Independent Sets from an Algebraic Perspective

We approach the basic problem of counting independent sets in a graph and, in particular, the problem of counting antichains in a finite partially ordered set, from an algebraic perspective. We derive structural considerations and complexity results. The use of algebraic methods in the study of discrete problems, in particular problems in graph theory, was pioneered by Richard Stanley [26] and László Lovász [20], from the combinatorics side, and Jürgen Herzog, Takayuki Hibi, Aron Simis, Wolmer Vasconcelos and Rafael Villarreal [17,25,24] from the commutative algebra side. The enumeration of independent sets has been approached using Reverse Search ( [10]), the Belief Propagation heuristic ( [5]) and Binary Decision Diagrams ( [27]), to name a few techniques. The main algebraic object we will use is the Hilbert Series of the initial monomial ideal associated with a graph. The problem of computing a Hilbert Series is NP-Complete ( [1]). There is a standard algorithm (first proposed in [21]) for computing the Hilbert Series of a quotient C[x]/I, where I is a homogeneous ideal in C[x]. There are some classes of ideals for which this algorithm finishes in time polynomial in the input, e.g. Borel ( [1]) and Borel-type ideals ( [14]). Open computer algebra systems (CoCoA [6], Singular [13], Macaulay2 [12]) implement the standard algorithm in subtly different ways. We suggest [18,Ch. 5] as a general reference on Hilbert Series. The connection between independent sets and commutative algebra is spearheaded by the following construction. Definition 1.1. Let G = (V, E) be a graph, with V = {v 1 , . . . , v n }. The edge ideal ( [25,24]) I G ⊆ C[x 1 , . . . , x n ] of G is defined as (1) I G = x i x j , for all (v i , v j ) ∈ E . This ideal links independent sets in G and certain monomials. If x α is a monomial not in I G (termed a standard monomial ), then it encodes an independent set S of G in this way: (2) This encoding is not one-to-one. For example, the monomials x 1 and x 2 1 represent the same independent set: {v 1 }. We introduce a slightly modified version of I G , with which we obtain a bijective encoding. Definition 1.2. Let G = (V, E) be a graph. We define the modified edge ideal I G of G as (3) I G = I G + x 2 i , for all v i . Notice that I G is zero-dimensional (the origin is the only root), and that its standard monomials are square-free, with the degree of a monomial equal to the size of the corresponding independent set. The number of independent (or stable) sets in G thus coincides with the k-vector space dimension of the quotient of the polynomial ring in n variables over any field k by the ideal I G . This dimension is computed in [6,12,13] using the additivity of the Hilbert function in short exact sequences. In Section 2 we recall the definition of the Hilbert function (see ( 4)) and we analyze the instantiation of the standard algorithm for computing the Hilbert Series for the ideals I G . Our main result in this section shows that the recursive calls simply correspond to counting independent sets of G that contain a pivot vertex, and those that do not contain it. In Section 3, we turn our attention to the subproblem of counting the antichains of a finite poset. We present the universal reduced Gröbner Basis for a family of zero-dimensional radical ideals derived from posets. In Section 4, we specialize our study in the case of Cohen-Macaulay bipartite graphs, corresponding to Cohen-Macaulay ideals I G . Using the characterization in [17], we show that counting independent sets in such graphs is equivalent to evaluating at 2 the independence polynomial of the comparability graph of a general finite poset. Section 5 contains our complexity study. We prove that antichain polynomials cannot be evaluated in polynomial time at any non-zero rational number t unless P = #P . When combined with the algebraic results from the previous sections we deduce Corollaries 5.4 and 5.5 on the intractability of the evaluation of Hilbert Series of initial ideals of zero-dimensional complete intersections and independence polynomials of Cohen-Macaulay bipartite graphs. We close with a few experimental observations in Section 6. We start by recalling a few definitions concerning Hilbert Series. Let M be a positively graded finitely generated C[x]-module (e.g. the quotient C[x]/I G for some graph G). We can write where M i is the subspace of M of degree i. The Hilbert Function (HF M ) of M maps i onto dim C (M i ). The Hilbert Series (HS M ) of M is the generating function If M = C[x]/I for a monomial ideal I, then HF M (i) is the number of standard monomials of degree i (that is, monomials which are not in I). If we take I = I G for some graph G, as we mentioned in the introduction, HF M (i) is then the number of independent sets of size i in G. In this case, the Hilbert Series of C[x]/I G is a polynomial, called the independence polynomial of G. As usual, we denote this polynomial by I(G, x). We refer the reader to [19] for a comprehensive survey of independence polynomias. The standard algorithm for computing HS M hinges on the following property. If we have a homogeneous exact sequence of finitely generated graded C[x]-modules HS M (z) = HS M (z) + HS M (z). Given a finitely generated graded C[x]-module M and f = 0 a homogeneous polynomial of degree d, we have the following multiplication sequence where ϕ is induced by multiplication by f . Here, (0 : and(-d) induces a degree shift, so that ϕ is a homogeneous map of degree 0. Rewriting equation ( 6) we obtain The polynomial f above is called a pivot. Actually, the standard algorithm does not directly compute the Hilbert Series. We can see in [18,Theorem 5.2.20] that in the case of the modified edge ideal I G , the Hilbert Series of M = C[x 1 , . . . , x n ]/I G has the form ( 9) where HN M (z) is called the Hilbert Numerator. The algorithm computes HN M (z), and the series is then obtained by dividing it by (1 -z) n . We reproduce the algorithm for computing the Hilbert Numerator of a monomial ideal (see [18,Theorem 5.3.7]). Choose a monomial p as pivot. Notice that the sets of generators of I G and I G described in (1) and (3) are minimal. The process of obtaining minimal sets of generators for the ensuing recursive calls can be optimized by performing careful interreductions. The choice of pivot must satisfy one condition. Namely, Here, deg(I) denotes the sum of the degrees of all the minimal monomial generators of I. Intuitively, this condition says that the recursive calls are made on "smaller" ideals, and shows that the algorithm terminates. The program CoCoA implements this algorithm, and uses a certain strategy for the choice of pivot in step 4. First, it chooses any variable x i appearing in the most number of generators of I G . Then it picks two random generators containing that variable. The pivot is the highest power of x i that divides both random generators. We present a specialized version of Algorithm 2.1, suited for the computation of the Hilbert Series of C[x]/I G for any graph G. This algorithm has an obvious graphical interpretation. The choice of step 4 corresponds to choosing a node v of the graph. The recursive calls of step 7 correspond to counting the independent sets of G that contain v (HS Colon ) and those that do not contain v (HS Plus ). return where k is the number of variables which appear squared in L. Choose a variable x i that appears squared in L. Colon ← a minimal set of monomial generators of ( L : x i ). Plus ← a minimal set of monomial generators of L, x i . return zHS Colon (z) + HS Plus (z) 8: end if Proof. Algorithm 2.3 differs from Algorithm 2.1 in two key steps. In step 1, the special version does not check coprimality, as is done in Algorithm 2.1. The other difference is in step 4: The specialized version chooses a variable, instead of an arbitrary monomial. We make a claim that helps us understand why this specialized version is correct. In every call to the algorithm, each of the n variables appears in L raised to the first or to the second power. Furthermore, in each call, L contains only the powers just mentioned and the "edge monomials" x i x j of G such that both x i and x j appear squared in L. This leads to an obvious graphical interpretation: The list L represents the subgraph of G induced by those variables that appear squared in L. We prove the correctness of the algorithm by showing that the choice of a pivot in Algorithm 2.1 must always yield a variable when applied to a modified edge ideal, and that the claim of the previous paragraph is true. When the algorithm is originally invoked, every variable appears squared in L. Besides the squares of variables, L contains the "edge monomials" x i x j for every edge (i, j) of G. This proves that the claim above holds in the first call. Assuming that the elements of L have the structure we claim, let us show that any choice of pivot yields a variable. Suppose that we employ any conceivable strategy for the choice of pivot, always subject to condition (10). The pivot p cannot be a multiple of any monomial in L. If it is, then Plus = L, and the decreasing total degree condition (10) is not satisfied. The pivot p must then be a product of variables that appear squared in L, but it must not be divisible by any "edge monomial." Suppose that the pivot is the product of at least two variables. That is, x i x j | p, where x 2 i and x 2 j are in L, and x i x j is not in L. Then Plus violates the decreasing total degree condition (10), because it has the same generators as L, plus p. If p = 1, then Colon = L, and this violates the decreasing total degree condition. The only valid choice is then p = x i , for some x i that appears squared in L. Once we know that the pivot is always a variable, we can show that the claim above holds for Plus and for Colon. In doing so, we also explain the second part of the theorem. The list of minimal monomial generators for Plus contains all the variables that were raised to the first power in L. Furthermore, it must also contain the pivot x i . The square of x i is not in Plus, because Plus is minimal, and the "edge monomials" that contained x i , are not present in Plus. The rest of the generators in L are unaffected. Therefore, we have that every variable appears in Plus either squared or raised to the first power, as we wanted to show. Plus corresponds to the graph obtained by removing the node that corresponds to x i and all the edges incident with it. The analysis of Colon is somewhat similar. To obtain a minimal set of monomial generators, we just cross out the pivot x i from every generator in L that contains it, and then eliminate multiples. If we had an "edge monomial" x i x j , then x j is in Colon. Therefore, the square of x j is no longer a generator, and all the "edge monomials" containing x j are also missing from Colon. Again, every variable appears either squared or raised to the first power. In this case, we remove the node corresponding to x i , all its adjacent nodes and all the edges incident with x i or with any node adjacent to x i . Let v be the node of G associated with the pivot x i . The combination step of the algorithm reflects the meaning of Colon and Plus: The independent sets of G are those of Plus (i.e. those do not that contain v) and those of Colon (i.e. those that contain v). The algorithm terminates when there are no more "edge monomials". Since all the generators are variables, or squares of variables, then they are pairwise coprime and satisfy the stopping criterion of Algorithm 2.1. A note is in order about the value returned in the base case. Algorithm 2.1 returns where d i is the degree of the i-th generator. Since in the specialized case the generators are of the form x i or x 2 i , expression (11) has the form where k is the number of variables that appear squared in L. According to formula (9), the value returned by Algorithm 2.3 is the Hilbert Series of C[x 1 , . . . , x n ]/I G . All these observations show that the graphical interpretation is accurate and that the specialized version is indeed correct. In this section, we study a family of zero-dimensional radical complete intersection polynomial ideals associated with posets, first proposed in [4]. Recall that a poset (or partially ordered set) is a set P , together with a (partial order) relation ≤ satisfying • a ≤ a, for all a ∈ P . We can associate to a poset P its comparability graph. Definition 3.1. Let P be a poset. The comparability graph G(P ) has the set P as nodes and there is an edge between two different nodes a and b if and only if a, b are comparable in P . A subset S of a poset P is an antichain if all the elements of S are pairwise incomparable in P . We write A(P ) for the set of antichains of P . Note that S ∈ A(P ) if and only if S is an independent set of G(P ). Definition 3.2. For any poset P we define the antichain polynomial A(P, x) by Thus, the k-th coefficient of A(P, x) equals the number of antichains of P with k elements and the cardinal |A(P )| is given by the evaluation A(P, 1). Given a finite poset (P, ≤), we define a polynomial ideal J P ⊂ C[x 1 , . . . , x n ] by: (13) x j , for all v i ∈ P . Lemma 3.3. Let P be a finite poset. Then the elements of V (J P ) are strings of 0's and 1's. Proof. Let a ∈ V (J P ). Suppose that an element v i ∈ P is minimal. Then x i -x 2 i ∈ J P , hence a i is 0 or 1. Now, take any v i , and assume that for every v j < v i we know that a j is 0 or 1. Note that x i -x i vj ≤vi x j = x i (1 -x i vj v i }. The map f is bijective, and its inverse g : A(P ) → V (J P ) is defined by g(S) = a , where a i = 1 if ∃ v j ∈ S such that v i ≤ v j and a i = 0 otherwise. Proof. It is clear from the definition of f that no pair of elements of the subset f (a) can be comparable for any a ∈ V (J P ), that is, that f (a) is indeed an antichain. Reciprocally, let S be an antichain of P and let a = g(S). We need to see that a i (1 -a i v k ≤vi a k ) = 0 for all i. This is clear if a i = 0. When a i = 1, there exists v j ∈ S with v j ≥ v i . By the transitivity of the order relation we deduce that a k = 1 for all v k ≤ v i and so the equation is satisfied. Let a be an element of V (J P ). Let S = f (a) and a = g(S). We want to show that a = a . Suppose that a i = 1. Then ∃ v j ∈ S such that v i ≤ v j , and therefore a i = 1. By a similar argument, if a i = 0, then a i = 0. We now describe the universal reduced Gröbner basis of J P = J P . Proposition 3.7. The universal, reduced Gröbner Basis of J P is the set Gb P of polynomials Proof. Lemma 3.3 shows that the elements of V (J P ) are strings of 0's and 1's. The polynomials x 2 i -x i are in Gb P , and therefore the elements of V (Gb P ) are also strings of 0's and 1's. Let x = (x i ) vi∈P be a string of 0's and 1's. x ∈ V (J P ) if and only if ∀ v i ∈ P, (x i = 0 ⇔ (∃ v j ≤ v i such that x j = 0)). But this is equivalent to x ∈ V (Gb P ). Then, Gb P is zero-dimensional, and contains a square-free univariate polynomial in each variable (gb i ). Therefore it is also radical. This shows that the ideal generated by Gb P coincides with J P . We now prove that Gb P is a Gröbner Basis for any monomial order <. Recall that, given < and a non-zero polynomial p, LT < (p) denotes the largest term of p, with respect to <. Clearly, LT < (gb i ) = x 2 i and LT < (gb (j,i) ) = x i x j . Given any two polynomials in the set Gb P , we show that their S-polynomial is divisible by the polynomials in (Gb P ). If we let p = x i x j -x i and q = x k x -x k be two polynomials in Gb P , all possible combinations of the indices i, j, k and boil down to the following non-trivial possibilities for (p, q) (with i, j, k, all different): (1) ( . In all three cases, since the leading monomials of p and q are coprime, S(p, q) is divisible by (p, q). (7) (x i x j -x i , x j x i -x j ). This can only hold if v i ≤ v j and v j ≤ v i , that is, v i = v j . In cases 4 and 5 above, we know that gb (j,k) and gb ( ,i) , respectively, are in Gb P , because a partial order relation is transitive. Therefore, Gb P is a Gröbner Basis. Finally, none of the polynomials are redundant, all of the leading coefficients are one, and the "other" monomial in each polynomial of Gb P has degree 1, so it cannot be divisible by any leading monomial of Gb P . Therefore, Gb P is a reduced universal Gröbner Basis of J P . We can count the antichains of P by studying J P . We have seen that |A(P )| = |V (J P )|. It is well-known ([8, Theorem 2.2.10]) that as J P is radical, it holds that The Hilbert Series algorithm could help us to compute dim C (C[x]/J P ), but it requires that the ideal J P be homogeneous, which is not the case. This is circumvented by considering an initial ideal of J P . If < is a monomial order and I is an ideal, the initial ideal of LT < (I) is defined by In particular, we have the following equality ( 17) Let Gb P be the (universal) Gröbner basis of J P in the statement of Proposition 3.7. By the definition of a Gröbner Basis, it holds that for any monomial order, LT < (J P ) = LT < (g), g ∈ Gb P . Note that this initial ideal has the same structure of the ideals I G in Section 2. In fact, it equals I G(P ) . Let G be a graph, and I G its edge ideal. We say that G is a Cohen-Macaulay Cohen-Macaulay rings and modules are extensively studied in [16], and the article [23] covers Cohen-Macaulay graphs. Not every graph is Cohen-Macaulay, of course. For example, the path of length three (see Figure 1) has the edge ideal It is not even equidimensional, since the zero set of J P3 consists of the plane x 2 = 0, together with the line x 1 = x 3 = 0. There is an equivalent characterization, given by the following result. , and the vertices V 1 = {x 1 , . . . , x n } and V 2 = {y 1 , . . . , y n } can be labeled in such a way that (1) (x i , y i ) ∈ E for all i = 1, . . . , n; (2) if (x i , y j ) ∈ E, then i ≤ j; (3) if (x i , y j ) and (x j , y k ) are edges, then (x i , y k ) is also an edge. There are two ways of seeing a bipartite Cohen-Macaulay graph G = (V 1 V 2 , E) as a poset. The obvious way is to set the following partial order on the nodes of G: x ≤ y if and only if x = y or x ∈ V 1 , y ∈ V 2 and (x, y) is an edge of G. That is, one chooses one of the parts as the "upper" one. The other way, which we will consider here, involves a different construction. Let G = (V 1 V 2 , E) be a bipartite Cohen-Macaulay graph. We define a poset P G as follows. The elements of P G are those of V 1 . Given x i and x j , we set x i ≤ x j if and only if the edge (x i , y j ) is in E. From the transitivity of bipartite Cohen-Macaulay graphs, we see that P G is a poset. Conversely, let P be a poset, with elements x 1 , . . . , x r . We build a bipartite graph G P = (V, E) as follows. We set V = V 1 V 2 , with V 1 = {x 1 , . . . , x r } and V 2 = {y 1 , . . . , y r }. We put the edges (x i , y i ) in E for all i, and we have the edge (x i , y j ) if and only if x i ≤ x j in P . In this case, the transitivity of ≤ ensures that G P is a bipartite Cohen-Macaulay graph. The following lemma is straightforward. Proof. The construction outlined above expands every element of the poset P G into a segment in the bipartite Cohen-Macaulay graph G. An antichain S of size k in P G gives rise to 2 k independent sets of size k in the bipartite graph G, since we can replace any x i ∈ S by either the node x i or the node y i of G. It is clear that any independent set of G can be seen in this way for a unique antichain S of P G . Is it classically known that it is not possible to count the number of antichains of a general poset (that is, to evaluate its antichain polynomial at 1) in polynomial time unless P = #P [22]. We extend this result in Theorem 5.3 to the evaluation at any non-zero rational number t, by a translation and specialization of [2, Theorem 2.2] to the context of finite posets. We then use our previous results to deduce in Corollaries 5.4 and 5.5 the hardness of evaluating the Hilbert function of initial ideals of zero-dimensional radical ideals and the independence polynomial of Cohen-Macaulay bipartite graphs. We start with some definitions. Definition 5.1. We define the lexicographic product poset P 1 [P 2 ] of two finite posets P 1 and P 2 as the set P 1 × P 2 , ordered by the relation ( of two graphs as the set G 1 × G 2 with (i, j) adjacent to (k, l) iff i is adjacent to k or if i = k and j is adjacent to l. It is easy to check that P 1 [P 2 ] is indeed a poset. Given a natural number m, denote by Km the poset given by the set {1, . . . , m}, ordered with the usual ≤ relation. The associated comparability graph is the complete graph K m in m nodes, whose independence polynomial equals I(K m , x) = 1 + mx. It is straightforward to check that the comparability graph of the lexicographic product P 1 [P 2 ] of two posets equals the lexicographic product G(P 1 )[G(P 2 )] of the respective comparability graphs. We therefore have: We are now ready to prove the following theorem: Theorem 5.3. Evaluating the antichain polynomial of any finite poset P at any non-zero rational number t is #P -hard. Proof. We mimic the arguments in [2, Theorem 2.2]. Suppose, on the contrary, that given any poset P on n vertices, there exists an O(n k )-algorithm to compute A(P, t) for some constant k. Then, given a poset P with n vertices, consider the posets P [ Km ] for m = 1, . . . , n + 1. It follows from Lemma 5.2 that we can use the reasoning in [2, Theorem 2.2] to deduce that that A(P, mt) = A(P [ K] m , t). In fact, by [3, Theorem 1]), A(P [ K] m , t) = A(P, A( Km , t) -1) = A(P, mt). As the posets P [ Km ], m = 1, . . . , n + 1 can be constructed in polynomial time from the data of P , it would be possible to compute A(P, mt) in polynomial time for m = 1, . . . , n + 1. But then, the coefficients i j of A(P, x) = n j=0 i j x j would be computed in polynomial time by solving the (n + 1) × (n + 1) linear system with invertible matrix M = (M ij ) given by M ij = (jt) i-1 , i, j = 1, . . . , n + 1. It follows that the number of antichains |A(P )| of P would be computable in polynomial time by adding j i j . But this counting problem is #P -complete [22]. Combining this complexity results with the algebraic results of the previous sections, we have the following two corollaries. Corollary 5.4. No algorithm can evaluate the Hilbert Series at a fixed non-zero rational number t in polynomial time when applied to initial ideals of radical zerodimensional complete intersections, unless #P = P . Proof. By the results of Sections 2 and 3, the Hilbert Series of the initial ideal I G(P ) of the radical zero-dimensional ideal J P associated to any poset, equals the antichain polynomial A(P, x). The result follows from Theorem 5.3. Corollary 5.5. There can be no polynomial algorithm to evaluate at any non-zero rational number t the independence polynomial of bipartite Cohen-Macaulay graphs unless #P = P . Proof. By Lemma 4.4, the independence polynomial I(G, p) of a bipartite Cohen-Macaulay graph G and the antichain polynomial A(P G , x) of its associated poset P G are related by the equality I(G, x) = A(P G , 2x). So, any polynomial algorithm to evaluate I(G, t/2) in polynomial time for any Cohen-Macaulay graph G, would allow us to evaluate A(P, t) in polynomial time for any poset by Lemma 4.3. The result now follows from Theorem 5.3. We tested the three Computer Algebra Systems CoCoA, Singular and Macaulay 2. The examples we used were the posets consisting of the power set of {1, . . . , n}, ordered by inclusion (Boolean lattice). Of the three systems, only CoCoA managed to count the antichains for n = 7. These numbers (called Dedekind numbers) are known for n up to 13. However, those computations required many hours of supercomputer time [15]. The strategy employed by CoCoA for the Hilbert Numerator algorithm seems to be generally good. We made some observations about it in [9]. We have also tested a recent software package, EdgeIdeals ( [11]). EdgeIdeals allows us to compute the Hilbert Series of a modified edge ideal of a graph G by computing the f -vector of the simplicial complex associated with the edge ideal of G. The simplicial complex also contains a description of the standard monomials of the modified edge ideal of G. The computation of both objects (the f -vector and the standard monomials) was faster using EdgeIdeals for the Boolean lattice, compared to the native Macaulay 2 implementation of hilbertSeries, for n up to 6.