite graphs, the perfect matching polytope can be described by the incidence matrix of a given graph which is a TUM. In the matroid intersection problem, the constraints of the common base polytope are a rank bound on every subset of the ground set. These constraints, in general, do not form a TUM. However, for every face of the polytope there exist two laminar families of subsets that form a basis for the tight constraints of the face. The incidence matrix for the union of two laminar families is TU (see [Sch03b,Theorem 41.11]). Other examples of families whose polytopes are defined by TU constraints are vertex covers of a bipartite graph, independent sets of a bipartite graph, edge covers of a bipartite graph. Since the constraint matrix defining the polytope (or any of its face) itself does not have to be TU for it to satisfy the condition above, the condition required in Theorem 2.3 on the polytope P (B) is quite weak and is well studied. Schrijver [Sch03a,Theorem 5.35] shows that this condition is sufficient to prove that the polytope is box-totally dual integral. The second volume of Schrijver's book [Sch03b] is gives excellent overview on polytopes that satisfy the condition of Theorem 2.3: • up hull of an r-arborescence polytope [ Our starting point is a reformulation of the approach in the recent past work [FGT16,GT17] in terms of certain lattices associated to polytopes. For each face F of P (B), we consider the lattice L F of all integer vectors parallel to F . We show that, if for each face F of P (B), the number of near-shortest vectors in L F is polynomially bounded then we can construct an isolating weight assignment for B with quasi-polynomially bounded weights; see Theorem 2.4. Our main technical contribution is to give a polynomial bound on the number of vectors whose length is no more than 3/2 times that of the shortest vector in L F when this lattice is the set of integral vectors in the null space of a TUM; see Theorem 2.5. This is in contrast for general lattices where the number of such near-shortest vectors could be exponential in the dimension.
The above result can be reformulated using the language of matroid theory: the number of near-shortest circuits in a regular matroid is polynomially bounded; see Theorem 2.6. In fact we show how Theorem 2.4 can be deduced from Theorem 2.6. One crucial ingredient in the proof of Theorem 2.6 is Seymour's decomposition theorem for regular matroids [Sey80]. Theorem 2.6 answers a question raised by Subramanian [Sub95] and can be viewed as a generalization of known results in case of graphic and cographic matroids, that is, the number of near-minimum length cycles in a graph is polynomially bounded (see [TK92,Sub95]) and the number of near-mincuts in a graph is polynomially bounded; see [Kar93].
Thus, not only do our results make significant progress in derandomizing the isolation lemma for combinatorial polytopes, our structural results about the number of near-shortest vectors in lattices and near-shortest matroids should be of independent interest and raise the question: to what extent are they generalizable? 2 Our Results

Isolating a Vertex in a Polytope
For a weight function w : E → Z, consider its extension to the any subset S ⊆ E as follows: A weight function w : E → Z is called isolating for a family B ⊆ 2 E of subsets of E, if the minimum weight set in B, with respect to w, is unique. In other words, the set arg min S∈B w(S) is unique. The Isolation Lemma of Mulmuley, Vazirani, and Vazirani [MVV87] asserts that a random weight function is isolating with a good probability for any B.
Lemma 2.1 (Isolation Lemma). Let w be a random weight function on the set E where for each e ∈ E, the weight w(e) is chosen independently and uniformly at random from the set {1, 2, . . . , 2|E|}. Then for any family B of subsets of E, the weight function w is isolating with probability at least 1/2.
The question of derandomizing the Isolation Lemma asks for a deterministic construction of an isolating weight function with weights polynomially bounded in the set size |E|. Here, we view the isolation question for B as an isolation over a corresponding polytope P (B), which is defined as follows. For a set S ⊆ E, its characteristic vector x S ∈ R E is defined as For any family of sets B ⊆ 2 E , the polytope P (B) ⊂ R E is defined as the convex hull of the characteristic vectors of the sets in B, The isolation question for a family B is equivalent to constructing a weight vector w ∈ Z E such that w, x has a unique minimum over P (B). The property we need for our isolation approach is in terms of a totally unimodular matrix.
Definition 2.2 (Totally unimodular matrix). A matrix A ∈ R n×m is said to be totally unimodular (TU), if every square submatrix has determinant 0 or ±1.
Our main theorem gives an efficient quasi-polynomial isolation for a family B when each face of the polytope P (B) lies in the affine space defined by a TU matrix. weight assignments on E with weights bounded by m O(log m) with the following property: Let B ⊆ 2 E be a family of sets. Suppose that for any face F of the polytope P (B) there exists a TU matrix A F such that the affine space spanned by F is given by Then for the family B, one of the weight assignments in W is isolating.

Short vectors in lattices associated to polytopes
Our starting point towards proving Theorem 2.3 is a reformulation of the isolation approach of [FGT16,GT17]. We define a lattice corresponding to each face of the polytope P (B). The isolation approach works when this lattice has a small number of short vectors. For any face F of P (B), consider the lattice of all integral vectors parallel to F , The length of the shortest nonzero vector of a lattice is denoted by where · denotes the ℓ 1 -norm. We prove the following theorem that asserts that if, for all faces F of P (B), the number of near-shortest vectors in L F is small, then we can efficiently isolate over P (B).
Theorem 2.4. Let P (B) ⊂ R E be a 0-1 polytope such that there exists a constant c > 1, such that for any face F of P (B), we have where m = |E|. Then one can construct a set of m O(log m) weight functions with weights bounded by m O(log m) such that at least one of them is isolating for B.
The main ingredient of the proof of Theorem 2.3 is to show that the hypothesis of Theorem 2.4 is true when the lattice L F is the set of all integral vectors in the nullspace of a TU matrix. For any n × m matrix A we define a lattice L(A) as follows: Proof of Theorem 2.3. Let F be a face of the polytope P (B) and let A F be the TU matrix associated with F . Thus A F x = b F defines the affine span of F . In other words, the set of vectors parallel to F is precisely the solution set of A F x = 0 and the lattice L F is given by L(A F ). Theorem 2.5 implies the hypothesis of Theorem 2.4 for any L F = L(A F ), when the matrix A F is TU.

Short circuits in regular matroids
The proof of Theorem 2.5 is combinatorial and uses the language and result from matroid theory. We refer the reader to Section 4 for preliminaries on matroids; here we just recall a few basic definitions. A matroid is said to be represented by a matrix A, if its ground set is the column set of A and its independent sets are the sets of linearly independent columns of A. A matroid represented by a TU matrix is said to be a regular matroid. A circuit of a matroid is a minimal dependent set. The following is one of our main results which gives a bound on the number of short circuits in a regular matroid, which, in turn, implies Theorem 2.5. Instead of size, we allow weights to the elements of the ground set and present a more general result.
Theorem 2.6. Let M = (E, I) be a regular matroid with m = |E| ≥ 2 and w : E → N be a weight function. Let r be an even number. Suppose M does not have any circuit C with w(C) ≤ r. Then |{ C : C circuit in M and w(C) ≤ 3r/2 }| ≤ 150 m 5 .
Remark 2.7. An extension of this result would be to give a polynomial bound on the number of circuits of weight at most αr for any constant α. Our current proof does not extend to this setting. This will be the subject of an upcoming work.
Organization of the rest of the paper In Section 3, we present a proof of Theorem 2.4. In Section 4.1, we present a basic introduction to matroids. In Section 4.2, we describe some well-known properties of regular matroids which will be key to the proof of Theorem 2.6. Section 5 describes how Theorem 2.5 follows from Theorem 2.6. Finally, in Section 6, we prove Theorem 2.6.
3 Isolation via the Lattices Associated to the Polytope: Proof of Theorem 2.4 This section is dedicated to a proof of Theorem 2.4. That is, we give a construction of an isolating weight assignment for a family B ⊆ 2 E assuming that for each face F of the corresponding polytope P (B), the lattice L F has small number of short vectors. First, let us see how the isolation question for a family B translates in the polytope setting. For any weight function w : E → Z, we view w as a function on P (B). That is, we define an extension of the weight function w to R E . For That is, we can consider w as a vector in Z E , and the weight of a vector x is the inner product with w. Note that w, x B = w(B), for any B ⊆ E. Thus, we get the following.
Claim 3.1. A weight function w : E → Z is isolating for a family B if and only if the function w, x has a unique minimum over the polytope P (B).
Observe that for any w : E → Z, the points that minimize w, x in P (B) will form a face of the polytope P (B). The idea is to build the isolating weight function in rounds. In every round, we slightly modify the current weight function to get a smaller minimizing face. Our goal is to significantly reduce the dimension of the minimizing face in every round. We stop when we reach a zero-dimensional face, i.e., we have a unique minimum weight point in P (B).
The following claim asserts that if we modify the current weight function on a small scale, then the new minimizing face will be a subset of the current minimizing face. In the following, we will denote the size of set E by m.
Claim 3.2. Let w : E → Z be a weight function and F being the face of P (B) that minimizes w, x . Then, for any weight function w ′ : E → {0, 1, . . . , N − 1}, the face F ′ that minimizes the function mN w + w ′ , x is contained in F .
Proof. Consider any vertex x ∈ P (B) of the face F ′ . Since it minimizes the function mN w+w ′ , x , we can say for any other vertex y of P (B), In other words, (1) Since x and y are vertices of P(B), we know On the other hand, if | mN w, x − y | is nonzero then it is at least mN and thus dominates | w ′ , x − y |. Hence, for (1) to hold, it must be that In other words, w, x ≤ w, y . Thus, x belongs to the face F . We have shown that every vertex Thus, in each round, we will add a new weight function to the current function using a smaller scale and try to get a subface with significantly smaller dimension. Henceforth, N will be sufficiently large number bounded by poly(m).
for some x 1 , x 2 ∈ F and α ∈ R.
The following claim will be useful for designing the new weight function.
The last equality holds because x 1 and x 2 lie on the face minimizing w and thus, w, x 1 = w, x 2 .
Let F 0 be the face minimizing the current weight function w 0 . Let v be a vector parallel to F 0 . Now, we choose a new weight function w ′ ∈ {0, 1, . . . , N − 1} E such that Let us define w 1 := mN w 0 + w ′ and let F 1 be the face minimizing w 1 . Clearly, w 1 , v = 0 and thus from Claim 3.4, v is not parallel to F 1 . This implies that F 1 is strictly contained in F 0 . To ensure that F 1 is significantly smaller than F 0 , we choose a many vectors parallel to F 0 , say v 1 , v 2 , . . . , v k , and construct a weight function w ′ such that for all i ∈ [k] we have w ′ , v i = 0. The following well-known lemma says that such a w ′ can be constructed easily (see, for example, [FKS84]). It actually constructs a list of weight functions such that one of them has the desired property.
Lemma 3.5. Given m, k, t, let q = mk log t. In time poly(m, k, log t) one can construct a set of weight vectors w 1 , w 2 , . . . , w q ∈ {0, 1, 2, . . . , q} m such that for any set of vectors Proof. First define w := (1, t, t 2 , . . . , t m−1 ). Clearly, w, v i = 0 for each i, because each coordinate of v i is less than t. To get a weight vector with small coordinates, we go modulo small numbers. We consider the following weight vectors w j for 1 ≤ j ≤ q: We claim that this set of weight vectors has the desired property. We know that Note that the product W is bounded by t mk . On the other hand, it is known that lcm(2, 3, . . . , q) > 2 q = t mk for all q ≥ 7 [Nai82]. Thus, there must exist a 2 ≤ j ≤ q such that j does not divide W . In other words, for all which is the desired property.
There are two things to note about this lemma: (i) it is black-box in the sense that we do not need to know the set of vectors {v 1 , v 2 , . . . , v k }. (ii) We do not know a priori which function will work in the given set of functions. So, one has to try all possibilities.
The lemma tells us that we can ensure w ′ , v = 0 for polynomially many vectors v whose coordinates are polynomially bounded. It is the foundation of our strategy that was also used by [FGT16,GT17]. Below, we formally give the weight construction. Recall that for a face F , the lattice L F is the set of all integral vectors parallel to F .
To prove Theorem 2.4, let c be the constant in the assumption of the theorem. Let N = m O(1) be a large enough number and p = log We argue the face F p of P (B) that minimizes w p is a point. First observe that F i ⊂ F i−1 , for each i (Claim 3.2). By definition, it follows that This is same as The following claim proves that w p is isolating.
Claim 3.7. The face F p is a point.
Proof. Suppose not then there will be at least two vertices of F p , say y 1 , y 2 ∈ {0, 1} E . Clearly, y 1 − y 2 ∈ L Fp and y 1 − y 2 ≤ m ≤ c p+1 . This contradicts Claim 3.6.
Bound on the weights. To bound the weights in vector w p , we bound w ′ i for each i. From Claim 3.6, we have that λ( Recall that we have to ensure w ′ i , v = 0 for vectors v in the above set. From Lemma 3.5, one can construct w ′ i with weights bounded by m O(1) (the parameter t is less than m here). Clearly, the weights in

Matroids and Circuits
In Section 4.1 we recall some basic definitions and well-known facts about matroids (see, for example, [Oxl06]). In Section 4.2 we describe Seymour's decomposition theorem for regular matroids.

Matroids preliminaries
In this section, we recall some basic definitions and well-known facts about matroids (see [Oxl06,Sch03b]). A subset I of E is said to be independent, if I belongs to I and dependent otherwise. An inclusionwise maximal independent subset of E is a base of M . An inclusionwise minimal dependent set is a circuit of M .
We define some special classes of matroids.
Definition 4.2 (Linear, binary, and regular matroid). A matroid M = (E, I) with m = |E| is linear or representable over some field F, if there is a matrix A ∈ R n×m over F, for some n, such that the collection of subsets of the columns of A that are linearly independent over F is identical It is well known that regular matroids can be characterized in terms of TU matrices. Moreover, the two definitions of circuits, Definition 5.1 for TU matrices and Definition 4.1 for regular matroids, coincide.  Proof. We show the second claim. If u ∈ R E is a circuit of A, then the columns in A corresponding to the set supp(u) are minimally dependent. Thus, the set supp(u) is a circuit of matroid M . In the other direction, a circuit C ⊆ E of matroid M is a minimal dependent set. Thus, the set of columns of A corresponding to C is minimally dependent. Hence, there is a unique vector u (up to sign) in L(A) with its support being C.
Two special classes of regular matroids are graphic matroids and their duals, cographic matroids.
Recall that an edge cut of a graph G = (V, E) is a set C ⊆ E such that G − S has more connected components than G. The symmetric difference to two cycles in a graph is a disjoint union of cycles. The analogous statement is true for binary matroids.
Fact 4.6. Let M be binary. If C 1 and C 2 are circuits of M , then the symmetric difference C 1 △C 2 is a disjoint union of circuits.
A crucial role in our argument is to bound the number of short circuits. The following theorem is a version of Theorem 2.6 applied to the cases of graphic and cographic matroids. 2. If G has no cuts of size ≤ r then number of cuts in G with size at most αr/2 is bounded by m α for α ≥ 2 [Kar93].
We define two operations on matroids.  2. The classes of regular matroids, graphic matroids, and cographic matroids are minor closed.
For a characterization of regular matroids, we will need a specific matroid R 10 , first introduced by [Bix77]. It is a matroid with 10 elements in the ground set represented over GF (2) by the following matrix.

Seymour's Theorem and its variants
The main ingredient for the proof of Theorem 2.6 is a theorem of Seymour [Sey80,Theorem 14.3] that shows that every regular matroid can be constructed from piecing together three kinds of matroids -graphic matroids, cographic matroids, and a certain matroid R 10 of size 10. This piecing together is done via a matroid operations called 1-sum, 2-sum and 3-sum. These operations are defined for binary matroids.
Definition 4.11 (Sum of two matroids [Sey80], see also [Oxl06]). Let M 1 = (E 1 , I 1 ) and M 2 = (E 2 , I 2 ) be two binary matroids, and let S = E 1 ∩ E 2 . The sum of M 1 and M 2 is the matroid denoted by M 1 △M 2 . It is defined over the ground set E 1 △E 2 such that the circuits of M 1 △M 2 are minimal non-empty subsets of E 1 △E 2 that are of the form C 1 △C 2 , where C i is a (possibly empty) disjoint union of circuits of M i , for i = 1, 2.
From the characterization of the circuits of a matroid [Oxl06, Theorem 1.1.4], it can be verified that the sum M 1 △M 2 is indeed a matroid. We are only interested in three special sums: Definition 4.12 (1, 2, 3-sums). Let M 1 = (E 1 , I 1 ) and M 2 = (E 2 , I 2 ) be two binary matroids and E 1 ∩ E 2 = S. Let m 1 = |E 1 |, m 2 = |E 2 |, and s = |S|. Let furthermore m 1 , m 2 < |E 1 △E 2 | = m 1 + m 2 − 2s. The sum M 1 △M 2 is called a • 1-sum, if s = 0, • 2-sum, if s = 1 and S is not a circuit of M 1 , M 2 , M * 1 or M * 2 , • 3-sum, if s = 3 and S is a circuit of M 1 and M 2 that does not contain a circuit of M * 1 or M * 2 .
Note that the condition m 1 , m 2 < m 1 + m 2 − 2s implies that m 1 , m 2 ≥ 2s + 1 (3) To get an intuition, we describe how the sum operation looks like for graphic matroids. For two graphs G 1 and G 2 , their 2-sum G is obtained as follows: for an edge (u 1 , v 1 ) in G 1 and an edge (u 2 , v 2 ) in G 2 , identify the two edges, that is, identify u 1 with u 2 and v 1 with v 2 . Finally, G is the union of G 1 and G 2 where the edge (u 1 , v 1 ) = (u 2 , v 2 ) is removed. Here, S is the set containing the single edge (u 1 , v 1 ) = (u 2 , v 2 ). It is instructive to see how a cycle C in G looks like. Either C is a cycle in G 1 or G 2 that avoids nodes u 1 , v 1 , u 2 , v 2 , or it is a union of a path u 1 v 1 in G 1 and a path v 2 u 2 in G 2 . Equivalently, it is a symmetric difference C = C 1 △C 2 , for two cycles C 1 in G 1 and C 2 in G 2 , such that C 1 and C 2 both contain the common edge (u 1 , v 1 ) = (u 2 , v 2 ). Analogously, the sum operation for two binary matroids is defined such that any circuit C of the sum M 1 △M 2 is either a circuit in M 1 or M 2 that avoids the elements in S, or it is C = C 1 △C 2 , for circuits C 1 and C 2 of M 1 and M 2 , respectively, such that both C 1 and C 2 contain a common element from S.
From the definition of M 1 △M 2 the following fact follows easily.
Fact 4.13. Let C i be a disjoint union of circuits of M i , for i = 1, 2. If C 1 △C 2 is a subset of E 1 △E 2 then it is a disjoint union of circuits of M 1 △M 2 .
In particular, it follows that for i = 1, 2, any circuit Further, for 1-sums, circuits are easy to characterize. Thus, if one is interested in the number of circuits, one can assume that the given matroid is not a 1-sum of two smaller matroids.

Definition 4.15 (Connected matroid).
A matroid M is connected if it cannot be written as a 1-sum of two smaller matroids.
A characterization of circuits in a 2-sum or 3-sum is not as easy. Seymour [Sey80, Lemma 2.7] provides a unique representation of the circuits for these cases.
Lemma 4.16 (Circuits in a 2-or 3-sum, [Sey80]). Let C 1 and C 2 be the set of circuits of M 1 and M 2 , respectively. Let M be a 2-or 3-sum of M 1 and M 2 . For S = E 1 ∩ E 2 , we have |S| = 1 or |S| = 3, respectively. Then for any circuit C of M , one of the following holds: 1. C ∈ C 1 and S ∩ C = ∅, or 2. C ∈ C 2 and S ∩ C = ∅, or 3. there exist unique e ∈ S, C 1 ∈ C 1 and C 2 ∈ C 2 such that S ∩ C 1 = S ∩ C 2 = {e} and C = C 1 △C 2 .
Seymour proved the following decomposition theorem for regular matroids.
However, to prove Theorem 2.6, we need a refined version of Seymour's theorem that was proved by Truemper [Tru98]. The difference from Seymour's theorem is that it decomposes a regular matroid as a sum of a smaller regular matroid and a graphic, cographic, or the R 10 matroid. In contrast, Seymour's theorem just guarantees a decomposition into two regular matroids neither of which may be graphic, cographic or R 10 . The theorem we write here slightly differs from the one by Truemper [Tru98, Lemma 11.3.18]. A proof of Theorem 4.18 can be found in Appendix A.
Theorem 4.18 (Truemper's decomposition for regular matroids, [Tru98]). Let M be any connected regular matroid, that is not graphic or cographic and is not isomorphic to R 10 . Letẽ be a fixed element of the ground set of M . Then M is a 2-sum or 3-sum of M 1 and M 2 , where M 1 is a graphic or cographic matroid, or a matroid isomorphic to R 10 and M 2 is a regular matroid that containsẽ.

A Bound on the Number of Short Vectors in Lattices: Proof of Theorem 2.5
In this section, we show that Theorem 2.5 follows from Theorem 2.6. We define a circuit of a matrix and show that it is sufficient to upper bound the number of short circuits of a TU matrix. We argue that this, in turn, is implied by a bound on the number of short circuits of a regular matroid.
We have the following properties of circuits.
Fact 5.2. Let A be a matrix.
1. Any vector v ∈ L(A) is either a multiple of a circuit of A or there is a circuit u of A with supp(u) supp(v).

Let
A be a TU matrix. Then every circuit of A has its coordinates in {−1, 0, 1} (see [Onn10,Lemma 3.18]). Now, we define a notion of conformality, which will allow us to show that any sufficiently small vector in L(A) is a circuit.
Observe that for vectors u and v with u ⊑ v, we have The following lemma follows from [Onn10, Lemma 3.19].
Lemma 5.4. Let A be a TU matrix. Then for any nonzero vector v ∈ L(A), there is a circuit u of A that is conformal to v.
We use the lemma to argue that any small enough vector in L(A) must be a circuit.
Lemma 5.5. Let A be a TU matrix. Then any nonzero vector v ∈ L(A) with v < 2λ(L(A)) is a circuit of A.

Proof. Suppose for contradiction that v is not a circuit. From Lemma 5.4, there is a circuit u of
Since both u and v − u are nonzero vectors in L(A), we know that u ≥ λ(L(A)) and v − u ≥ λ (L(A)). This together with Equation (5) implies that v ≥ 2λ(L(A)), which is contradiction.
To prove Theorem 2.5, let A be TU matrix. By Lemma 5.5, it suffices to bound the number of near-shortest circuits of A. Further, from Theorem 4.3, the two definitions of circuits, for a TU matrix and for a corresponding regular matroid, coincide. Moreover, the size of a circuit of a regular matroid is same as the ℓ 1 -norm of the corresponding circuit of the associated TU matrix. Now Theorem 2.5 follows by from Theorem 2.6 when we define the weight of each element being 1.

A Bound on the Number of Short Circuits in Regular Matroids:
Proof of Theorem 2.6 In this section, we prove our main technical tool: in a regular matroid, the number of circuits that are close to the shortest circuit is polynomially bounded (Theorem 2.6). The proof argues along the decomposition provided by Theorem 4.18. First, we need to show a bound on the number of circuits for the two special cases -graphic and cographic matroids.

Base Case: Graphic and cographic matroids
We actually prove a lemma for graphic and cographic matroids that does more -it gives an upper bound on the number of circuits that contain a fixed element of the ground set.
Lemma 6.1. Let M = (E, I) be a graphic or cographic matroid, where |E| = m ≥ 2, and w : E → N be a weight function. Let R ⊆ E with |R| ≤ 1 (possibly empty) and r be an even number. If there is no circuit C in M such that w(C) ≤ r and C ∩ R = ∅, then, for any integer α ≥ 2, the number of circuits C such that R ⊆ C and w(C) ≤ αr/2 is at most (2(m − |R|)) α .
Proof. Part 1: M graphic. (See [TK92,Sub95] for a similar argument as in this case.) Let G = (V, E) be the graph corresponding to the graphic matroid M . By the assumption of the lemma, any cycle C in G such that C ∩ R = ∅ has weight w(C) > r. Consider a cycle C in G with R ⊆ C and w(C) ≤ αr/2. Let the edge sequence of the cycle C be (e 1 , e 2 , e 3 , . . . , e q ) such that if R is nonempty then R = {e 1 }. We choose α edges of the cycle C as follows: Let i 1 = 1 and for j = 2, 3, . . . , α, define i j to be the least index (if one exists) such that Note that some of these paths might be empty. By the choice of i j we know that w(p j ) ≤ r/2 for j = 1, 2, . . . , α − 1. Combining (6) with the fact that w(C) ≤ αr/2, we obtain that w(p α ) < r/2. We associate the ordered tuple of oriented edges (e i 1 , e i 2 , . . . , e iα ) with the cycle C.
Claim 6.2. For two distinct cycles C, C ′ in G, such that both contain R and w(C), w(C ′ ) ≤ αr/2, the two associated tuples (defined as above) are different.
Proof. For the sake of contradiction, assume that the associated tuples are same for both the cycles. Thus, C and C ′ pass through (e i 1 , e i 2 , . . . , e iα ) with the same orientation of these edges. Further, there are α paths connecting them, say p 1 , p 2 , . . . , p α from C and p ′ 1 , p ′ 2 , . . . , p ′ α from C ′ . Since C and C ′ are distinct, for at least one j, it must be that p j = p ′ j . However, since the starting points and the end points of p j and p ′ j are same, p j ∪ p ′ j contains a cycle C ′′ . Moreover, since w(p j ), w(p ′ j ) ≤ r/2, we can deduce that w(C ′′ ) ≤ r. Finally, since neither of p j and p ′ j contain e 1 , we get C ′′ ∩ R = ∅. This is a contradiction.
Since, each cycle C with w(C) ≤ αr/2 and R ⊆ C is associated with a different tuple, the number of such tuples upper bounds the number of such cycles. We bound the number of tuples depending on whether R is empty or not.
• When R is empty, the number of tuples of α oriented edges is at most (2m) α .
• When R = {e 1 }, the number of choices for the rest of the α − 1 edges and their orientation is a most (2(m − 1)) α−1 .
Part 2: M cographic. Let G = (V, E) be the graph corresponding to the cographic matroid M and let n = |V |. Recall from Fact 4.5 that circuits in cographic matroids are minimal edge cuts in G. By the assumption of the lemma, any cut C in G with R ∩ C = ∅ has weight w(C) > r. We want to give a bound on the number of cuts C ⊆ E such that w(C) ≤ αr/2 and R ⊆ C.
We argue similar to the probabilistic construction of a minimum cut of Karger [Kar93]. The basic idea is to contract randomly chosen edges. Contraction of an edge e = (u, v) means that all edges between u and v are deleted and then u is identified with v. Note that we get a multi-graph that way: if there were two edges (u, w) and (v, w) before the contraction, they become two parallel edges after identifying u and v. The contracted graph is denoted by G/e. The intuition behind contraction is, that randomly chosen edges are likely to avoid the minimum cut edges.
The following algorithm implements the idea. It does k ≤ n contractions in the first phase and then chooses a random cut within the remaining nodes of the contracted graph in the second phase that contains the edges of R.
randomly choose e ∈ E \ R with probability w(e)/w(E \ R) 3 G ← G/e 4 R ← R ∪ {new parallel edges to the edges in R} Selection 5 Among all possible cuts C in the obtained graph G with R ⊆ C, choose one uniformly at random and return it.
Let C ⊆ E be a cut with w(C) ≤ αr/2 and R ⊆ C. We want to give a lower bound on the probability that Small Cut outputs C. Let G 0 = G and G i = (V i , E i ) be the graph after the i-th contraction, for i = 1, 2, . . . , k. Note that G i has n i = n − i nodes since each contraction decreases the number of nodes by 1. Let R i denote the set R after the i-th contraction. That is, if R = {e 1 }, then R i contains all edges parallel to e 1 in G i . In case that R = ∅, also R i = ∅. Note that in either case R i ⊆ C.
Conditioned on the event that no edge in C has been contracted in iterations 1 to i, the probability that an edge from C is contracted in the (i + 1)-th iteration is at most We know that w(C \ R i ) ≤ w(C) ≤ αr/2. For a lower bound on w(E i \ R i ), consider the graph G ′ i obtained from G i by contracting the edges in R i . The number of nodes in G ′ i will be n ′ i = n − i− |R| and its set of edges will be E i \ R i . For any node v in G ′ i , consider the set δ(v) of edges incident on v in G ′ i . The set δ(v) forms a cut in G ′ i and also in G. Note that δ(v) ∩ R = ∅, as the edge in R has been contracted in G ′ i . Thus, we can deduce that w(δ(v)) > r. By summing this up for all nodes in G ′ i , we obtain Therefore the probability that an edge from C is contracted in the (i + 1)-th iteration is This bound becomes greater than 1, when i > n − α − |R|. This is the reason why we stop the contraction process after k = n − α − |R| iterations. The probability that an edge from C is not contracted in any of the rounds is After n − α − |R| contractions we are left with α + |R| nodes. We claim that the number of possible cuts on these nodes that contain R is 2 α−1 . This is obvious in case when R = ∅. When R = {e 1 }, then the number of possible cuts on α + 1 nodes that contain e 1 is again 2 α−1 . We choose one of these cuts randomly. Thus, the probability that C survives the contraction process and is also chosen in the selection phase is at least Note that in the end we get exactly one cut. Thus, the number of cuts with weight ≤ αr/2 and R ⊆ C must be at most (n − |R|) α , which is bounded by (2(m − |R|)) α .

General regular matroids
In this section, we prove our main result about regular matroids. For the inductive step, let M = (E, I) be a regular matroid with |E| = m > 10 and assume that the theorem holds for all smaller regular matroids. Note that M cannot be R 10 since m > 10. We can also assume that matroid M is neither graphic nor cographic, otherwise the bound follows from Lemma 6.1. By Theorem 4.17, matroid M can be written as a 1-, 2-, or 3-sum of two regular matroids M 1 = (E 1 , I 1 ) and M 2 = (E 2 , I 2 ). We define In case that M is the 1-sum of M 1 and M 2 , we have S = ∅, and therefore m = m 1 + m 2 . By This proves the theorem in case of a 1-sum. Hence, in the following it remains to consider the case that M cannot be written as a 1-sum. In other words, we may assume that M is connected (Definition 4.15). Now we can apply Theorem 4.18 and assume that M is a 2-or 3-sum of M 1 and M 2 , where M 1 is a graphic, cographic or the R 10 matroid, and M 2 is a regular matroid.
We define for i = 1, 2 and e ∈ S C i,e := { C : C is a circuit of M i and C i ∩ S = {e} } , By Facts 4.9 and 4.10, matroid M ′ 1 is graphic or cographic, and M ′ 2 is regular. Recall from Lemma 4.16 that any circuit C of M can be uniquely written as C 1 △C 2 such that one of the following holds: • C 1 = ∅ and C 2 ∈ C ′ 2 .
Thus, we will view each circuit C of M as C 1 △C 2 and consider cases based on how the weight of C is distributed among C 1 and C 2 . Recall that the weight function w is defined on E = E 1 △E 2 . We extend w to a function on E 1 ∪ E 2 by defining w(e) = 0, for e ∈ S. Now, for the desired upper bound, we will divide the set of circuits of M with weight at most 3r/2 into three cases.
In the following, we will derive an upper bound for the number of circuits in each of the three cases. Then the sum of these bounds will be an upper bound on the number of circuits in M . We will show that the sum is less than 150 m 5 .
On the other hand, when m 1 = 2s + 1, the number of circuits can be at most 2 m 1 −s ≤ 2 4 , which is again bounded by T 0 .
Case 2: w(C 1 ) ≤ r/2 The main point why we distinguish case 2 is that here C 1 is uniquely determined.
Claim 6.3. For any e ∈ S, there is at most one circuit C 1 ∈ C 1,e with w(C 1 ) ≤ r/2.
Proof. For the sake of contradiction, assume that there are two circuits C 1 , C ′ 1 ∈ C 1,e , with w(C 1 ), w(C ′ 1 ) ≤ r/2. By Fact 4.6, we know that C 1 △C ′ 1 is a disjoint union of circuits in M 1 . Note that C 1 ∩ S = C ′ 1 ∩ S = {e}, and hence (C 1 △C ′ 1 ) ∩ S = ∅. Thus, C 1 ∆C ′ 1 is in fact a disjoint union of circuits in M . Let C be a subset of C 1 △C ′ 1 that is a circuit. For the weight of C we have This is a contradiction because M has no circuit of weight r.
Thus, as we will see, it suffices to bound the number of circuits C 2 in M 2 . Let C * e be the unique choice of a circuit provided by Claim 6.3 (if one exists) for element e ∈ S. For the ease of notation, we assume in the following that there is a C * e for every e ∈ S. Otherwise we would delete any element e ∈ S from M 2 for which no C * e exists, and then would consider the resulting smaller matroid. It might actually be that we thereby delete all of S from M 2 .
We define a weight function w ′ on E 2 as follows: , if e ∈ S, w(e), otherwise.
We now have that any circuit C of Case 2 can be written as C * e △C 2 , for some e ∈ S, or C = C 2 when C 1 = ∅. Because C * e is unique, the mapping C → C 2 is injective for circuits C of Case 2. Moreover, we have w(C) = w ′ (C 2 ). This follows from the definition in case that C = C 2 . In the other case, we have w(C) = w(C * e △C 2 ) = w(C * e ) + w(C 2 ) = w ′ (C 2 ).
For the equalities, recall that w(e) = 0 for e ∈ S. We conclude that the number of circuits C 2 in M 2 with w ′ (C 2 ) ≤ 3r/2 is an upper bound on the number of Case 2 circuits C of M with w(C) ≤ 3r/2. Now, to get an upper bound on the number of circuits in M 2 , we want to apply induction hypothesis. We need the following claim.
Claim 6.4. There is no circuit C 2 in M 2 with w ′ (C 2 ) ≤ r.
Proof. For the sake of contradiction let C 2 be such a circuit. We show that there exists a circuit C ′ in M with w(C ′ ) ≤ r. This would contradict the assumption of the lemma.
Case(i): C 2 ∩ S = ∅. Then C 2 ∈ C ′ 2 itself yields the contradiction because it is a circuit of M and w(C 2 ) = w ′ (C 2 ) ≤ r.
Case(iv): C 2 ∩ S = {e 1 , e 2 , e 3 }. Since S is a circuit, it must be the case that C 2 = S. Since C * e 1 , C * e 2 , C * e 3 and S constitute all the circuits of M 1 , the set C * e 1 △C * e 2 △C * e 3 △S contains a circuit C ′ of M 1 (Fact 4.13). Since {e i } = C * e i ∩ S, for i = 1, 2, 3, we know that S ∩ C ′ = ∅. Thus, C ′ ∈ C ′ 1 is a circuit of M . Since w(e 1 ) = w(e 2 ) = w(e 3 ) = 0, we obtain that This proves the claim.
By Claim 6.4, we can apply the induction hypothesis for M 2 with the weight function w ′ . We get that the number of circuits C 2 in M 2 with w ′ (C 2 ) ≤ 3r/2 is bounded by T 1 := 150 m 5 2 .
As mentioned above, this is an upper bound on the number of circuits C in M with w(C) ≤ 3r/2 in Case 2.
Case 3: w(C 1 ) > r/2 Since w(C) = w(C 1 ) + w(C 2 ) ≤ 3r/2, we have w(C 2 ) < r in this case. We also assume that C 2 = ∅. Hence, there is an e ∈ S such that C 1 ∈ C 1,e and C 2 ∈ C 2,e . Let T 2 be an upper bound on the number of circuits C 1 ∈ C 1,e with w(C 1 ) ≤ 3r/2, for each e ∈ S. Let T 3 be an upper bound on the number of circuits C 2 ∈ C 2,e with w(C 2 ) < r, for each e ∈ S. Because there are s choices for the element e ∈ S, the number of circuits C = C 1 △C 2 with w(C) ≤ 3r/2 in Case 3 will be at most To get an upper bound on the number of circuits in C 1,e and C 2,e , consider two matroids M 1,e and M 2,e . These are obtained from M 1 and M 2 , respectively, by deleting the elements in S \ {e}. The ground set cardinalities of these two matroids are m 1 − s + 1 and m 2 − s + 1. We know that for i = 1, 2, any circuit C i of M i,e with e ∈ C i is in C ′ i and hence, is a circuit of M . Therefore, there is no circuit C i of M i,e with e ∈ C i and w(C i ) ≤ r. Using this fact, we want to bound the number of circuits C i of M i,e with e ∈ C i . We start with M 1,e .
Claim 6.5. An upper bound on the number of circuits C 1 in M 1,e with e ∈ C 1 and w(C 1 ) ≤ 3r/2 is T 2 := min{8(m 1 − s) 3 , 2 m 1 −s } (9) Proof. Recall that the decomposition of M was such that M 1 is graphic, cographic or the R 10 matroid. Case(i). When M 1 is graphic or cographic, the matroid M 1,e falls into the same class by Fact 4.9. In this case, we apply Lemma 6.1 to M 1,e with R = {e} and α = 3 and get a bound of 8(m 1 − s) 3 . The number of circuits containing e is also trivially bounded by the number of all subsets that contain e, which is 2 m 1 −s . Recall that the ground set of M 1,e has cardinality m 1 − s + 1. Thus, we get Equation (9).
Case(ii). When M 1 is the R 10 matroid, then the cardinality of M 1,e , that is m 1 − s + 1, is at most 10. In this case again, we use the trivial upper bound of 2 m 1 −s . One can verify that when m 1 − s + 1 ≤ 10 then 2 m 1 −s ≤ 8(m 1 − s) 3 . Thus, we get Equation (9).
Next, we want to bound the number of circuits C 2 in M 2,e with e ∈ C 2 and w(C 2 ) < r. This is done in Lemma 6.7 below, where we get a bound of T 3 := 32(m 2 − s) 2 .
To finish Case 3, we now have T 2 = min{8(m 1 − s) 3 , 2 m 1 −s }, By Equation (8), the number of circuits in Case 3 is bounded by s T 2 T 3 .
This proves the claim.

Summing up Cases 1, 2 and 3
Finally we add the bounds on the number of circuits of Case 1, 2 and 3. The total upper bound we get is This completes the proof of Theorem 2.6, except for the bound on T 3 that we show in Lemma 6.7. Now we move on to prove Lemma 6.7, which completes the proof of Theorem 2.6. The lemma is similar to Theorem 2.6, but differs in two aspects: (i) we want to count circuits up to a smaller weight bound, that is, r, and (ii) we have a weaker assumption that there is no circuit of weight at most r that does not contain a fixed element e.
Lemma 6.7. Let M = (E, I) be a connected, regular matroid with ground set size m ≥ 2 and w : E → N be a weight function on E. Let r be an even number and letẽ ∈ E be any fixed element of the ground set. Assume that there is no circuit C in M such thatẽ ∈ C and w(C) ≤ r. Then, the number of circuits C in M such thatẽ ∈ C and w(C) ≤ r is bounded by 32(m − 1) 2 .
Proof. We closely follow the proof of Theorem 2.6. We proceed again by an induction on m, the size of the ground set E.
For the base case, let m ≤ 10. There are at most 2 m−1 circuits that containẽ. This number is bounded by 32(m − 1) 2 , for any 2 ≤ m ≤ 10.
For the inductive step, let M = (E, I) be a regular matroid with |E| = m > 10 and assume that the theorem holds for all smaller regular matroids. Since m > 10, matroid M cannot be R 10 . If M is graphic or cographic, then the bound of the lemma follows from Lemma 6.1. Thus, we may assume that M is neither graphic nor cographic.
By Theorem 4.17, matroid M can be written as a 1-, 2-, or 3-sum of two regular matroids M 1 = (E 1 , I 1 ) and M 2 = (E 2 , I 2 ). We use the same notation as Theorem 2.6, The case that M is a 1-sum of M 1 and M 2 is again trivial. Hence, we may assume that M is connected. By Theorem 4.18, M is a 2-sum or a 3-sum of M 1 and M 2 , where M 1 is a graphic, cographic or the R 10 matroid, and M 2 is a regular matroid containingẽ. For i = 1, 2 and e ∈ S, define C i,e := { C : C is a circuit of M i and C i ∩ S = {e} } .
Also the weight function w is extended on S by w(e) = 0, for any e ∈ S.
We consider again cases based on how the weight of C is distributed among C 1 and C 2 . Note thatẽ is in M 2 and we are only interested in circuits C that containẽ. Hence, we haveẽ ∈ C 2 . Therefore we do not have the case where C 2 = ∅.
This proves the claim.
This gives us the desired bound and completes the proof of Lemma 6.7.