Recognizing Graph Theoretic Properties with Polynomial Ideals
Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect -colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Gröbner bases, toric algebra, convex programming, and real algebraic geometry.
In his well-known survey , Noga Alon used the term polynomial method to refer to the use of nonlinear polynomials when solving combinatorial problems. Although the polynomial method is not yet as widely used as its linear counterpart, increasing numbers of researchers are using the algebra of multivariate polynomials to solve interesting problems (see for example [2, 12, 13, 17, 19, 23, 24, 32, 31, 35, 36, 38, 43] and references therein). In the concluding remarks of , Alon asked whether it is possible to modify algebraic proofs to yield efficient algorithmic solutions to combinatorial problems. In this paper, we explore this question further. We use polynomial ideals and zero-dimensional varieties to study three hard recognition problems in graph theory. We show that this approach can be fruitful both theoretically and computationally, and in some cases, result in efficient recognition strategies.
Roughly speaking, our approach is to associate to a combinatorial question (e.g., is a graph -colorable?) a system of polynomial equations such that the combinatorial problem has a positive answer if and only if system has a solution. These highly structured systems of equations (see Propositions 1.1, 1.3, and 1.4), which we refer to as combinatorial systems of equations, are then solved using various methods including linear algebra over finite fields, Gröbner bases, or semidefinite programming. As we shall see below this methodology is applicable in a wide range of contexts.
In what follows, denotes an undirected simple graph on vertex set and edges . Similarly, by we mean that is a directed graph with arcs . When is undirected, we let
consist of all possible arcs for each edge in . We study three classical graph problems.
Let be an undirected simple graph on vertices . Fix a positive integer , and let be a field with characteristic relatively prime to . The polynomial system
has a common zero over (the algebraic closure of ) if and only if the graph is -colorable.
Depending on the context, the fields we use in this paper will be the rationals , the reals , the complex numbers , or finite fields with a prime number.
Hilbert’s Nullstellensatz [11, Theorem 2, Chapter 4] states that a system of polynomial equations with coefficients in has no solution with entries in its algebraic closure if and only if
Thus, if the system has no solution, there is a Nullstellensatz certificate that the associated combinatorial problem is infeasible. We can find a Nullstellensatz certificate of a given degree or determine that no such certificate exists by solving a system of linear equations whose variables are in bijection with the coefficients of the monomials of (see  and the many references therein). The number of variables in this linear system grows with the number of monomials of degree at most . Crucially, the linear system, which can be thought of as a -th order linear relaxation of the polynomial system, can be solved in time that is polynomial in the input size for fixed degree (see [34, Theorem 4.1.3] or the survey ). The degree of a Nullstellensatz certificate of an infeasible polynomial system cannot be more than known bounds , and thus, by searching for certificates of increasing degrees, we obtain a finite (but potentially long) procedure to decide whether a system is feasible or not (this is the NulLA algorithm in [34, 14, 13]). The philosophy of “linearizing” a system of arbitrary polynomials has also been applied in other contexts besides combinatorics, including computer algebra [18, 25, 37, 44], logic and complexity , cryptography , and optimization [30, 28, 29, 39, 40, 41].
As the complexity of solving a combinatorial system with this strategy depends on its certificate degree, it is important to understand the class of problems having small degrees . In Theorem 2.1, we give a combinatorial characterization of non-3-colorable graphs whose polynomial system encoding has a degree one Nullstellensatz certificate of infeasibility. Essentially, a graph has a degree one certificate if there is an edge covering of the graph by three and four cycles obeying some parity conditions on the number of times an edge is covered. This result is reminiscent of the cycle double cover conjecture of Szekeres (1973)  and Seymour (1979) . The class of non-3-colorable graphs with degree one certificates is far from trivial; it includes graphs that contain an odd-wheel or a 4-clique  and experimentally it has been shown to include more complicated graphs (see [34, 13, 15]).
In our second application of the polynomial method, we use tools from the theory of Gröbner bases to investigate (in Section 3) the detection of Hamiltonian cycles of a directed graph . The following ideals algebraically encode Hamiltonian cycles (see Lemma 3.8 for a proof).
Let be a simple directed graph on vertices . Assume that the characteristic of is relatively prime to and that is a primitive -th root of unity. Consider the following system in :
Here, denotes those vertices which are connected to by an arc going from to in . The system has a solution over if and only if has a Hamiltonian cycle.
We prove a decomposition theorem for the ideal generated by the above polynomials, and based on this structure, we give an algebraic characterization of uniquely Hamiltonian graphs (reminiscent of the one for -colorability in ). Our results also provide an algorithm to decide this property. These findings are related to a well-known theorem of Smith  which states that if a -regular graph has one Hamiltonian cycle then it has at least three. It is still an open question to decide the complexity of finding a second Hamiltonian cycle knowing that it exists .
Finally, in Section 4 we explore the problem of determining the automorphisms of an undirected graph . Recall that the elements of are those permutations of the vertices of which preserve edge adjacency. Of particular interest for us in that section is when graphs are rigid; that is, . The complexity of this decision problem is still wide open . The combinatorial object will be viewed as an algebraic variety in as follows.
Let be a simple undirected graph and its adjacency matrix. Then is the group of permutation matrices given by the zeroes of the ideal generated from the equations:
The last three sets of equations say that is a permutation matrix, while the first one ensures that this permutation preserves adjacency of edges (). ∎
In what follows, we shall interchangeably refer to as a group or the variety of Proposition 1.4. This real variety can be studied from the perspective of convexity. Indeed, from Proposition 1.4, consists of the integer vertices of the polytope of doubly stochastic matrices commuting with . By replacing the equations in (1) with the linear inequalities , we obtain a polyhedron which is a convex relaxation of the automorphism group of the graph. This polytope and its integer hull have been investigated by Friedland and Tinhofer [48, 20], where they gave conditions for it to be integral. Here, we uncover more properties of the polyhedron and its integer vertices .
Our first result is that is quasi-integral; that is, the graph induced by the integer points in the 1-skeleton of is connected (see Definition 7.1 in Chapter 4 of ). It follows that one can decide rigidity of graphs by inspecting the vertex neighbors of the identity permutation. Another application of this result is an output-sensitive algorithm for enumerating all automorphisms of any graph . The problem of determining the triviality of the automorphism group of a graph can be solved efficiently when is integral. Such graphs have been called compact and a fair amount of research has been dedicated to them (see [8, 48] and references therein).
Next, we use the theory of Gouveia, Parrilo, and Thomas , applied to the ideal of Proposition 1.4, to approximate the integer hull of by projections of semidefinite programs (the so-called theta bodies). In their work, the authors of  generalize the Lovász theta body for polyhedra to generate a sequence of semidefinite programming relaxations computing the convex hull of the zeroes of a set of real polynomials [33, 32]. The paper  provides some applications to finding maximum stable sets  and maximum cuts . We study the theta bodies of the variety of automorphisms of a graph. In particular, we give sufficient conditions on for which the first theta body is already equal to (in much the same way that stable sets of perfect graphs are theta-1 exact [21, 33]). Such graphs will be called exact. Establishing these conditions for exactness requires an interesting generalization of properties of the symmetric group (see Theorem 4.6 for details). In addition, we prove that compact graphs are a proper subset of exact graphs (see Theorem 4.4). This is interesting because we do not know of an example of a graph that is not exact, and the connection with semidefinite programming may open interesting approaches to understanding the complexity of the graph automorphism problem.
2 Recognizing Non-3-colorable Graphs
In this section, we give a complete combinatorial characterization of the class of non-3-colorable simple undirected graphs with a degree one Nullstellensatz certificate of infeasibility for the following system (with ) from Proposition 1.1:
This polynomial system has a degree one () Nullstellensatz certificate of infeasibility if and only if there exist coefficients such that
Our characterization involves two types of substructures on the graph (see Figure 1). The first of these are oriented partial-3-cycles, which are pairs of arcs , also denoted , in which (the vertices induce a 3-cycle in ). The second are oriented chordless 4-cycles, which are sets of four arcs , denoted , with (the vertices induce a chordless 4-cycle).
For a given simple undirected graph , the polynomial system over encoding the -colorability of
has a degree one Nullstellensatz certificate of infeasibility if and only if there exists a set of oriented partial -cycles and oriented chordless -cycles from such that
for all and
where denotes the set of cycles in in which the arc appears. Moreover, the class of non-3-colorable graphs whose encodings have degree one Nullstellensatz infeasibility certificates can be recognized in polynomial time.
We can consider the set in Theorem 2.1 as a covering of by directed edges. From this perspective, Condition 1 in Theorem 2.1 means that every edge of is covered by an even number of arcs from cycles in . On the other hand, Condition 2 says that if is the directed graph obtained from by the orientation induced by the total ordering on the vertices , then when summing the number of times each arc in appears in the cycles of , the total is odd.
Note that the 3-cycles and 4-cycles in that correspond to the partial 3-cycles and chordless 4-cycles in give an edge-covering of a non-3-colorable subgraph of . Also, note that if a graph has a non-3-colorable subgraph whose polynomial encoding has a degree one infeasibility certificate, then the encoding of will also have a degree one infeasibility certificate.
The class of graphs with encodings that have degree one infeasibility certificates includes all graphs containing odd wheels as subgraphs (e.g., a -clique) .
If a graph contains an odd wheel, then the encoding of -colorability of from Theorem 2.1 has a degree one Nullstellensatz certificate of infeasibility.
Assume contains an odd wheel with vertices labelled as in Figure 2 below. Let
Figure 2 illustrates the arc directions for the oriented partial 3-cycles of . Each edge of is covered by exactly zero or two partial 3-cycles, so satisfies Condition 1 of Theorem 2.1. Furthermore, each arc is covered exactly once by a partial 3-cycle in , and there is an odd number of such arcs. Thus, also satisfies Condition 2 of Theorem 2.1. ∎
A non-trivial example of a non-3-colorable graph with a degree one Nullstellensatz certicate is the Grötzsch graph.
Figure 3 illustrates the arc directions for the 4-cycles of . Each edge of the graph is covered by exactly two 4-cycles, so satisfies Condition 1 of Theorem 2.1. Moreover, one can check that Condition 2 is also satisfied. It follows that the graph has no proper 3-coloring. ∎
We now prove Theorem 2.1 using ideas from polynomial algebra. First, notice that we can simplify a degree one certificate as follows: Expanding the left-hand side of (3) and collecting terms, the only coefficient of is and thus for all . Similarly, the only coefficient of is , and so for all . We thus arrive at the following simplified expression:
Now, consider the following set of polynomials:
The elements of are those polynomials that can appear in a degree one certificate of infeasibility. Thus, there exists a degree one certificate if and only if the constant polynomial 1 is in the linear span of ; that is, , where is the vector space over generated by the polynomials in .
We next simplify the set . Let be the following set of polynomials:
If we identify the monomials as the arcs , then the polynomials (2) correspond to oriented partial 3-cycles and the polynomials (9) correspond to oriented chordless 4-cycles. The following lemma says that we can use instead of to find a degree one certificate.
We have if and only if .
The polynomials (6) above can be split into two classes of equations: (i) or and (ii) and . Thus, the set consists of
Observe that . We can eliminate the polynomials (13) as follows. For every , is the only polynomial in containing the monomial and thus the polynomial cannot be present in any nonzero linear combination of the polynomials in that equals 1. We arrive at the following smaller set of polynomials, which we label .
So far, we have shown if and only if .
Next, we eliminate monomials of the form . There are 3 cases to consider.
Case 1: but and . In this case, the monomial appears in only one polynomial, , so we can eliminate all such polynomials.
Case 2: , . Graphically, this represents a 3-cycle in the graph. In this case, the monomial appears in three polynomials:
Using the first polynomial, we can eliminate from the other two:
Note that both of these polynomials correspond to two of the arcs of the -cycle .
Case 3: , and . We have
As before we use the first polynomial to eliminate the monomial from the second:
We can now eliminate (21); thus, the original system has been reduced to the following one, which we label as :
Note that if and only if .
The monomials and with always appear together and only in the polynomials (25) in the expression . Thus, we can eliminate the monomials and with by choosing one of the polynomials (25) and using it to eliminate the expression from all other polynomials in which it appears. Let be such that and . We can then eliminate the monomials and as follows:
The system has the property that if and only if , and thus, if and only if as required ∎
We now establish that the sufficient condition for infeasibility is equivalent to the combinatorial parity conditions in Theorem 2.1.
There exists a set of oriented partial 3-cycles and oriented chordless 4-cycles satisfying Conditions 1. and 2. of Theorem 2.1 if and only if .
Assume that . Then there exist coefficients such that . Let ; then, . Let be the set of oriented partial 3-cycles where together with the set of oriented chordless 4-cycles where . Now, is the number of polynomials in of the form (2) or (9) in which the monomial appears, and similarly, is the number of polynomials in of the form (2) or (9) in which the monomial appears. Thus, implies that, for every pair and , either
, , and or
, , and .
In either case, we have . Moreover, since , there must be an odd number of the polynomials of the form in . That is, case 2 above occurs an odd number of times and therefore, as required.
Conversely, assume that there exists a set of oriented partial 3-cycles and oriented chordless 4-cycles satisfying the conditions of Theorem 2.1. Let be the set of polynomials where and the set of polynomials where together with the set of polynomials where . Then, implies that every monomial appears in an even number polynomials of . Moreover, since , there are an odd number of polynomials appearing in . Hence, and . ∎
Combining Lemmas 2.4 and 2.5, we arrive at the characterization stated in Theorem 2.1. That such graphs can be decided in polynomial time follows from the fact that the existence of a certificate of any fixed degree can be decided in polynomial time (as is well known and follows since there are polynomially many monomials up to any fixed degree; see also [34, Theorem 4.1.3]).
Finally, we pose as open problems the construction of a variant of Theorem 2.1 for general -colorability and also combinatorial characterizations for larger certificate degrees .
Characterize those graphs with a given -colorability Nullstellensatz certificate of degree .
3 Recognizing Uniquely Hamiltonian Graphs
Throughout this section we work over an arbitrary algebraically closed field , although in some cases, we will need to restrict its characteristic. Let us denote by the Hamiltonian ideal generated by the polynomials from Proposition 1.3. A connected, directed graph with vertices has a Hamiltonian cycle if and only if the equations defined by have a solution over (or, in other words, if and only if for the algebraic variety associated to the ideal ). In a precise sense to be made clear below, the ideal actually encodes all Hamiltonian cycles of . However, we need to be somewhat careful about how to count cycles (see Lemma 3.8). In practice can be treated as a variable and not as a fixed primitive -th root of unity. A set of equations ensuring that only takes on the value of a primitive -th root of unity is the following:
We can also use the cyclotomic polynomial , which is the polynomial whose zeroes are the primitive -th roots of unity.
We shall utilize the theory of Gröbner bases to show that has a special (algebraic) decomposition structure in terms of the different Hamiltonian cycles of (this is Theorem 3.9 below). In the particular case when has a unique Hamiltonian cycle, we get a specific algebraic criterion which can be algorithmically verified. These results are Hamiltonian analogues to the algebraic -colorability characterizations of . We first turn our attention more generally to cycle ideals of a simple directed graph . These will be the basic elements in our decomposition of the Hamiltonian ideal , as they algebraically encode single cycles (up to symmetry).
When has the property that each pair of vertices connected by an arc is also connected by an arc in the opposite direction, then we call doubly covered. When is presented as an undirected graph, we shall always view it as the doubly covered directed graph on vertices with arcs .
Let be a cycle of length in , expressed as a sequence of arcs,
For the purpose of this work, we call a doubly covered cycle if consecutive vertices in the cycle are connected by arcs in both directions; otherwise, is simply called directed. In particular, each cycle in a doubly covered graph is a doubly covered cycle. These definitions allow us to work with both undirected and directed graphs in the same framework.
Definition 3.1 (Cycle encodings).
Let be a fixed primitive -th root of unity and let be a field with characteristic not dividing . If is a doubly covered cycle of length and the vertices in are , then the cycle encoding of is the following set of polynomials in :
If is a directed cycle of length in a directed graph, with vertex set , the cycle encoding of is the following set of polynomials:
Definition 3.2 (Cycle Ideals).
The polynomials are computationally useful generators for cycle ideals. (Once again, see  for the relevant background on Gröbner bases and term orders.)
The cycle encoding polynomials are a reduced Gröbner basis for the cycle ideal with respect to any term order with .
In particular, since reduced Gröbner bases (with respect to a fixed term order) are unique, it follows that cycle encodings are canonical ways of generating cycle ideals (and thus of representing cycles by Lemma 3.6).
Having explicit Gröbner bases for these ideals allows us to compute their Hilbert series easily.
The Hilbert series of for a doubly covered cycle or a directed cycle is equal to (respectively)
The naming of these ideals is motivated by the following result; in words, it says that the cycle is encoded as a complete intersection by the ideal .
The following hold for the ideal .
if is direct