The Fine-Grained Complexity of Computing the Tutte Polynomial of a Linear Matroid

03/07/2020
by   Andreas Björklund, et al.
Ericsson
aalto
0

We show that computing the Tutte polynomial of a linear matroid of dimension k on k^O(1) points over a field of k^O(1) elements requires k^Ω(k) time unless the #ETH—a counting extension of the Exponential Time Hypothesis of Impagliazzo and Paturi [CCC 1999] due to Dell et al. [ACM TALG 2014]—is false. This holds also for linear matroids that admit a representation where every point is associated to a vector with at most two nonzero coordinates. Moreover, we also show that the same is true for computing the Tutte polynomial of a binary matroid of dimension k on k^O(1) points with at most three nonzero coordinates in each point's vector. These two results stand in sharp contrast to computing the Tutte polynomial of a k-vertex graph (that is, the Tutte polynomial of a graphic matroid of dimension k—which is representable in dimension k over the binary field so that every vector has exactly two nonzero coordinates), which is known to be computable in 2^k k^O(1) time [Björklund et al., FOCS 2008]. Our lower-bound proofs proceed in three steps: (i) a classic connection due to Crapo and Rota [1970] between the number of tuples of codewords of full support and the Tutte polynomial of the matroid associated with the code; (ii) an earlier-established #ETH-hardness of counting the solutions to a bipartite (d,2)-CSP on n vertices in d^o(n) time; and (iii) new embeddings of such CSP instances as questions about codewords of full support in a linear code. We complement these lower bounds with a matching upper-bound algorithm design that computes the Tutte polynomial of a linear matroid of dimension k on k^O(1) points in k^O(k) arithmetic operations in the base field.

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1. Introduction

1.1. Matroids and the Tutte polynomial

A matroid is a tuple , where is a finite set of points, and is a nonempty set of subsets of called the independent sets of the matroid with the following two properties:

  1. every subset of an independent set is an independent set; and

  2. for any two independent sets and with , there exists an such that is an independent set.

Matroids generalize fundamental combinatorial and algebraic notions such as graphs and linear independence in vector spaces; for an introduction, cf. Welsh [28] and Oxley [22].

A matroid is linearly representable (briefly, linear) over a field if it can be described by a matrix of rank , where the number of rows is the dimension of the matroid, and the columns are indexed by the points of the matroid with . For any subset of the columns, let us write denote the matrix obtained by restricting to the columns indexed by . We write for the rank of over . The independent sets of a linear matroid are the sets for which

; that is, the subsets of linearly independent vectors. We say that a matroid is

binary if it is linearly representable over the two-element field.

The Tutte polynomial of a linear matroid is the integer-coefficient polynomial in two indeterminates any defined by

(1)

This generalisation of the Tutte polynomial from graphs to matroids was first published by Crapo [7], although it already appears in Tutte’s thesis; Farr [12] gives an historical account of the Tutte polynomial and its generalizations. Brylawski [6]—foreshadowed by Tutte [24, 25]—showed that the Tutte polynomial is a universal invariant for deletion–contraction recurrences, and thus captures a wealth of combinatorial counting invariants; cf. Biggs [3], Godsil and Royle [14], and Welsh [29] for a detailed account. Among these connections, most relevant to our present work is the connection of the Tutte polynomial to linear codes in coding theory, cf. Sect. 1.3 for a discussion.

In 2008, Björklund et al. [4] showed that if the matroid is graphic; that is, when the matrix is an incidence matrix of an undirected graph over the binary field, then the Tutte polynomial can be computed in time . Due to universality of the Tutte polynomial, it would be highly serendipitous to obtain a similar running time for a larger class of matroids.

1.2. Our results—fine-grained hardness of the Tutte polynomial

In this paper, we prove that such a running time for two natural ways of extending the graphic case to a larger class of linear matroids would have unexpected consequences in the fine-grained complexity of counting. Namely, we relate the complexity of computing Tutte polynomials of linear matroids to the Counting Exponential Time Hypothesis (#ETH)—cf. Sect. 2.2 for a precise statement—of Dell et al. [10], which relaxes the Exponential Time Hypothesis (ETH) of Impagliazzo and Paturi [16].

Our first main theorem shows that under #ETH one cannot extend the graphic case—that is, binary field with at most two nonzero entries in every column of —to moderately large field sizes without super-exponential scalability in .

Theorem 1 (Hardness of Tutte polynomial of a linear matroid under #ETH).

Assuming #ETH, there is no deterministic algorithm that computes in time the Tutte polynomial of a given linear matroid of dimension with points over a field of size . Moreover, this holds even when every column of has at most two non-zero entries.

Our second main theorem shows that under #ETH one cannot extend the graphic case to more general matrices even over the binary field without super-exponential scalability in .

Theorem 2 (Hardness of Tutte polynomial of a binary matroid under #ETH).

Assuming #ETH, there is no deterministic algorithm that computes in time the Tutte polynomial of a given linear matroid of dimension with points over the binary field. Moreover, this holds even when every column of the matrix has at most three non-zero entries.

We complement these hardness results with a deterministic algorithm design for linear matroids, but with super-exponential scalability in the dimension .

Theorem 3 (An algorithm for linear matroids).

There exists a deterministic algorithm that computes the Tutte polynomial of a given linear matroid of dimension with points over a -element field in time and space.

Previously, the hardness of the Tutte polynomial has been studied restricted to the graphic case from a number of angles, including the #P-hardness results of Jaeger, Vertigan and Welsh [18] (see also Welsh [29]), the counting inapproximability results of Goldberg and Jerrum [15], the fine-grained hardness results of Dell et al. [10] under #ETH, as well as the fine-grained dichotomy results of Brand, Dell, and Roth [5].

1.3. Key techniques—linear codes and sparse algebraic constraint satisfaction

Let us now give a high-level discussion of the key techniques employed. We proceed to prove Theorems 1 and 2 by utilizing known connections between linear codes and the Tutte polynomial. Towards this end, let us recall some basic terminology. A linear code of length and dimension over a finite field is a -dimensional subspace of the -dimensional vector space ; the elements of are called codewords. Such a code can be represented by a generator matrix of rank , with the interpretation that any linear combination with is a codeword of . The support of a codeword is the set of nonzero coordinates. For a nonempty set of codewords, the combined support is defined by . The combined support is full if .

Our two lower bounds use the following famous connection between the Tutte polynomial and code words of full combined support due to Crapo and Rota [8]:

Theorem 4 (The Critical Theorem; Crapo and Rota [8]).

Let be a positive integer and let be a linear code with a generator matrix . Then, the number of -tuples of codewords in with full combined support is .

Consider a linear code with generator matrix . Theorem 4 with implies that the number of codewords of with full support can be obtained as the evaluation of the Tutte polynomial at a single point. Our proof of Theorem 1 will crucially rely on this connection. In essence, the property of the codeword having full support corresponds to being a solution of a system of linear homogeneous inequations over , one inequation for each column of . Geometrically each such inequation can be viewed as a constraint that forces

to lie not on a particular hyperplane through the origin, and a system of such constraints forces

to lie properly inside a chamber of an arrangement of hyperplanes through the origin. The crux of our proof of Theorem 1 is to show via a sequence of lemmas that the task of computing the total volume of these hyperplane chambers is hard under #ETH, even in the case when every hyperplane is defined by a vector with at most two nonzero entries; this technical result may be of independent interest.

To prove Theorem 2, we will invoke Theorem 4 for larger values of the parameter to access the codewords of full support in an extension code. In more precise terms, let be a base code with generator matrix . For a positive integer , we obtain the extension code of the base code by embedding elementwise into to obtain the generator matrix of . Theorem 4 applied to the base code with this implies that the number of codewords of the extension code with full support can be obtained as the evaluation of the Tutte polynomial  of the base code at a single point. This is because for every -tuple of codewords in with full combined support and for , we can build a unique so that is a codeword of with full support. Indeed, can be represented as the polynomial quotient ring in the indeterminate , where is an irreducible polynomial of degree over , and we can build the scalars in this representation as for . This representation also shows that the reverse transform is possible: from every codeword of full support in , we can construct a unique -tuple of codewords in with full combined support. Hence, their cardinalities are the same. Thus, we can rely on a Tutte polynomial of the generator matrix of the base code to access the count of full-support codewords for the extension code. In particular, the base code can be over the binary field, which enables establishing hardness under #ETH for the binary field. The crux of our proof of Theorem 2 is to establish hardness under #ETH for systems of linear homogeneous sum-inequations with for all , even in the case when for at most three . In particular, sum-inequations are representable over the binary field, which enables our hardness reductions under #ETH as a sequence of lemmas culminating in Theorem 2.

Let us conclude this section with a brief discussion of related work and techniques. First, our combinatorial techniques on instances of constraint satisfaction problems are influenced by earlier hardness results, such as the seminal work of Traxler [23]. Similarly, the work of Kowalik and Socala [19] demonstrates how to bridge between combinatorial and sparse algebraic constraints in the form of generalized list colorings. Earlier work on -form tight lower bounds under ETH includes e.g. the work on Cygan et al. [9] on graph embedding problems. Finally, our present focus is on tuples of codewords of full support in a linear code via Theorem 4; dually, words of least positive support size determine the minimum distance of the code, a quantity which is also known to be hard to compute; cf. Vardy [26].

1.4. Organization

The rest of this paper is organized as follows. Section 2 proves our main lower-bound theorems, Theorem 1 and Theorem 2. Section 3 presents our upper-bound algorithm design with super-exponential scalability in the dimension.

2. Lower bounds

This section proves our two main lower-bound theorems, Theorem 1 and Theorem 2. We start with preliminaries on constraint satisfaction problems, the counting exponential time hypothesis and sparsification, and then proceed to develop the technical preliminaries and tools needed to transform combinatorial CSP instances into appropriately restricted algebraic versions that can then be accessed in a coding-theoretic context.

2.1. Constraint satisfaction problems

For nonnegative integers , , , and , a constraint satisfaction problem instance with parameters —or briefly, a -CSP instance—consists of variables and constraints such that

  1. associated with each variable , there is an at-most--element set , the domain of ; and

  2. associated with each constraint , there is an -tuple of distinct variables as well as a set of permitted combinations of values for the variables.

We say that the parameter is the domain size of the variables and the parameter is the arity of the constraints. We may omit the parameters and and simply refer to a -CSP instance if this is convenient.

We say that a -CSP instance is satisfiable if there exists a satisfying assignment such that for every it holds that assigns a permitted combination of values to the constraint —that is—we have ; otherwise, we say that is unsatisfiable. Let us write for the set of all satisfying assignments of .

Let us write -CSP for the task of deciding whether a given -CSP instance is satisfiable. Similarly, let us write -CSP for the task of counting the number of satisfying assignments to a given -CSP instance.

A constraint where all but one combination of values to the variables is permitted is called a clause. Instances consisting of clauses over variables with a binary domain are said to be in conjunctive normal form (CNF). We refer to instances in CNF with arity as -CNF, where the parameter is the length of the clauses.

2.2. The counting exponential-time hypothesis and sparsification

No efficient algorithm is known for solving constraint satisfaction problems in the general case. As such, we will establish our present hardness results under the following hypothesis of Dell et al. [10], which relaxes the Exponential Time Hypothesis of Impagliazzo and Paturi [16].

Hypothesis 5 (Counting exponential time hypothesis (#ETH); Dell et al. [10]).

There exists a constant such that there is no deterministic algorithm that solves a given -variable instance of -CNF in time .

We will also need a counting-variant of the Sparsification Lemma of Impagliazzo, Paturi, and Zane [17] due to Dell et al. [10] (see also Flum and Grohe [13]).

Lemma 6 (Counting sparsification; Dell et al. [10]).

For , there exists a computable function and a deterministic algorithm that, for and an -variable -CNF instance given as input, in time computes -CNF instances , each over the same variables and variable domains as , such that

  1. ;

  2. where the union consists of disjoint sets; and

  3. each variable occurs in at most clauses of .

2.3. Hardness of bipartite CSPs

It will be convenient to base our main hardness reductions on CSPs whose constraints have the topology of a bipartite graph. Towards this end, this section presents variants of well-known (e.g. cf. Traxler [23]) hardness reductions that have been modified to establish hardness in the bipartite case.

In more precise terms, let us say that a CSP instance with arity is graphic. Indeed, it is immediate that we can view the constraints of such an instance as the edges of a (directed) graph whose vertices correspond to the variables of the instance. We say that a graphic CSP instance is bipartite if this graph is bipartite.

Lemma 7 (Hardness of bipartite #CSP under #ETH).

Assuming , there is a constant such that there is no deterministic algorithm that solves a given bipartite -CSP instance in time .

Proof.

Let be the constant in Hypothesis 5 and let be a -variable instance of -CNF. Select a positive integer so that . Run the sparsification algorithm in Lemma 6 on to obtain in time the #-CNF instances with .

Let us transform into a bipartite -CSP instance with . Without loss of generality we may assume that every variable occurs in at least one clause. Let us assume that consists of clauses over variables with domains , respectively. By Lemma 6, we have . Let us write the support of and for the permitted values of .

The construction of is as follows. For each clause with , introduce a variable with domain into . For each variable with , introduce a variable with domain into . For each clause with and each , introduce a constraint with support and permitted combinations into . In total thus has variables and constraints. It is also immediate that has domain size , arity , and bipartite structure as a graph. Furthermore, since every variable of occurs in at least one clause, it is immediate that there is a one-to-one correspondence between and . The transformation from to is clearly computable in time .

To reach a contradiction, suppose now that there is a deterministic algorithm that solves a given bipartite -CSP instance in time for a constant with . Then, we could use this algorithm to solve each of the instances for in total time for a constant . But since , this means that we could solve each of the instances , and thus the #3-CNF instance by Lemma 6, in similar total time, which contradicts Hypothesis 5. ∎

The next lemma contains a well-known tradeoff that amplifies the lower bound on the running time by enlarging the domains of the variables.

Lemma 8 (Hardness amplification by variable aggregation under #ETH).

Assuming ETH, there is no deterministic algorithm that solves a given bipartite -CSP instance in time .

Proof.

We establish hardness via Lemma 7. Let be a bipartite

-CSP instance. Without loss of generality—by padding with extra variables constrained to unique values—we may assume that (i) the variables of

are , (ii) every constraint of has support of the form for some , and (iii) . Let be a constant whose value is fixed later and let . Group the variables into pairwise disjoint sets of at most variables each. Similarly, group the variables into pairwise disjoint sets of at most variables each.

Let us construct from a bipartite CSP instance with as follows. The variables of are and so that the domain of each variable is the Cartesian product of the domains of the underlying variables of . The constraints of are obtained by extension of the constraints of as follows. For each constraint with support in , let and be the unique indices with and , and introduce a constraint with support into ; set the permitted values of this constraint so that they force a permitted value to the variables and as part of the variables and but otherwise do not constrain the values of and . This completes the construction of . It is immediate that is bipartite and that holds. Furthermore, has variables, each with domain size at most , and constraints; that is, constraints. Choosing , we have for all large enough . The transformation from to is clearly computable in time .

To reach a contradiction, suppose now that there is a deterministic algorithm that solves a given bipartite -CSP instance in time . Then, we could use this algorithm to solve , and hence by , in time , which contradicts Lemma 7. ∎

2.4. Linear inequation systems and chambers of hyperplane arrangements

We are now ready to introduce our main technical tool, namely CSPs over finite fields whose constraints are of a special geometric form. (For preliminaries on finite fields, cf. e.g. Lidl and Niederreiter [20].) More precisely, let us write for the finite field with elements, a prime power, and let be variables taking values in . For , , and , we say that the constraint

(2)

is a (linear) inequation of arity (or weight) . We say that the inequation is homogeneous if and inhomogeneous otherwise. We say that the inequation is a sum-inequation if for all we have .

Previously, the complexity of inequations of low arity has been studied for example by Kowalik and Socala [19] under the terminology of generalized list colorings of graphs. We also remark that for one can view (2) geometrically as the constraint that a point does not lie in the hyperplane defined by the coefficients and ; accordingly, a system of constraints of this form is satisfied by a point if and only if lies properly inside a chamber of the corresponding hyperplane arrangement, and the task of counting the number of such points corresponds to determining the total volume of the chambers in . (Cf. Orlik and Terao [21], Dimca [11], and Aguiar and Mahajan [1] for hyperplane arrangements.)

Here our objective is to establish that systems of inequations are hard to solve under #ETH already in the homogeneous case and for essentially the smallest nontrivial arity, using our preliminaries on bipartite CSPs to enable the hardness reductions. We start with inequations of arity two in the following section, and proceed to sum-inequations of arity three in the next section.

2.5. Homogeneous inequation systems of arity two

Our first goal is to show that counting the number of solutions to a homogeneous inequation system of arity two over a large-enough field is hard under #ETH.

It will be convenient to start by establishing hardness of modular constraints of arity two, and then proceed to the homogeneous case over by relying on the cyclic structure of the multiplicative group of . The modular setting will also reveal the serendipity of our work with bipartite CSPs. Towards this end, let , , and be variables taking values in , the integers modulo . We say that an inequation of arity two over is special modular if it is one of the following forms: (i) , (ii) , or (iii) for and . A CSP instance over is special modular if all of its constraints are special modular.

Lemma 9 (Hardness of special modular systems under #ETH).

Assuming ETH, there is no deterministic algorithm that in time solves a given special modular #-CSP instance over with .

Proof.

We establish hardness via Lemma 8. Let be a bipartite -CSP instance. Without loss of generality—by padding with extra variables constrained to unique values—we may assume that the variables of are and every constraint of has support of the form for some . Furthermore, by relabeling of the domains as necessary, we can assume that all variables have domain and all variables have domain with . Let us now construct a special modular CSP instance as follows. Let . Introduce the variables , , and into so that each variable has domain . For each , force modulo by introducing special modular constraints of type (ii) into . For each , force modulo by introducing special modular constraints of type (iii) into . We observe that the introduction of these constraints into forces that for all we have modulo , and the values of and modulo are uniquely determined by the difference modulo . Finally, for each constraint of with support of the form for some , use at most special modular constraints of type (i) to force the values of and to the permitted pairs of values. It is immediate that ; indeed, each satisfying assignment to corresponds to exactly satisfying assignments to , one for each possible choice of value to . Furthermore, is computable from in time . We also observe that has variables, constraints, domain size , and arity .

To reach a contradiction, suppose now that there is a deterministic algorithm that in time solves a given special modular #-CSP instance over with . Then, we could use this algorithm to solve , and hence by , in time , which contradicts Lemma 8. ∎

We are now ready to establish hardness of homogeneous inequation systems of arity two over for large-enough . For arithmetic in , we tacitly assume an appropriate irreducible polynomial and a generator for the multiplicative group of are supplied as part of the input. (For algorithmics for finite fields, cf. e.g. von zur Gathen and Gerhard [27].)

Lemma 10 (Hardness of homogeneous inequation systems of arity two under #ETH).

Assuming ETH, there is no deterministic algorithm that in time solves a given #-CSP instance with the structure of a homogeneous inequation system over with .

Proof.

We proceed via Lemma 9. Let be a special modular #-CSP instance with variables taking values in for . Let us construct a homogeneous inequation system over with as follows. Let be a generator for the multiplicative group of . Introduce into the variables , , and , each taking values in . Introduce the homogeneous inequations , , and for all into . By the cyclic structure of the multiplicative group of , we have that to arbitrary nonzero values of the variables , , in , there correspond unique integers , , modulo such that , , and for all . Furthermore, under this correspondence, each special modular constraint over corresponds to the homogeneous inequation of arity over . The special modular constraints and have similar correspondence. We can thus complete the construction of by inserting the constraints corresponding to the constraints of into ; in particular, we have . The transformation from to is clearly computable in time . It thus follows from Lemma 9 that, assuming #ETH, there is no deterministic algorithm that in time solves a given #-CSP instance with the structure of a homogeneous inequation system over with . ∎

2.6. Homogeneous sum-inequation systems of arity three

We now proceed to look at homogeneous inequation systems with -coefficients on the variables; that is, we establish under #ETH the hardness of counting the number of solutions to a homogeneous sum-inequation system of low arity. Bipartiteness in the input of the reduction will again be serendipitous in achieving low arity.

We will require the following preliminaries on sets with additive structure. For an Abelian group , we say that a subset is a Sidon set if for any of which at least three are different, it holds that . An Abelian group is elementary Abelian if all of its nontrivial elements have order for a prime . The additive group of a finite field is elementary Abelian.

Lemma 11 (Existence of Sidon sets; Babai and Sós [2, Corollary 5.8]).

Elementary Abelian groups of order have Sidon sets of size .

We are now ready for the main result of this section.

Lemma 12 (Hardness of homogeneous sum-inequation systems of arity three under #ETH).

Assuming ETH, there is no deterministic algorithm that in time solves a given #-CSP instance with the structure of a homogeneous sum-inequation system over with .

Proof.

We proceed via Lemma 8. Let be a bipartite -CSP instance. Without loss of generality—by padding with extra variables constrained to unique values—we may assume that the variables of are and every constraint of has support of the form for some . Furthermore, by relabeling of the domains as necessary, we can assume that all variables and have domain with .

Let us construct a homogeneous sum-inequation system over with as follows. Introduce the variables , , , , , and , each taking values over , into . In total there are thus variables.

We introduce six different types of homogeneous sum-inequations into . Let be an arbitrary but fixed injective map.

First, inequations of type (i) force the variables , , to take pairwise distinct values; this can be forced with homogeneous sum-inequations of arity .

Second, inequations of type (ii) force the variables to take pairwise distinct values; this can be forced with homogeneous sum-inequations of arity .

Third, for each , we force the equality by introducing homogeneous sum-inequations —let us call these inequations of type (iii)—one inequation for each .

Fourth, inequations of type (iv) force the variables to take values in the set of values of the variables ; together with (i), this can be forced with homogeneous sum-inequations and for all , , and .

Fifth, inequations of type (v) force the variables to take values in the set of values of the variables ; together with (i), this can be forced with homogeneous sum-inequations and for all , , and .

Sixth, for each constraint with support in for some , and letting be the set of permitted values for the constraint, introduce the homogeneous sum-inequations for each ; let us call these inequations of type (vi).

This completes the transformation from to , which is clearly computable in time . We observe that has domain size , arity , variables, and constraints.

Next we claim that for all large enough we have for a positive-integer-valued function of the parameters . Indeed, let be the total number of solutions to the system of inequations consisting of the variables , , , and all the inequations of types (i), (ii), and (iii). Recalling that , from Lemma 11 we have that for all large enough the additive group of contains a Sidon set of size . Assign each element of this Sidon set to exactly one of the variables to conclude that the sums are distinct for all . Assign the remaining variables to distinct values in one of the possible ways to conclude that . Fix one of the solutions. Inequations of type (iv) are by definition satisfied if and only if for all we have that takes a value in the set of values for . Similarly, inequations of type (v) are by definition satisfied if and only if for all we have that takes a value in the set of values for . Consider any such assignment to and for . Suppose that and for . Then, since inequations of type (iii) are satisfied. Suppose now has a constraint with support and permitted values . By construction, we have that the inequations of type (vi) originating from this constraint are satisfied if and only if . Thus, we have as claimed.

To reach a contradiction, suppose that there is a deterministic algorithm that in time solves a given #-CSP instance with the structure of a homogeneous sum-inequation system over with . Let be a bipartite -CSP instance and take . First, use the assumed algorithm to the system of inequations consisting of the variables , , , and all the inequations of types (i), (ii), and (iii). The algorithm returns as the solution. Then, construct from and use the algorithm on to get as the solution. Divide by to obtain . Since the total running time of this procedure is , we obtain a contradiction to Lemma 8. ∎

We are now ready to complete our proofs of Theorem 1 and Theorem 2.

2.7. Proof of Theorem 1

We will rely on Lemma 10 and Theorem 4. Let be #-CSP instance with the structure of a homogeneous inequation system over with . Take and construct a matrix so that each column of corresponds to a unique homogeneous inequation of ; in particular, every column of has at most two nonzero entries. For all we have that has full support if and only if . Theorem 4 with thus implies that . Since , we have . Furthermore, . An algorithm that computes the Tutte polynomial in time would thus enable us to compute in time and thus contradict Lemma 10 under #ETH.

2.8. Proof of Theorem 2

We will rely on Lemma 12 and Theorem 4. Let be a #-CSP instance with the structure of a homogeneous sum-inequation system over with .

Construct a matrix with so that each column of corresponds to a unique sum-inequation of ; in particular, every column of has at most three nonzero entries. Recalling the construction in Sect. 1.3, extend elementwise from to to obtain . For all we have that