Faster Algorithms via Waring Decompositions

We show that decompositions of certain polynomials as sums of powers of linear forms yield faster algorithms for some algebraic problems. Such a decomposition is known as a Waring decomposition. Our results are: 1. We give a black-box algorithm for computing the sum of the coefficients of the degree-k multilinear monomials in a polynomial over a field of characteristic zero. Our algorithm runs in time O^*(n^k/2) and space poly(n). This solves an open problem of Koutis and Williams [1]. 2. We give a randomized O^*((3e/2)^k) ≈ O^*(4.08^k) time, polynomial space, black-box algorithm for detecting multilinear monomials of degree k over a field of characteristic zero. This improves on the O^*(4.32^k) time, exponential space algorithm given in [2]. We note that there is a O^*(2^k) lower bound on our approach.

Authors

• 4 publications
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Black Box Absolute Reconstruction for Sums of Powers of Linear Forms

We study the decomposition of multivariate polynomials as sums of powers...
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Rigid continuation paths II. Structured polynomial systems

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1 Introduction

The Waring rank of a homogeneous -variate degree- polynomial , denoted , is the minimum such that

 f=ℓd1+⋯+ℓdr, (1)

for some linear forms . The study of Waring rank is a classical problem in algebraic geometry and invariant theory, with pioneering work done in the second half of the 19th century by A. Clebsch, J.J. Sylvester, and T. Reye, among others [IK99, Introduction]. It has enjoyed a recent resurgence of popularity within algebraic geometry [IK99, Lan12] and has connections in computer science to the limiting exponent of matrix multiplication [CHI18], the Mulmuley-Sohoni Geometric Complexity Theory program [BIP19], and several other areas in algebraic complexity [Lan17, EGOW18]. This paper adds parameterized algorithms to this list, showing that several methods in this area (color-coding methods [AYZ95, AG07, HWZ08], the group-algebra/determinant sum approach [Kou08, Wil09, Bjö10], and inclusion-exclusion methods) fundamentally result from rank upper bounds for a specific family of polynomials. In a situation analogous to that of , better explicit upper bounds on the Waring rank of these polynomials yield faster algorithms for certain problems in a completely black-box manner, and lower bounds on the Waring rank of these polynomials imply barriers such algorithms face.

This connection should not come as a complete surprise, as many algorithms work by solving a question about the coefficients of some efficiently-computable “generating polynomial” determined by the input. The insight of this paper, which has been largely unexploited, is that in general this is a question about Waring rank.

Let denote the elementary symmetric polynomial of degree in variables. We will study the following questions:

Question 1.

What is , the minimum Waring rank among all with the property that ?222Here .

Question 2.

What is , the the minimum Waring rank among all with the property that ?

Question 3.

For , what is , the minimum Waring rank among all with the property that and the nonzero coefficients of are in the range ?

We now illustrate the algorithmic relevance of these questions with a new and very simple -time and -space algorithm for exactly counting simple cycles (i.e., closed walks with no repeated vertices) of length in an -vertex graph. This is the fastest polynomial space algorithm for this problem, improving on a -time algorithm of Fomin et al. [FLR12] which in turn improved on a -time algorithm of Vassilevska Williams and Williams [VW09].

Given a directed graph , let be the symbolic matrix with entry equal to the variable if there is an edge from vertex to vertex , and zero otherwise. By the trace method,

Now we denote by the partial differential operator . The algorithm is based on two easy observations:

Observation 4.

The number of simple cycles of length in equals .

Observation 5.

If , where for , then for all ,

 g(∂x)f=d!r∑i=1aif(ci,1,…,ci,n).

It is immediate that we can compute the number of simple cycles in of length using evaluations of . Now, it was recently shown in [Lee16] that

 R(en,d)≤(n≤⌊d/2⌋)\coloneqq⌊d/2⌋∑i=0(ni).

Explicitly, for and , define the indicator function if , and otherwise. Then for odd,

 2d−1d!⋅en,d=∑S⊂[n]|S|≤⌊d/2⌋(−1)|S|(n−⌊d/2⌋−|S|−1⌊d/2⌋−|S|)(δS,1x1+δS,2x2+⋯+δS,nxn)d.

(A similar formula holds for even.) It follows that the number of length- simple cycles in equals

 12d−1∑S⊂[n]|S|≤⌊d/2⌋(−1)|S|(n−⌊d/2⌋−|S|−1⌊d/2⌋−|S|)fG(δS,1,…,δS,n). (3)

This gives a closed form for the number of length- simple cycles in that is easily seen to be computable in the stated time and space bounds. This algorithm is much simpler, both computationally and conceptually, than those of previous approaches.333We note that the use of inclusion-exclusion (or “Möbius inversion” [Ned09]) in numerous exact-counting algorithms, such as Ryser’s formula for computing the permanent [Rys64] and algorithms for counting Hamiltonian cycles [KGK77] and set packings [BH06], implicitly relies on a natural but suboptimal bound on ; namely the one given by Equation 5 below. We elaborate on this in Example 15.

The above argument shows something very general: given as a black-box, we can compute (that is, the sum of the coefficients of the multilinear monomials in ) using queries. This answers a “significant” open problem asked by Koutis and Williams [KW09] in a completely black-box way.444An alternate solution to this problem was given contemporaneously in [ACDM18]. Moreover, it follows from a special case of our Theorem 6 that any algorithm must make [Lee16] queries to compute in the black-box setting:

Theorem 6.

Fix and let be given as a black-box. The minimum number of queries to needed to compute is , assuming unit-cost arithmetic operations.

In light of this lower bound, one might next ask for a ) approximation of . This prompts our main algorithmic result, which is based on an answer to Question 3:

Theorem 7.

Let be given as a black-box. There is a randomized algorithm which given any computes a number

such that with probability 2/3,

 (1−ε)⋅en,d(∂x)f

This algorithm runs in time and uses space. Here is the maximum bit complexity of on the domain .

The algorithm and the proof behind Theorem 7 are simple and can be found in Section 4. Applying this theorem to to the graph polynomial , an algorithm for approximately counting simple cycles of length is immediate.555In fact, Theorem 7 gives the fastest polynomial space algorithm for approximately counting cycles that we are aware of. More generally, we have the following:

Theorem 8.

Let and be graphs where , , and has treewidth . There is a randomized algorithm which given any computes a number such that with probability ,

 (1−ε)⋅Sub(H,G)

This algorithm runs in time . Here denotes the number of subgraphs of isomorphic to .

This is the fastest known algorithm for approximately counting subgraphs of bounded treewidth, improving on a -time algorithm of Alon et al. [ADH08] which in turn improved on a -time algorithm of Alon and Gutner [AG07]. The first parameterized algorithm for a variant of this problem was given by Arvind and Raman [AR02] and has runtime . In the case that has pathwidth , a recent algorithm of Brand et al. [BDH18] runs in time . We stress that this application is only a motivating example – Theorem 7 is extremely general and also applies to counting set partitions and packings [BH06], dominating sets [KW09], repetition-free longest common subsequences [BBDS12], and functional motifs in biological networks [GS13].

In the rest of this section we outline our approach. This will suggest a path to derandomize and improve the base of the exponent in Theorem 7 (and hence Theorem 8) from to . Specifically, we raise the following question:

Question 9.

Is ?

Prior to this work it was believed [KW15] that a derandomization of polynomial identity testing would be needed to obtain, for instance, a deterministic -time algorithm just for detecting simple paths of length in a graph. On the contrary, an explicit affirmative answer to the above question would give a -time deterministic algorithm for approximately counting simple paths.

Remark 10.

A focus on approximating in the case that and are real stable has recently led to several advances in algorithms and combinatorics; see e.g. [Gur06]. In particular, a result of Anari et al. [AOGSS17] shows that in this case can be approximated (up to a factor of ) deterministically in polynomial time given black-box access to . This paper shows that the general (i.e., unstable) case raises interesting questions as well.

1.1 Our Approach and Connections to Previous Work

To continue with the previous example, note that the graph polynomial is supported on a multilinear monomial if and only if contains a cycle of length . This motivates the following problem of well-recognized algorithmic importance [Gur04, Kou08, Wil09]:

Problem 11.

Given black-box access to over , decide if is supported on a multilinear monomial.

It is not hard to see that any algorithm for computing , where is supported on exactly the set of degree- multilinear monomials, can be used to solve Problem 11 with one-sided error (Proposition 22). This suggests studying upper bounds on (Question 1) as an approach to solve Problem 11. Perhaps surprisingly though, it turns out that several known methods in parameterized algorithms can be understood as giving constructive upper bounds on , and better upper bounds to would improve upon these methods. For example, the seminal color-coding method of Alon, Yuster, and Zwick [AYZ95] can be recovered from an upper bound on of , and an improvement to color-coding given by Hüffner et al. [HWZ08] follows from an upper bound on of (Remark 59). The group-algebra/determinant sum approach of [Wil09, Kou08, Bjö10] reduces to answering a generalization of Question 1 (see Definition 48) in the case that the underlying field is not but of characteristic 2. (In Theorem 52 we give the essentially optimal upper bound of for this variant, which in turn can be used to recover [Wil09, Kou08, Bjö10]). Prior to this work, no connection of this precision between these methods was known.

Question 1 provides insight into lower bounds on previous methods as well. For example, the bounds on given in [Lee16] directly yield asymptotically sharper lower bounds than those given by Alon and Gutner [AG09, Theorem 1] on the size of perfectly balanced hash families used by exact-counting color-coding algorithms (Theorem 72). Curiously, this improvement is ultimately a consequence of Bézout’s theorem in algebraic geometry. Question 1 and a classical lower bound on Waring rank (Theorem 16) explain why disjointness matrices arose in the context of lower bounds on color-coding [AG09] and the group-algebra approach [KW09]: they are the partial derivatives matrices of the elementary symmetric polynomials.

Our main answers to Question 1 are the following. By our Theorems 58, 41 and 28, it follows that

 2d−1≤A(n,d)≤min(6.75d,O(4.075dlogn)).

Perhaps surprisingly, this gives an upper bound on independent of . On the negative side, our lower bound on rules out Question 1 as an approach to obtain algorithms faster than for Problem 11; moreover, we show in Theorem 24 that there is also a lower bound of on the number of queries needed to solve Problem 11 with one-sided error.

It is easily seen by creftypecap 5 that constructive upper bounds on yield deterministic algorithms for determining if is supported on a multilinear monomial in the case that has nonnegative real coefficients (as, e.g., the graph polynomial has), and constructive upper bounds on yield deterministic algorithms for approximating . This broadly generalizes the use of color-coding in designing approximate counting and deterministic decision algorithms.

Our bounds on also hold for . Remarkably, we show in Example 67 that if then . It follows from our Theorem 28 and Theorem 58 that

 2d−1≤Aε(n,d)≤O(4.075dε−2logn),

and from our Corollary 36 that for all and – unlike , depends on . As an aside, it is immediate that

 R––(en,d)≤limε→0Aε(n,d)≤R(en,d),

where denotes the Waring border rank of , i.e., the minimum such that there exists a sequence of polynomials of Waring rank at most converging to in the Euclidean (or equivalently, Zariski) topology.

1.2 Paper Overview

For ease of exposition, we work over unless specified otherwise. Most of our theorems can be extended to infinite (or sufficiently large) fields of arbitrary characteristic by replacing the polynomial ring with the ring of divided power polynomials (see [IK99, Appendix A]). Except for in Section 4, we assume that arithmetic operations can be performed with infinite precision and at unit cost.

In Section 2 we introduce concepts related to Waring rank (in particular the Apolarity Lemma) in order to better understand the following problems:

Problem 12.

Compute .

Compute a approximation of (assuming ).

Determine if .

The fundamental connection between Waring rank and Problem 12 is given by our Theorem 6. Using similar ideas, we show that at least queries are required to test if with one-sided error in Theorem 24. We then introduce the new concepts of support rank, -support rank, and nonnegative support rank, which give upper bounds on the complexity of randomized and deterministic algorithms for Problems 12, 12 and 12

. A related notion of support rank for tensors has previously appeared in the context of

and quantum communication complexity [CU13, BCZ17, WGE16], but we are unaware of previous work on support rank in the symmetric (polynomial) case. In the case when these notions are related to the well-studied concepts of sign rank, zero-nonzero rank, and approximate rank of matrices [BDYW11, ALSV13].

In Section 3 we study and its variants. We start in Section 3.1 by proving negative results, showing that (Theorem 28), and that for sufficiently large , (Proposition 33) and (Corollary 31). Using bounds on the -rank

of the identity matrix

[Alo03], we show in Corollary 36 that for ,

 Ω(logn⋅ε−2/log(ε−1))≤Aε(n,2)≤O(logn⋅ε−2).

While it may at first seem like we are splitting hairs by focusing on particular values of , we will later show in Example 67 that, for example, proving that for sufficiently large would yield improved upper bounds on for all and .

Curiously, our lower bound on is a consequence of the classical Cayley-Salmon theorem in algebraic geometry, and our general lower bound on ultimately follows from Bézout’s theorem via [RS11]. On this note, we show in Proposition 30 that Question 1 is equivalent to a question about the geometry of linear spaces contained in the Fermat hypersurface .

The rest of Section 3 is focused on general upper bounds on and its variants. Proposition 38 will give a simple explanation as to why determinant sums (as in the title of [Bjö10]) can be computed in a parameterized way: for all matrices and , the Waring rank of

 ∑α∈{0,1}n|α|=ddet(AαBα)xα11⋯xαnn (4)

is at most . A special case of this example is used in Theorem 41 to show that . In order to improve this, it would suffice to find a better upper bound on the Waring rank of a single polynomial: the determinant of a symbolic Hankel matrix. We show in Theorem 43 that the method of partial derivatives cannot give lower bounds on the Waring rank of this polynomial better than .

Next we define rank for polynomials over a field of arbitrary characteristic – as it is, our definition of rank is not valid in positive characteristic (example: try to write as a sum of squares of linear forms over a field of characteristic two). Using this we define , which equals when . We note in Theorem 49 that . Theorem 52 shows that this lower bound is essentially optimal when , as then ; specifically, this rank upper bound holds for Equation 4 in the case that . This is a simple consequence of the fact that the permanent and the determinant agree in characteristic 2. We explain in this section how the group-algebra approach of [Kou08, Wil09] and the basis of [Bjö10] reduce to a slightly weaker fact than this upper bound. A precise connection between support rank and a certain “product-property” of abelian group algebras critical to [Kou08, Wil09] is given by Theorem 53.

In Section 3.3 we present a method for translating upper bounds on for some fixed and into upper bounds on for all and (Theorem 66). This method also allows us to recursively bound for fixed (Theorem 56). This approach can be seen as a vast generalization of color-coding methods, and is based on a direct power sum operation on polynomials and a combinatorial tool generalizing splitters that we call a perfect splitter. We use this to show that in Theorem 58.

In Section 4 we give applications of the previous section. We start by giving the proof Theorem 7, which is then used to prove Theorem 8. We end with an improved lower bound on the size of perfectly-balanced hash families in Theorem 72.

We conclude by giving several standalone problems.

2 Preliminaries and Methods

We use multi-index notation: for , we write , where . For , we let and . We then define , and similarly . Given we say that if for all . We denote by the differential operator , and we let . We let denote the hypersurface defined by . For , we let . For , the ideal of polynomials in vanishing on is denoted by . The ideal generated by is denoted by . Given an ideal we let denote the subspace of of degree- polynomials.

The set of matrices with entries in a field is denoted by . For a matrix and a multi-index , we let be the matrix whose first columns are the first column of , next columns are the second column of , etc. We let , denote the degree- determinant and permanent polynomials, respectively. Recall that the permanent is defined by

 perd=∑σ∈Sdd∏i=1xi,σ(i),

where denotes the symmetric group on letters.

The subsequent theorems are classical and easily verified. The first is the crux of this paper. The second shows that Waring rank is always defined (i.e., finite).

Theorem 13.

Let and let .

[IK99, Lemma 1.15(i)] For all ,

 f(∂x)r∑i=1ℓji=d!r∑i=1f(ℓ∗i)ℓj−di.

[CGLM08, Lemma 3.5] For all , .

Theorem 14.

[IK99, Corollary 1.16] .

Importantly, Theorems 13 and 13 imply that can be computed with queries in Problem 12, as noted in creftypecap 5. We will show in the next subsection that this is optimal, even if we are allowed to query adaptively.

Example 15.

The following Waring decomposition of is easily seen by inclusion-exclusion:

 d!⋅en,d=∑α∈{0,1}n|α|≤d(−1)|α|+d(n−|α|d−|α|)(n∑i=1αixi)d. (5)

In fact, this decomposition is synonymous with inclusion-exclusion in many exact algorithms, as we now illustrate. For , let

 ProdA\coloneqq(A1,1x1+⋯+A1,nxn)⋯(An,1x1+⋯+An,nxn)∈Snn.

It is easily seen that the coefficient of in equals the permanent of . In other words, . It follows directly from Theorem 13 and Equation 5 that

 per(A)=∑α∈{0,1}n(−1)|α|+nProdA(α),

which is Ryser’s formula for computing the permanent [Rys64]. As another example, applying Theorem 13 and Equation 5 to the closed-walk generating polynomial Equation 2, one finds that the number of Hamiltonian cycles in equals

 ∑α∈{0,1}n(−1)|α|+ntr(AnG)(α),

which was first given in [KGK77] and rediscovered several times thereafter [Kar82, Bax93]. As a third example, let , where for all . Note that that the coefficient of in equals the number of ordered partitions of into of the sets . Therefore the number of such partitions equals

 ∑α∈{0,1}kr(−1)|α|+krPartS1,…,Sm(α),

which was given in [BH06, BHK09]. The fastest known algorithms for computing the permanent and counting Hamiltonian cycles and set partitions follow from the straightforward evaluation of the above formulas. A similar perspective on these algorithms appeared earlier in [Bar96].

Understanding these algorithms from the perspective of Waring decompositions is extremely insightful, and was our initial motivation. For example, it is clear from the above argument that any Waring decomposition of yields an algorithm for the above problems – there is nothing special about Equation 5. This immediately raises the question: what is ? This was only answered recently in [RS11], where a lower bound on the degree of a form’s apolar subscheme was used to show that .666A lower bound of can be shown easily using the method of partial derivatives, presented in the next subsection. This lower bound shows that the above algorithms are, in a restricted sense, optimal. Similar observations have been made in [Gur08, Gly13].

Although the Waring decomposition of Equation 5 is essentially optimal in the case when , it is far from optimal in general. Indeed, Equation 5 only shows that , whereas it was shown in [Lee16] that for odd, , and for even,

 (n≤d/2)−(n−1d/2)≤R(en,d)≤(n≤d/2).

2.1 Apolarity and the Method of Partial Derivatives

Fix . For integers such that , let be given by

 Catg(u,v)(f)\coloneqqf(∂x)g.

These maps, called catalecticants, were first introduced by J.J. Sylvester in 1852 [Syl52]. Their importance is due in large part to the following method for obtaining Waring rank lower bounds, known as the method of partial derivatives in complexity theory [Lan17, Section 6.2.2].

Theorem 16.

[IK99, pg. 11] For all and integers such that ,

 R(g)≥rank(Catg(u,v)).
Remark 17.

As a matrix, has columns, indexed by the degree- monomials in , and rows, indexed by the degree- monomials in . Therefore the best rank lower bound Theorem 16 can give is , which is obtained when . In contrast, it is known [Lan12, Section 3.2] that the rank for almost all is at least (with respect to a natural distribution on forms), so the method of partial derivatives is far from optimal. Finding methods for proving better lower bounds is a significant barrier and a topic of great interest from both an algebraic-geometric and complexity-theoretic perspective; see [Lan17, Section 10.1] and [EGOW18].

Example 18.

It is a classical fact from linear algebra that for , . Explicitly, this says that can be written as a sum of at most squares of linear forms if and only if the matrix has rank at most . Hence Waring rank can be viewed as a higher dimensional generalization of symmetric matrix rank.

Let be the set of degree- forms annihilating under the differentiation action. The next fact is known as the Apolarity Lemma in the Waring rank literature.

Lemma 19.

[Tei14, Theorem 4.2] Let be pairwise linearly independent. Then for all , if and only if .

A complete answer to the complexity of Problem 12 is now in hand.

Proof of Theorem 6.

The upper bound is immediate from Theorem 13. To prove the lower bound we first show the following: for any pairwise linearly independent points where , there exists a such that but . If this were not the case, there exist pairwise linearly independent points such that . But this implies that has rank at most by the Apolarity Lemma, a contradiction.

So now given any , suppose that our algorithm queries at , which can be assumed to be pairwise linearly independent. By the above argument, there exists some such that for all , and hence the algorithm cannot distinguish from , but at the same time . ∎

2.2 Support Rank, Nonnegative Support Rank, and ε-Support Rank

We now introduce variants of Waring rank of algorithmic relevance.

Definition 20.

The support rank and nonnegative support rank of are given by

 Rsupp(f) \coloneqqmin(R(g):g∈Snd,supp(g)=supp(f)), R+supp(f) \coloneqqmin(R(g):g∈R≥0[x1,…,xn]d,supp(g)=supp(f)). Furthermore, if f∈R≥0[x1,…,xn]d, the ε-support rank of f is given by Rεsupp(f) \coloneqqmin(R(g):g∈R[x1,…,xn]d,∀α∈Nnd,(1−ε)⋅∂αf≤∂αg≤(1+ε)⋅∂αf).

Note that condition in the definition of is simply that the coefficient of in is bounded by a factor of times the coefficient of in .

Roughly speaking, support rank corresponds to decision algorithms, nonnegative support rank to deterministic decision algorithms, and -support rank to deterministic approximate counting algorithms. This is now formalized.

Definition 21.

For and , a -support intersection certification algorithm with one-sided error is an algorithm which, given any as a black-box, outputs on all instances where , and correctly outputs with probability at least on all instances where .

Proposition 22.

For all and , there is a -support intersection certification algorithm with one-sided error that makes queries.

For a fixed and all given as a black-box, there is a deterministic algorithm that decides if using queries.

For a fixed and all given as a black-box, there is a deterministic algorithm that computes a -approximation to using queries.

Proof.
1. [label=.]

2. Let , where . Let be indeterminates. Note that is not identically zero in if and only if . Then by choosing uniformly at random from , will evaluate to zero whenever , and whenever this does not evaluate to zero with probability at least by the Schwartz-Zippel lemma. By Theorem 13, can be computed using queries, and the conclusion follows.

3. If both and have nonnegative coefficients, then if and only if . The result follows from Theorem 13.

4. This is immediate from Theorem 13.∎

It follows from a variation of the proof of Theorem 6 that Proposition 22 is optimal for monomials:

Proposition 23.

For all and all , any -support intersection certification algorithm with one-sided error makes at least queries.

Proof.

The upper bound follows from Theorem 13; in fact, this shows that we can compute exactly using queries.

For the lower bound, given any where , suppose a support intersection certification algorithm queries at pairwise linearly independent points , where . Then by the Apolarity Lemma, there exists a such that but (see the proof of Theorem 6). Note that the condition that is equivalent to saying that . Therefore there exists some such that . But note that for all , and hence the algorithm cannot distinguish between and . Since the algorithm has no false negatives, it must always give the incorrect answer on . We conclude by the matching upper and lower bounds on given in [CCG11]. ∎

Theorem 24.

Any -support intersection certification algorithm with one-sided error makes at least queries.

Proof.

Suppose for contradiction that such an algorithm made fewer queries. Then given as a black-box, we run this algorithm with access to . By definition, this algorithm always answers correctly if the coefficient of is zero, and answers correctly with probability at least if this coefficient is nonzero. But this gives an -support intersection certification algorithm with one-sided error making fewer than queries. Since [RS11], this contradicts LABEL:{montest}. ∎

3 Support Ranks of Elementary Symmetric Polynomials

We are now ready to study and its variants, which we now recall.

Problem 25.

Determine , and .

Obviously and for all , . It follows from [RS11] that and from [Lee16] that ; the latter turns out to be arbitrarily far from optimal, however.

We will be interested in Problem 25 as goes to infinity. To facilitate this, we adopt the notation , defining and analogously. We will show in Proposition 27 that , and are nondecreasing in , in Proposition 38 that is finite for each , and in Corollary 36 that is infinite for and .

For notational convenience, we define

 E(n,d) \coloneqq{f∈Snd:supp(f)=supp(en,d)}, E+(n,d) \coloneqq{f∈E