Combinatorial Nullstellensatz modulo prime powers and the Parity Argument

We present new generalizations of Olson's theorem and of a consequence of Alon's Combinatorial Nullstellensatz. These enable us to extend some of their combinatorial applications with conditions modulo primes to conditions modulo prime powers. We analyze computational search problems corresponding to these kinds of combinatorial questions and we prove that the problem of finding degree-constrained subgraphs modulo $2^d$ such as $2^d$-divisible subgraphs and the search problem corresponding to the Combinatorial Nullstellensatz over $\mathbb{F}_2$ belong to the complexity class Polynomial Parity Argument (PPA).


Introduction
In this paper, we are interested in combinatorial and computational problems in connection with Alon's Combinatorial Nullstellensatz ( [1]) which is a landmark theorem in algebraic combinatorics. The following corollary is often used implicitly in applications, see [1]. The goal of this paper is to give similar theorems for problems modulo arbitrary prime powers: we prove that if the number m of variables is sufficiently large, the corollary also holds modulo arbitrary prime powers. We develop a general method for the Combinatorial Nullstellensatz-type proofs, where the polynomials are modulo prime powers instead of primes. As an application, we extend the following theorem of Olson ([2]) and its generalization by Alon, Friedland and Kalai [3].
Let us be given a prime p, nonnegative integers d 1 ≥ d 2 ≥ · · · ≥ d n and sets Q 1 , Q 2 , . . . , Q n such that each of them contains zero and Q i ⊆ Z p d i for every i = 1, . . . , n. Let us denote (d 1 , d 2 , . . . , d n ) by d and (Q 1 , Q 2 , . . . , Q n ) by Q.
where card p (Q) denotes the number of distinct elements in Q modulo p.
Whereas Theorem 4 does not seem to be a strong estimation because of card p (Q) ≤ p, no better estimation has been known thus far.
It is worth noting that for d 1  Motivated by these questions, in this paper, we give analogous theorems modulo arbitrary prime powers instead of primes, extending Corollary 2, and give improved bounds on F (d, Q).

Complexity aspects
As an application of Olson's theorem, Alon, Friedland and Kalai [3] discussed the following extremal graph theoretic question. Given a prime power p d and an integer n, the problem is to determine the smallest value of m such that for every graph on n vertices and m edges, there exists a nonempty p ddivisible subgraph, that is, a nonempty subset of edges such that the number of edges incident to every vertex is divisible by p d . Conversely, determine the maximum number of edges a graph can have without containing a nonempty p d -divisible subgraph. The exact answer was given in [3], see Theorem 21.
A natural question is to determine the computational complexity of finding such a subgraph if the graph has sufficiently large number of edges. For the case p d = 2, the problem is equivalent to finding a cycle in a graph. In this case, there exists a polynomial time algorithm, but the problem is open in all other cases.
Due to various applications of the Combinatorial Nullstellensatz, it is also a natural question to determine the computational complexity of the corresponding search problem. An open question by West [5] is about the complexity of the Combinatorial Nullstellensatz over F 2 = {0, 1}. He conjectures that the corresponding search problem belongs to the complexity class Polynomial Parity Argument (PPA) defined by Papadimitriou [4]. This complexity class contains such computational search problems that the existence of a solution can be proved by so-called parity argument: Every finite graph has an even number of odd-degree nodes. In this paper, we verify his conjecture.

Main results
Now we present the first main result of this paper: the extension of Corollary 2 for arbitrary prime powers. This theorem also implies Theorem 3 and Theorem 4.
We will prove this theorem in Section 3. It is easy to check that Theorem 6 implies Corollary 2: let Theorem 3 and Theorem 4 will follow from Theorem 6 via the following general estimation for F (d, Q) which we will prove in Section 4.
In Section 4, we will give a general and constructive bound for price(B), which gives a strictly stronger estimation for F (d, Q) than that one in Theorem 4. We also show a wide class where this estimation is tight.
In the rest of the paper, we analyze the related computational questions. In Section 6, we will prove that the 2 d -divisible subgraph problem belongs to the complexity class Polynomial Parity Argument (PPA). We reduce the 2 d -divisible subgraph problem to the search problem of the Combinatorial Nullstellensatz over F 2 and in Section 5, we verify West's conjecture: the search problem of the Combinatorial Nullstellensatz over F 2 is also in PPA, if the polynomial is given in a general form such as in most of the applications.
In Section 7, we focus on degree-constrained subgraphs modulo prime powers, and we will prove an analogous theorem for Shirazi-Verstraëte theorem [7].
The proof of Theorem 6 presented here is similar to the proof of Theorem 4 in [3]. Alon et al. used a similar polynomial to the one in Equation (1), however, they used only the special construction of Equation (2) instead of arbitrary polynomials. Now we extend it to arbitrary integer-valued polynomials h and we can use more than one polynomial at the same time.
We apply the following corollary of Gregory-Newton formula for integer-valued polynomials. [ . , x m ] be the following polynomial: The degree of the constructed polynomial Ψ r (f ) is at most r · deg(f ). It is worth noting that if In the following lemma, we can obtain the benefit of these definitions: h(f (x)) can be written as a polynomial with integer coefficients.
Proof. Let c = f (s). Then, the number of terms in Ψ r (f ) that are 1 at s is precisely c r , the other terms are 0. So Now we are ready to prove Theorem 6.
Proof of Theorem 6. To simplify the notation, let C i be the complementary set of Q i , that is, Due to the definitions, there exists a set of polynomials H i which covers C i with the total degree price(C i ).
Let us consider the following polynomial in F p [x 1 , . . . , x m ]: where c is a nonzero constant to be defined later.
The degree of the first part of the polynomial is completing the proof.

The generalization of Olson's theorem: estimation for F (d, Q)
Let us now derive Theorem 7 from Theorem 6.
We are ready to show that Theorem 7 implies Theorem 4 and its special case, Theorem 3.
Let d be arbitrary, and 0 ∈ Q ′ ⊆ Z p d be a set of distinct integers modulo p. Then, let For every integer T , in the product q ∈Q ′ (T − q), at least p d−r − 1 numbers are divisible by p r for more than k r times and let Proof. We construct a polynomial that covers a complete p r -residue system modulo p r+1 with price p r .
Let q 1 , q 2 , . . . , q p r be a complete p r -residue system and let For every integer T , the integers T − q 1 , T − q 2 , . . . , T − q p r also form a complete residue system modulo p r , so in the product Hence, the product is divisible by p δ and h(T ) is an integer-valued polynomial.

Further, h(T ) is divisible by p if and only if the factor which is divisible by
Then, by Definition 12, the statement immediately follows: one can cover the integers that are divisible by p d−1 with k such conditions. These conditions also covers other residues k times, so such residues are not covered by the conditions that appear more than k times. These remaining residues are inB modulo p d−1 and they can be covered with κ(B).

A special case when Theorem 7 is tight
Here we show a special case when the theorem is tight. This statement shows a wide class where Theorem 7 and hence Theorem 6 give tight estimation. In general, tightness is not yet known. This result also shows cases when Theorem 4 gives strictly weaker estimation than the one in Theorem 7.
The proof is by induction on d. Let B = Z p d \Ω. Further, let d ′ = d−1 and let Ω ′ be the R ′ = R\{d−1}- Moreover, these bounds are tight: , then there exists integers a ij such that the proper nontrivial subset does not exist. Let a ij be −1 where , and zero otherwise. However, in the range −p di , . . . , 0, the largest integer of the R i -zero sets modulo p di is r ∈Ri and −κR i ≤ j∈J a ij ≤ 0, hence, no nonempty subset exists that fulfills the condition (♣). The complexity of finding such a vector whose existence is guaranteed by the Combinatorial Nullstellensatz depends on the input form of the given polynomial.

Complexity aspects of the Combinatorial Nullstellensatz
It is easy to check that the problem belongs to P if the polynomial is given explicitly as the sum of monomials. First, we can replace the term x . . x i k , because these are equal due to the fact 0 t = 0, 1 t = 1 in F 2 . Substitute 0 and 1 to x 1 : let g(x 2 , . . . , x n ) = f (0, x 2 , . . . , x n ) and h(x 2 , . . . , x n ) = f (1, x 2 , . . . , x n ). If in f the coefficient of x 2 x 3 . . . x m is nonzero, then in g the coefficient of x 2 x 3 . . . x m will be also nonzero. If it is zero, in h the coefficient of x 2 x 3 . . . x m will be nonzero. Then, substitute 0 and 1 to x 2 and in one of them the coefficient of x 3 x 4 . . . x m will be nonzero, and so on. Finally, we obtain a constant nonzero polynomial, and this means that for this substitution s ∈ F m 2 , f (s) = 0 holds. It is worth noting that a similar polynomial time algorithm can be obtained over arbitrary finite field, if the polynomial is given explicitly.
However, if the polynomial is given as the sum of products of polynomials (such as in most of the applications), the problem is not known to be solvable in polynomial time. An open question in [5] is about the complexity of the Combinatorial Nullstellensatz conjecturing that the problem over F 2 belongs to the class Polynomial Parity Argument (PPA) defined by Papadimitriou in [4].
In this section, we verify this conjecture: we prove that the Combinatorial Nullstellensatz over F 2 is in PPA if the polynomial is given as the sum of products of polynomials. Consequently, the applications given in Sections 6 and 7 also belong to PPA.
Roughly speaking, the class PPA is a subclass of the semantic class TFNP, the set of all total search problems. A search problem is called total if the corresponding decision problem is trivial, that is, for every feasible input, there exists a solution. A total problem is usually equipped with a mathematical proof showing that it belongs to TFNP, so the problems can be classified based on their proof styles.
The complexity class PPA is the class of all search problems whose totality is proved using the parity argument: Every finite graph has an even number of odd-degree nodes.
This class PPA can be defined with a canonical complete problem, the End Of The Line. Hence, a computational search problem is in PPA if and only if it is reducible to the problem End Of The Line.
In this problem, we are given a graph G = (V, E) on exponentially many nodes. It can be assumed that each node has an unique code from Σ n , that is V ⊆ Σ n . The edges of the graph are described by a polynomial time algorithm in n. This polynomial time pairing function is the following.
For an undirected graph G = (V, E), the function φ : if it satisfies the following conditions: if vw is not an edge of G, let φ(v, w) = * . Otherwise, it outputs a node w ′ = φ(v, w) such that w ′ is also connected to v and φ(v, φ(v, w)) = w holds. Furthermore, for every v, at most one such node w exists with property φ(v, w) = w.
It means that φ pairs up the neighbours of an input node v: for an even-degree node v, it pairs its neighbours completely, and for an odd-degree node v, φ pairs all but one neighbours. The task is to find an odd-degree node v and a node w such that φ(v, w) = w. This node w verifies that v is an odd-degree node.
The problem End Of The Line can be defined as follows.
End of the Line.
Input: an undirected finite graph G = (V, E) in the above way. The edges of the graph is described by a polynomial time pairing function.lists for a node its neighbours. Furthermore, a node ε is given which has odd number of edges and a node δ which shows it: φ(ε, δ) = δ.

Find:
another node v which has odd number of edges and a node w which give the certificate φ(v, w) = w.
In order to prove problems belonging to PPA, we give reductions to the problem End Of The Line.
It is worth noting that this problem is computationally equivalent to the problem in which the nodes have at most two neighbours, a node of degree one is given and the task is to find another node which has exactly one incident edge. (Instead of the polynomial time pairing function, a polynomial time algorithm is given which outputs the neighbours of an input node.) It is easy to see that this is an easier problem, however, Papadimitriou showed that they are computationally equivalent.
In [4], Papadimtriou shows that the following computational problem Chévalley MOD 2 belongs to the class PPA. The required vector exists due to Chévalley's following theorem.  We construct a graph, the nodes correspond to the vectors and the terms. The nodes with odd degree correspond to the vectors x such that f (x) = 0 and an extra node w. As we mentioned, we have to present a pairing function. It can be done easily at the terms, but it is more complicated at the vectors.
The main idea here is the following.
We call the polynomials mi j=1 p ij as the blocks of the input polynomial f . Each term is the product of monomials from the given polynomials of a block, so each term in the ith block can be represented by an (m i + 1)-tuple of integers: (i, a i,1 , . . . , a i,mi ). The first coordinate shows the block the term belongs to, and the other coordinates show the monomials the term is product of: it is the product of a i,j th monomials of p ij . (Note that the same term might have more than one occurrence and these occurrences are represented by different tuples.) In the next proof, we will pair up these tuples.

Remark 19. In a standard PPA-type problem definition it is required that the assumptions of the problem should be in NP. If the input is feasible, we have to return a solution, but if the input is infeasible, we have to return a polynomial certificate of infeasibility.
It is easy to check that the assumptions in the definition of the Combinatorial Nullstellensatz over F 2 are in NP. In the case of an infeasible input, we can give the following certificate: the index i such that mi j=1 deg(p ij ) > n or two occurrences of the term x 1 x 2 . . . x m for which the polynomial time pairing function fails.
Proof of Theorem 18. We shall construct a graph Γ whose odd-degree nodes precisely correspond to appropriate vectors s such that f (s) = 0 and furthermore, we add an extra node w: the standard leaf. For such node corresponding to a vector x that f (x) = 0 holds, we should pair up the terms such that t(x) = 1. Suppose that the term t is represented by (i, a i1 , . . . , a ij , . . . , a i,mi ). Denote its block by g = mi j=1 p ij . If g(x) = 0, then there is an index j such that p ij (x) = 0. Pick the smallest such j. There is an even number of monomials of p ij such that p ij (x) = 1. We pair these monomials by a pairing function φ i . Then the mate of term (i, a i1 , . . . , a ij , . . . , a i,mi ) is It is a more complicated case when g(x) = 1. Since f (x) = 0, there is an even number of indices l, such that m l j=1 p lj is 1 at x. We pair these blocks by a pairing function φ. So, for i and every j = 1, . . . , m i , p ij is 1 at x, and we can pair all but one monomials of p ij with p ij (x) = 1 by a pairing function φ ij .
One of them does not have a mate, denote its index by ω ij . If a ij = ω ij for all indices j, then we define its mate to be (φ(i), ω φ(i),1 , . . . , ω φ(i),m φ(i) ). Otherwise there is an index j such that a ij = ω ij . Pick the smallest such j. Then the mate of (i, a i1 , . . . , a i,mi ) is defined as (i, a i1 , . . . , φ ij (a ij ), . . . , a i,mi ).
Observe that this gives a bijection and a correct pairing function.
For such node corresponding to a vector x that f (x) = 1 holds, we should pair up all but one terms such that t(x) = 1. If t is a term of such block g = mi j=1 p ij that g(x) = 0 holds, it can be paired up similarly to the previous case. If g(x) = 1, we pair these blocks by a pairing function. One of them does not have a mate, denote its index by Ω. If g is not the block with index Ω, the pairing can be similar to the previous case. If g is the block with index Ω, we can pair up the terms similarly to the previous case, only the term t represented by (Ω, ω Ω1 , . . . , ω Ω,m Ω ) does not have a mate. So we paired up all but one neighbours of the node corresponding to the vector x.
Finally, we pair up the terms which are connected to the extra node w. These are the terms x 1 x 2 . . . x m .
Due to the assumptions, there is a polynomial time pairing function which can pair up all but one terms . . x m , so it can pair up the nodes which are connected to the extra node w.
We presented a polynomial algorithm that computes the mate of an edge out of a node, so the proof is complete.

Complexity of finding divisible subgraphs
Proof in [3].
Because of d 1 ≥ d 2 ≥ · · · ≥ d n ≥ 1, j∈J b ij = j∈J a ij is divisible by p dn for every i = 1, . . . , n − 1, hence j∈J a nj should be divisible by p dn and we are done.
Theorem 21 (Alon, Friedland, Kalai, [3]). For the maximum number of edges of a graph G on n vertices that contains no nontrivial p d -divisible subgraph, if p is an odd prime.
Proof in [3]. Here, we only prove the direction ≤ of the equality. In We only have to check that this reduction is a polynomial reduction. In the proofs the size of the constructed polynomial is O(2 d · d · n + m), which can be bounded O(nm log(m)) due to condition m > n · (2 d − 1), so the reduction is polynomial. Similarly, one can check that Even-Sum Olson MOD 2 d is also polynomially reducible to the Combinatorial Nullstellensatz over F 2 .
The reduction in the proof of Theorem 21 immediately implies the following.
In Louigi's problem, given are a graph G = (V, E) and forbidden sets F (v) ⊆ N for every v ∈ V .
By an F -avoiding subgraph we mean a subgraph ∅ = E ′ ⊆ E such that for every v ∈ V the number of incident edges of E ′ is not in F (v). Shirazi and Verstraëte [7] proved the following theorem. We give a new proof using our techniques.
Proof. Let p be a prime greater than the maximum degree in G. For the node v i ∈ V , let Q i = Z p \F (v i ) and a ij = 1 if the node v i is incident to the edge e j ∈ E, and 0 otherwise. Due to the conditions, vi∈V price(Z p \Q i ) = vi∈V |Z p \Q i | = v∈V |F (v)| < |E|, so according to Theorem 7, there exists a subset J, which corresponds to a nontrivial F -avoiding subgraph.
Note that, in [7], the authors also proved their theorem via the Combinatorial Nullstellensatz, but in a different way via polynomials over R.
One may ask a version of Louigi's problem modulo prime powers: given are a prime power p d , a graph G = (V, E) and forbidden sets modulo p d : F (v) ⊆ Z p d for every v ∈ V . By an F -avoiding subgraph modulo p d we mean a subgraph ∅ = E ′ ⊆ E such that for every v ∈ V the number of incident edges of E ′ is not congruent to any number in F (v) modulo p d . We can show the following. Similarly to the proofs of Theorems 22, 23, the proofs of Theorems 25, 7, 6 imply the following.
Theorem 26. Degree-constrained subgraph modulo 2 d is polynomially reducible to the Combinatorial Nullstellensatz over F 2 . Consequently, Degree-constrained subgraph modulo 2 d is in PPA.