We study the problem of approximating the matching polynomial of a graph. This polynomial has a parameter , called the edge activity. A matching of a graph is a set such that each vertex is contained in at most one edge in . We denote by the set of all matchings of . The matching polynomial is given by
This polynomial is also referred to as the partition function of the monomer-dimer model in statistical physics.
Here is what is known about approximating this polynomial. We first describe the case where is positive and real. This is a natural case, and is the case where the first complexity-theoretic results were obtained. We next describe the more general case, where is a complex number. There are many reasons for considering the more general case. The parameter is defined to be complex, rather than real, in the classic paper of Heilmann and Lieb [Heilmann1972]. Furthermore, it has recently been shown [physics] that the quantum evolution of a system originally in thermodynamic equilibrium is equivalent to the partition function of the system with a complex parameter. As [physics] explains, recent discoveries in physics make it possible to study thermodynamics in the complex plane of physical parameters — so complex parameters are increasingly relevant. As we will see in this paper, it is beneficial to study partition functions with complex parameters even when one is most interested in the real case — the reason is that the generalisation sheds light on “what is really going on” with complexity bottlenecks, and on appropriate potential functions. Here is the summary of known results in both cases.
When the edge activity is a positive real number: For any positive real number , Jerrum and Sinclair [JSPermanent, Corollary 4.4] gave an FPRAS for approximating . Using the correlation decay technique, Bayati et al. [Bayati] gave a (deterministic) FPTAS for the same problem for the case in which the degree of the input graph is at most a constant .
When the edge activity is a complex number: Known results are restricted to the case where is not a real number less than or equal to . In this case, there is a positive result, due to Patel and Regts [PR]. Using a method of Barvinok [BarvinokPermanent, barvinokbook] for approximating a partition function by truncating its Taylor series (in a region where the partition function has no zeroes), Patel and Regts [PR, Theorem 1.2] extended the positive result of Bayati et al. to the case in which is a complex number that is not a negative real that is less than , see also [barvinokbook, Section 5.1.7]). Patel and Regts obtained a polynomial time algorithm (rather than a quasi-polynomial time one) by developing clever methods for exactly computing coefficients of the Taylor series.
Our first contribution completes this picture by showing that for all and all real it is actually #P-hard to approximate on graphs with degree at most . We use the following notation to state our result more precisely. We consider the problems of multiplicatively approximating the norm of , and of computing its sign. Our first theorem shows that, for all and all rational numbers , it is -hard to approximate on bipartite graphs of maximum degree within a constant factor.
Let and be a rational number. Then, it is -hard to approximate within a factor of 1.01 on graphs of maximum degree , even when restricted to bipartite graphs with .
The number 1.01 in Theorem 1 is not important. It can be replaced with any constant greater than . In fact, for any fixed , the theorem, together with a standard powering argument, shows that it is #P-hard to approximate within a factor of .
Our second theorem shows that it is -hard to compute the sign of on bipartite graphs of maximum degree .
Let and be a rational number. Then, it is -hard to decide whether on graphs of maximum degree , even when restricted to bipartite graphs with .
We next explore whether the bound on the maximum degree of can be relaxed to a restriction on average degree. The notion of average degree that we use is the connective constant. Given a graph , and a vertex , let be the number of -edge paths in that start from . The following definition is taken almost verbatim from [connconst2, connconst1].111The only difference between Definition 3 and the corresponding definitions in [connconst2, connconst1] is the addition of the terminology “profile ” which will be used to state our hardness results in a strong form (the results in [connconst2, connconst1] were algorithmic which is why this handle on the constants and was not required).
Definition 3 ([connconst2, connconst1]).
Let be a family of finite graphs and let , and be positive real numbers. The connective constant of is at most with profile if, for any graph in and any vertex in , it holds that for all .
Sinclair, Srivastava, Štefankovič and Yin [connconst2, Theorem 1.3] showed that, for fixed , when is a positive real, the correlation decay method gives an FPTAS for approximating on graphs with connective constant at most (without any bound on the maximum degree of ). The run-time of their algorithm is , where is the number of vertices of and is the relative error.
Our next result shows that, in striking contrast to the bounded-degree case, the algorithmic result of Sinclair et al. cannot be extended to negative reals, even if . Given positive real numbers and and a real number , let be the set of graphs with connective constant at most and profile .
There exist a dense set of values on the negative real axis such that the following holds for any real numbers and all .
It is #P-hard to approximate within a factor 1.01 on graphs ,
it is #P-hard to decide whether on graphs .
Both of these results hold even when restricted to bipartite graphs with .
The algorithmic contribution of our paper is to show that, despite the hardness result of Theorem 4, correlation decay gives a good approximation algorithm for any complex value that does not lie on the negative real axis when the input graph has bounded connective constant. It is interesting that we are able to use correlation decay to get a good approximation. for all non-real complex values . Our result is the only known approximation in this setting. In particular, it is not known how to obtain such a result using the method of Patel and Regts [PR]. In order to describe our result, we use the following notation. Given a complex number , let denote the principal value of its argument in the range and denote its norm. Our result is the following.
Let , and be positive real numbers and let be any fixed edge activity. Then there is an algorithm which takes as input an -vertex graph and a rational and produces an output for some complex number with . The running time of the algorithm is where and .
Let , and be positive real numbers and let be any fixed edge activity. Then, for any rational and any positive rational , there are polynomial-time algorithms to take as input a graph and approximate within a multiplicative factor of and within an additive error .
In order to prove Theorem 5, showing correlation decay for complex , we use geodesic distances in the complex plane in the metric defined by an appropriate density function. Correlation decay for complex activities has been analysed in the context of the hard-core model (see Harvey, Srivastava and Vondrák [Piyush])222Note that Harvey et al were actually working with the mutivariate hard-core polynomial – this causes interesting complications which will not be relevant for this paper. They also extend their method (for the hard-core polynomial, in their region) to graphs of unbounded degree that have bounded connective constant.. The region in the complex plane in which the authors of [Piyush] worked allowed them to measure distances using the norm instead of requiring geodesic distances. An alternative approach was given by Peters and Regts [Peters], again in the context of the hard-core model, where they showed contraction within the basin of an attracting fixpoint using the theory of complex dynamical systems.
Let be a complex number and be an arbitrary graph. Recall that is the set of matchings of . For a matching , we denote by the set of matched vertices in the matching . For a vertex in , we also define
Thus, is the contribution to the partition function from those matchings such that is matched in , while is the contribution to the partition function from those matchings such that is not matched in .
We will use the following result about the location of the zeroes of the matching polynomial.
Theorem 7 ([Heilmann1972], see, e.g., [barvinokbook, Theorem 5.1.2]).
Let be an integer and be a graph of maximum degree . Then, for all complex that do not lie on the interval of the negative real axis, it holds that .
Let be an integer and be a real number. Then, for all graphs of maximum degree it holds that .
For our approximation algorithm of Theorem 5, given a graph with and a vertex , we will be interested in the quantity
The algorithm will be based on the following result by Godsil.
Theorem 9 ([godsil81]).
Let . Let be a graph and let be one of its vertices. Let be the self-avoiding walk tree of rooted at . Then,
3 FPTAS for graphs with bounded connective constant
In this section, we prove Theorem 5. Consider .
We will use the correlation decay technique of Weitz [Weitz], which we adapt for use with complex activities. We review the basic idea behind the technique (see, e.g., [Bayati, connconst1, connconst2]). For a graph (of bounded connective constant), we first express as a telescoping product
where is an arbitrary enumeration of the vertices of the graph and is the graph obtained from by deleting the vertices . In light of (1), we can therefore focus on approximating the value for a graph and vertex . Using Godsil’s Theorem (cf. Theorem 9), it in turn suffices to approximate . This might seem as a somewhat simpler task given that is a tree; the caveat however is that the tree is prohibitively large, so in order to be able to perform computations efficiently we need to truncate the tree. The correlation decay technique analyses the approximation error introduced by this truncation process by recursively tracking the error using tree recurrences.
In the case of matchings, for a tree and a vertex in , we can write a recursion for as follows. If is the only vertex in , then (since the only possible matching is the empty set and thus ). Otherwise, let be the trees of and let be the neighbours of in , respectively. Then, we have that
Hence, we need to evaluate the recurrence
with base case .
To show the decay of correlations, one wants to show that after applying the recurrence starting from two different sets of values at , the two computed values at will be “closer” than were the initial values at the ’s. This leads us to define a notion of distance. Often straightforward distances do not suffice to show decay of correlations, and distances defined via a “potential” function are used. We adapt this notion to the complex plane.
3.1 Metrics for measuring the error in the complex plane
We use a distance metric based on conformal density functions (see [Conformal] for details).
Definition 10 (Length, Distance, Metric).
Let be a simply connected open subset of and let be a function (called conformal density). The length with respect to of a path333Following [Conformal], paths are assumed to be continuous and piecewise continuously differentiable. is defined as
The distance with respect to between two points , denoted , is the infimum of the lengths of the paths connecting to (that is, and ). We will refer to the metric induced by the distance function as the (conformal) metric given by .
Suppose is the right-complex half-plane, that is iff , and . The metric is the Poincaré metric in the half-plane (usually one takes the upper-complex half-plane) and the distance between and is
We first quantify one-level correlation decay.
Let be a simply connected open subset of , be a conformal density function, and be the metric given by . Let and be conjugate exponents, that is, , where .
Suppose that is an integer and is a holomorphic map. Let and and let and . Assume that there exists a real such that for any
Let . For , let be a path connecting to of length . W.l.o.g. is re-parameterized to uniform speed, that is, for a.e. we have
We now define a path connecting to :
Let denote the length of and denote the function . Then, using the triangle inequality and (5), we have
By Hölder’s inequality and condition (3), for any , we have
Integrating this for between 0 and 1 and combining with (6), we obtain
Taking we obtain
Now, given a rooted tree, our goal will be to bound the correlation decay at the root when we truncate the tree at depth . Let be a finite tree rooted at a vertex and let be a subset of the leaves of . Let . We will have a family of maps where will be a symmetric map of arity (which will be the recurrence applied to a vertex of the tree with children). Let be an arbitrary assignment of values in to the vertices of . Let also be the “initial” value ( corresponds to the starting point of the recurrences). For a vertex in and an initial value , we define the quantity recursively as follows.
We can now study the sensitivity of to the assignment . The following lemma is the analogue of [connconst1, Lemma 3] for the complex plane and will be used to apply the correlation decay technique for graphs of bounded connective constant.
Let be a simply connected open subset of and be a conformal density function. For let be symmetric holomorphic maps. Suppose that there exists a real and conjugate exponents and such that for every integer and all it holds that
Then, the following holds for any initial value and any finite tree rooted at .
Let be a subset of the leaves of and consider two arbitrary assignments and . Then
where , and is the distance of from the root .
The proof is close to [connconst1, Proof of Lemma 3], the only difference is we have to use the metric induced by , which we denote . In particular, for an arbitrary vertex in , we use to denote the subset of that belongs to the subtree of rooted at . Then, we will show that
We show this by induction. When is a leaf of and , we have that and (9) holds trivially. When , then , and , so (9) holds by the definition of . For the inductive case, suppose that neither is a leaf of nor belongs to and that (9) holds for the children of . For , set , and observe that
By the inductive hypothesis, we also have that
proving (9). Notice that for any we have
The lemma follows from this and (9) (applied to ). ∎
3.2 Applying the method for matchings
Suppose that . We will parameterise as
Note that, in the choice of , we used the assumption that is not a negative real number. Let be the right complex half-plane, that is, the set of complex such that , and note that . We will also transform the space in which the quantities live using the map . In the transformed space, the recurrence (2) becomes
where if corresponds to a leaf then (we refer to this as the initial ). Let
The following lemma shows that the set is closed under application of the recurrence (11).
Suppose that and . Then, for given by (11), we have that as well. In fact, we have that .
Since , we have that . Using that , we have that and therefore . This yields that . Moreover, using again that and , we have
and hence . It follows that .
To prove the stronger bound on , note by the triangle inequality that
We next go on to show the required contraction properties for an appropriate function . This will largely be based on the following lemma from [connconst2].
Let and be positive real numbers. For , let . Let , and . Then, for arbitrary it holds that
The lemma follows from the derivations in [connconst2] as follows. Let as defined in equation (2) of [connconst2]. Then, Lemma 7 (see also Definition 10) of [connconst2] shows that the expression
constrained to for any fixed , is maximized for and for , for some .
Then, (14) can be bounded from above by
where the univariate . From Lemma 9 (see also Definition 11) of [connconst2] we get that for all ,
where the left-hand side is maximized for . So,
Plugging this bound and the bound obtained in (15) into the expression from the lemma, we get
It remains to prove that the right-hand side is equal to , which will finish the proof. To see this, we use :
which is equivalent to the right-hand side of (16). ∎
Using Lemma 15, we can obtain the following in the complex plane.
Then, the following holds for all integer .
Consider the map given by . Then, for arbitrary we have
Note that , so we obtain that (18) is equivalent to
We have and therefore