PPP-Completeness and Extremal Combinatorics^††thanks: Part of this wok done while visiting R.B., L.F., P.H., and N.I.S. were visiting Bocconi University.

We explore new connections between classical theorems in extremal combinatorics and the complexity classes PPP [Pap94] and PWPP [Jeř16], i.e., the classes of search problems with totality guaranteed by the (weak) pigeonhole principle. We show that PPP and PWPP can be characterized via a number of new TFNP problems based on the following theorems.

Erdős-Ko-Rado [EKR61].

Any family of distinct pairwise-intersecting $k$-sets on a universe of size $m$ has size at most $\binom{m-1}{k-1}$.

Sperner [Spe28].

The largest antichain, i.e., a family of subsets such that no member is contained in any other, on a universe with $2n$ elements is unique and consists of all subsets of size $n$.

Cayley [Cay89].

There are exactly $n^{n-2}$ spanning trees of the complete graph on $n$ vertices.

Just as for Ramsey and Sunflower, the corresponding search problems are efficiently solvable when given explicit access to the family of objects and, again, their computational complexity is open when we consider implicit access to the structure, e.g., where the instance is given by a circuit that on input $i$ returns an encoding of the $i^{\text{th}}$ object in the collection.²²2Note that an implicit representation of the collection might not necessarily satisfy the assumptions of the underlying theorem. For instance, representing sets via characteristic vectors for Erdős-Ko-Rado does not ensure that they are actually $k$-sets or that they are distinct. Importantly, such a violation could allow evading the totality of the search problem. Nevertheless, we can ensure totality by allowing locally verifiable evidence of a malformed representation as a solution, e.g., an index not corresponding to a $k$-set or two indices corresponding to the same set. The totality of the problems we define follows from a common principle – the instances are given via an implicit representation of a sufficiently large collection of objects (e.g., subsets for Erdős-Ko-Rado) such that, by the corresponding theorem, there exists a small subset of these objects satisfying some efficiently verifiable property (e.g, a pair of disjoint subsets for Erdős-Ko-Rado).

In addition to the above completeness results, we define TFNP problems arising from the following two results in extremal combinatorics.

Mantel [Man07].

Any triangle-free graph on $n$ vertices has at most $n^{2}/4$ edges.

Ward-Szabo [WS95].

Any edge-coloring of the complete graph on $n$ vertices with $2\leq r\leq\sqrt{n}$ colors must contain a bichromatic triangle.

We show that variants of the corresponding problems are hard for PWPP and PPP. However, proving their inclusion in PWPP or PPP remains open and they join Ramsey and Sunflower as candidate problems that might define a new class above PWPP or PPP (see Section 1.5). An overview of our results in terms of weak and strong problems (see Section 1.3) is given in Table 1.

A long-standing open problem regarding Ramsey and Sunflower has been to determine their status with respect to the classes PWPP and PPP. For the most part, the most challenging part in establishing completeness for some syntactic subclass of TFNP lies in proving hardness (see, e.g., [DGP09, Meh18, FG18]). For subclasses of TFNP such as PPAD, PPA, and PLS, the inclusion in a subclass mostly follows from the existence of an inefficient yet structured algorithm for the problem at hand; for example, the chessplayer algorithm for PPA [Pap94] or the steepest descent algorithm for PLS [JPY88]. However, this methodology seems inapplicable for proving inclusion in PWPP or PPP as these classes do not exhibit any characterizing graph-theoretic structure that could capture some class of natural algorithms.

In contrast to many existing bounds in TFNP, our work does not make use of structured algorithms but instead makes use of encodings that translate between substructures and collisions in circuits. In order to establish inclusion in PWPP, we encode the objects of the collection using a “property-preserving encoding” that encodes the objects in a way that translates some specific relation into collisions. More precisely, we want an encoding function that is efficiently computable and (nearly) optimal, such that whenever two elements have the same encoding, these two elements give a solution to the original problem. While this technique is quite general, it is not always clear how to instantiate the encoding to get the desired collisions.

Consider, for example, the total search problem corresponding to the Erdős-Ko-Rado theorem for intersecting families of $n$-sets on a universe of size $2n$. An instance can be given by a Boolean circuit $C\colon\{0,1\}^{\left\lceil\log\left(\binom{2n-1}{n-1}\right)\right\rceil+1}\to\{0,1\}^{2n}$ representing a family of subsets of $[2n]$, i.e., $C(i)$ is the characteristic vector of the $i$-th $n$-set in the family. Suppose the outputs of $C$ define distinct $n$-sets. Since there are more than $\binom{2n-1}{n-1}$ of them, then, by the Erdős-Ko-Rado theorem, there must exist a pair of inputs mapped to disjoint $n$-sets by $C$. We define any such pair of inputs to be a solution.³³3To ensure the totality of the problem, we introduce additional solutions corresponding to succinct certificates that $C$ does not define a family of distinct $n$-sets, i.e., either an $i$ such that $C(i)$ is not of Hamming weight $n$ or a pair $i\neq j$ such that $C(i)=C(j)$.

When proving that the above total search problem is contained in the complexity class PWPP, at a high level, we want to encode the $n$-sets of the family using a shrinking circuit, in such a way that collisions correspond to disjoint sets. Observe that for $n$-sets in a universe of size $2n$, the only disjoint sets are complements and, hence, we get an equivalent instance of the problem if we map each set to either itself or its complement, arbitrarily. In our construction, we map each set $S$ to the representative not containing the element 1. That is, if $1\not\in S$, the set is left unchanged and, otherwise, it is mapped to its complement $\overline{S}$. Note that, by the pigeonhole principle, two sets that do not contain 1 must have a non-empty intersection since we work with $n$-subsets of $[2n]$. To obtain a shrinking circuit, we make use of Cover encodings (Section 3.1) that give an optimal encoding of all $n$-sets by considering their lexicographic order. Notice that if the input $S$ is not an $n$-set, we may map it arbitrarily to any $n$-set, as a collision, in this case, yields a solution to the instance of the above problem motivated by the Erdős-Ko-Rado theorem.

In contrast, the PWPP-hardness results for Ramsey and Sunflower follow an extremely elegant but rather direct (compared to other hardness results for subclasses of TFNP) technique of graph-hash product [Kra05, KNY19], which we illustrate on Ramsey. Recall that there are known randomized constructions of edge-colorings of the complete graph $K_{2^{n/4}}$ on $2^{n/4}$ vertices that do not contain a monochromatic clique of size $n/2$ [Erd47]. Given such an underlying edge-coloring of $K_{2^{n/4}}$ and a hash function $h$ mapping $n$-bit strings to $n/4$-bit strings, one can construct an edge-coloring of the complete graph on $2^{n}$ vertices by assigning to every edge $(u,v)\in\{0,1\}^{n}\times\{0,1\}^{n}$ the color of the edge $(h(u),h(v))\in\{0,1\}^{n/4}\times\{0,1\}^{n/4}$ from the underlying coloring. Since the underlying edge-coloring of $K_{2^{n/4}}$ does not contain a monochromatic clique of size $n/2$, it is easy to see that any monochromatic clique of size $n/2$ in the resulting edge-coloring of $K_{2^{n}}$ (guaranteed to exist by Ramsey’s theorem) must have been introduced via a collision in the hash $h$.

As noted by [KNY19], the structure of a PWPP-hardness proof using the graph-hash product is not restricted to total search problems corresponding to graph-theoretic theorems of existence; indeed, [KNY19] used the graph-hash product to prove also PWPP-hardness of Sunflower. On a high level, for a problem to be amenable to the graph-hash product technique, it is sufficient to be able to construct a collection of objects such that 1) it does not contain the desired substructure, 2) its size is at least a constant fraction of the threshold necessary for the existential theorem to apply,⁴⁴4This is a technical condition ensuring that we can reduce from a PWPP-complete variant of the problem of finding collisions in a shrinking hash. Note that it is easy to find collisions in functions that exhibit extreme shrinking. and 3) it can be efficiently indexed. Then, we can interpret the output of an appropriately shrinking hash $h$ as an index into the small collection of objects, and, for each index, we can efficiently compute and output the corresponding element in the collection. Again, since the small collection does not contain the desired substructure, all solutions of the instance constructed via graph-hash product must in some way result from a collision in the hash $h$.

For example, consider the total search problem arising from Sperner’s theorem on antichains – here, the threshold size is $\binom{2n}{n}$, meaning that if we have a family with strictly more than $\binom{2n}{n}$ distinct subsets of $[2n]$ then one subset from the family must be contained in another member of the family. It is straightforward to construct a family of subsets that does not contain the specific substructure (i.e., a subset that is included in another one) with size equal to the threshold size $\binom{2n}{n}$. It suffices to consider the family of all the $n$-subsets of $[2n]$. Similarly, for many other combinatorial problems we study, an adequate collection of objects can be found by looking at a collection of maximum size that does not contain the substructure.

We also show natural reductions between some of the problems we define (from Erdős-Ko-Rado to Sperner-Antichain for instance), which, in our opinion, highlights the relevance of these new problems and the fact that their definition is the correct one.

Up to this point, our discussion did not explicitly distinguish between the classes PWPP and PPP. However, our work highlights important structural differences between the two complexity classes. Recall that the class PWPP contains the search problems in TFNP whose totality can be proved using the weak pigeonhole principle: “In any assignment of $2n$ pigeons to $n$ holes there must be two pigeons sharing the same hole.”

This statement can be seen as a result in extremal combinatorics bounding the maximum number of pigeons that can be assigned to $n$ holes without two pigeons being sent to the same hole. More generally, we say that a theorem from extremal combinatorics is “weak” if it gives an upper bound (which may or may not be tight) on the maximum size of a collection of objects that does not contain some substructure (above, two pigeons sharing the same hole). On the contrary, we say that a theorem from extremal combinatorics is “strong” if it gives a tight upper bound on the maximum size of a collection of objects that does not contain some substructure, as well as some structural property about the maximum families without the substructure. For instance, the strong pigeonhole principle can be stated as: “In any assignment of $n$ pigeons to $n$ holes there is either a pigeon in the first hole or two pigeons sharing the same hole.” Note that it is exactly this formulation of the strong pigeonhole principle that defines the class PPP.

Many results in extremal combinatorics have a weak statement and a strong statement. For such results, we can define a problem corresponding to the weak statement, which often is related to PWPP, and a problem corresponding to the strong statement, which often is related to PPP. In this paper, all PWPP-hard problems correspond to a weak theorem in extremal combinatorics, while PPP-hard problems correspond to a strong theorems in extremal combinatorics. As an example, consider Cayley’s formula and note that the bound $n^{n-2}$ is tight. Hence, if we are given a collection of exactly $n^{n-2}$ distinct graphs on $n$ vertices, then either one of the graphs is not a spanning tree, or every spanning tree is in the collection. This observation induces a TFNP problem that we show to be PPP-complete.

Compared to the majority of subclasses of TFNP that have been extensively studied and are known to capture various total search problems from diverse domains of mathematics, PPP and PWPP might seem less expressive and the first non-trivial completeness results appeared only recently.

Sotiraki, Zampetakis, and Zirdelis [SZZ18] and Ban, Jain, Papadimitriou, Psomas, and Rubinstein [BJP⁺19] demonstrated that PPP contains computational problems from number theory and the theory of integral lattices. In particular, Sotiraki et al. showed PPP-completeness of a computational problem related to Blitchfeld’s theorem and PPP-completeness (resp. PWPP-completeness) of a problem motivated by the Short Integer Solution problem. Hubáček and Václavek [HV21] showed that some general formalizations of the discrete logarithm problem are complete for PWPP and PPP and, motivated by classical constructions of collision-resistant hashing, they characterized PWPP via the problem of breaking claw-free (pseudo-)permutations.

Our work suggests various interesting directions for future research:

• •

  We exploit the power of strong statements in extremal combinatorics for establishing PPP-completeness. The notorious lack of tight bounds for the Erdős-Rado sunflower lemma and Ramsey’s theorem implies that we have no strong version of these theorems, which may explain why showing the inclusion of the corresponding problems in, e.g., PPP has eluded researchers.

• •

  We introduced total search problems corresponding to Mantel’s theorem, Turán’s theorem, and Ward-Szabo’s theorem. In this work, we only prove hardness results for these problems but no inclusion results. Hence, it is still open whether they are complete for the classes PPP and PWPP, or whether they could define a new subclass of TFNP.

• •

  The Turán_r problem is defined in a similar fashion to Mantel, yet, unlike for Mantel, we currently do not have a proof of PPP-hardness for it. Thus, the question of PPP-hardness of Turán_r is immediate. Alternatively, it would be interesting to define a different PPP-hard problem in a natural way from Turán’s theorem.

• •

  Another exciting question is whether the efficient Baranyai assumption (4.18) holds, as well as whether it is possible to prove the inclusion results of the problems associated to the general version of Erdős-Ko-Rado’s theorem without that assumption. Showing reductions between general-Erdős-Ko-Rado_k and general-Erdős-Ko-Rado_l for $k\neq l$ without the efficient Baranyai assumption would also be intriguing.

• •

  Finally, we believe the problems $\textsc{General-Pigeon}_{k}^{m}$ deserve a more thorough investigation to further our understanding of the classes they define and their interrelation.

A search problem is defined by a binary relation $R\subseteq\{0,1\}^{*}\times\{0,1\}^{*}$ – a string $s\in\{0,1\}^{*}$ is a solution for an instance $x\in\{0,1\}^{*}$ if $(x,s)\in R$. A search problem defined by relation $R$ is total if for every $x$, there exists an $s$ such that $(x,s)\in R$. We define TFNP as the class of all total search problems that can be efficiently verified, i.e., there is a deterministic polynomial-time Turing machine that, given $(x,s)$, outputs 1 if and only if $(x,s)\in R$ and, for every instance $x$, there exists a solution $s$ of polynomial length in the size of $x$.

To avoid unnecessarily cumbersome phrasing throughout the paper, we define TFNP relations implicitly by presenting the set of valid instances $X\subseteq\{0,1\}^{*}$ recognizable in polynomial time (in the length of an instance) and, for each instance $i\in X$, the set of admissible solutions $Y_{i}\subseteq\{0,1\}^{*}$ for the instance $i$. It is then implicitly assumed that, for any invalid instance $i\in\{0,1\}^{*}\setminus X$, we define the corresponding solution set as $Y_{i}=\{0,1\}^{*}$.

Next, we recall the definitions of the complexity classes PWPP and PPP via their canonical complete problems weak-Pigeon and Pigeon.

Our reductions in Section 4 make use of Cover encodings [Cov73] that efficiently encode subsets of a specified size in optimal space: namely, we may encode every subset $S\subseteq\{0,1\}^{m}$ such that $|S|=k$ by considering the lexicographic order of all $\binom{m}{k}$ such sets (in fact we consider the lexicographic order over their characteristic vectors $\in\{0,1\}^{m}$), and mapping this into binary strings: this requires $\left\lceil\log\binom{m}{k}\right\rceil$ bits, which is optimal. We denote the encoding and decoding functions as follows, with $\alpha(k,m)=\left\lceil\log\binom{m}{k}\right\rceil$.

	$\displaystyle E_{\textsf{Cover}}^{k,m}:\{0,1\}^{m}$	$\displaystyle\rightarrow\{0,1\}^{\alpha(k,m)}$
	$\displaystyle D_{\textsf{Cover}}^{k,m}:\{0,1\}^{\alpha(k,m)}$	$\displaystyle\rightarrow\{0,1\}^{m}$

We set $E_{\textsf{Cover}}=E_{\textsf{Cover}}^{n,2n}$ and $D_{\textsf{Cover}}=D_{\textsf{Cover}}^{n,2n}$, and $\alpha=\alpha(n,2n)$. As described in [Cov73], these functions can be made efficient.

In Section 7, we need to encode the subsets of $[2^{n}]$ with 2 distinct elements in an injective way. Unfortunately, since the base set is large, we cannot use Cover encodings to do so. However, we can use the idea behind Cover encodings, that is to encode the subsets by their rank in the lexicographic order. Consider $(x,y)\in[2^{n}]\times[2^{n}]$, with $x<y$. What is its rank in the lexicographic order?
All subsets whose smallest element is smaller than $x$ have a lower rank. The number of such subsets is

	$\displaystyle(2^{n}-1)+(2^{n}-2)+\ldots+(2^{n}-x+1)$	$\displaystyle=\sum_{j=2^{n}-x+1}^{2^{n}-1}j$
		$\displaystyle=\sum_{j=1}^{2^{n}-1}j-\sum_{j=1}^{2^{n}-x}j$
		$\displaystyle=\frac{2^{n}(2^{n}-1)}{2}-\frac{(2^{n}-x)(2^{n}-x+1)}{2}$

All subsets whose smallest element is $x$ and whose second smallest element is smaller than $y$ also have a lower rank. There are exactly $y-x-1$ such subsets.
Hence, the rank of the subset $(x,y)$ in the lexicographic order is

$$\frac{2^{n}(2^{n}-1)}{2}-\frac{(2^{n}-x)(2^{n}-x+1)}{2}+y-x-1$$

Note that since there are $\binom{2^{n}}{2}<2^{2n-1}$ subsets of $[2^{n}]$ with 2 distinct elements, the rank of any subset $(x,y)$ with $x<y$ can be written in binary using $2n-1$ bits. Now, denote as $E_{lex}:\{0,1\}^{n}\times\{0,1\}^{n}\rightarrow\{0,1\}^{2n-1}$ the following circuit. On input $(x,y)$, it proceeds as follows.

1. 1.

   If $x=y$, it returns $0^{2n-1}$.

2. 2.

   If $x<y$, it computes and returns the binary encoding on $2n-1$ bits of $\frac{2^{n}(2^{n}-1)}{2}-\frac{(2^{n}-x)(2^{n}-x+1)}{2}+y-x-1$.

3. 3.

   If $x>y$, it computes and returns the binary encoding on $2n-1$ bits of $\frac{2^{n}(2^{n}-1)}{2}-\frac{(2^{n}-y)(2^{n}-y+1)}{2}+x-y-1$.

Note that $E_{lex}$ has polynomial size, and is injective on the set of subsets of $[2^{n}]$ with 2 distinct elements by construction.

In Section 6, we make use of Prüfer codes [Pru18] that give an efficiently computable bijection between the set of labelled spanning trees on $n$ vertices and the set of sequences of $n-2$ elements of $[n]$. They were originally used by Heinz Prüfer [Pru18] to prove Classical Theorem 5

We denote by $E_{\textsf{Prüfer}}$ a circuit that efficiently computes the Prüfer encoding of a spanning tree described by an element of $\{0,1\}^{\binom{n}{2}}$. Similarly, let $D_{\textsf{Prüfer}}$ be a circuit that efficiently computes the spanning tree associated with a Prüfer code. By looking at the algorithm to compute Prüfer encodings, it is clear that we can assume these circuits to have polynomial size. We also assume that $E_{\textsf{Prüfer}}$ outputs elements of the right form even on inputs which do not correspond to spanning trees. Consider the lexicographic order on $[n]^{n-2}$. Let $R$ be a circuit that efficiently computes the rank of an element of $[n]^{n-2}$, and let $\tilde{E}_{\textsf{Prüfer}}=R\circ E_{\textsf{Prüfer}}$. Given a spanning tree, $\tilde{E}_{\textsf{Prüfer}}$ returns the rank of its Prüfer code in the lexicographic order.

Let $R^{\prime}$ be a circuit which on input $x$ computes the sequence of $[n]^{n-2}$ whose rank in the lexicographic order is $x$. Let $\tilde{D}_{\textsf{Prüfer}}=D_{\textsf{Prüfer}}\circ R^{\prime}$. Given a rank, $\tilde{D}_{\textsf{Prüfer}}$ returns the spanning tree whose Prüfer code has the corresponding rank in the lexicographic order. Note that $\tilde{D}_{\textsf{Prüfer}}$ and $\tilde{E}_{\textsf{Prüfer}}$ both have polynomial size. Now, if $\beta=\lceil(n-2)\log(n)\rceil$, then $\tilde{E}_{\textsf{Prüfer}}:\{0,1\}^{\binom{n}{2}}\rightarrow\{0,1\}^{\beta}$, $\tilde{D}_{\textsf{Prüfer}}:\{0,1\}^{\beta}\rightarrow\{0,1\}^{\binom{n}{2}}$. By construction, we have the following.

Catalan factorization [EK99] is an encoding of subsets of $[2n]$ that allows us to decompose the partially ordered set $(2^{[2n]},\subseteq)$ into $\binom{2n}{n}$ chains and to move efficiently within each chain to find a canonical representative, namely the only $n$-subset of the chain.

Let $x\in\{0,1\}^{2n}$ be a bitmap representing an element of $[2n]$. We introduce a new symbol $z$, and construct the Catalan factorization as follows. We temporarily record for each symbol whether or not it is underlined.

1. 1.

   Underline the leftmost substring that starts with a non-underlined 1, followed by a (possibly empty) sequence of underlined symbols, and ends in a non-underlined 0. If no such substring exists, go to step 3.

2. 2.

   Go to step 1.

3. 3.

   Record the number $k$ of non-underlined 1’s.

4. 4.

   Replace all non-underlined symbols in $x$ with $z$, and let $x^{\prime}\in\{0,1,z\}^{2n}$ be the resulting string (with underlinings removed).

5. 5.

   Output $(x^{\prime},k)$.

We denote the output of the Catalan factorization as $E_{\textsf{Catalan}}(x)\in\{0,1,z\}^{2n}\times[2n]$. We say $x^{\prime}=\tilde{E}_{\textsf{Catalan}}(x)$ is the Catalan string of $x$. If $x^{\prime}\in\{0,1,z\}^{2n}$ and $m$ is the number of $z$’s in $x^{\prime}$, then for any $l\leq m$, we define $D_{\textsf{Catalan}}(x^{\prime},l)$ as the string obtained from $x^{\prime}$ by replacing the $l$ last $z$’s by $1$ and the rest by 0.

We also denote by $D_{\textsf{Catalan}}^{(l)}:\{0,1,z\}^{2n}\rightarrow\{0,1\}^{2n}$ the map $x^{\prime}\mapsto D_{\textsf{Catalan}}(x^{\prime},l)$. If on input $x^{\prime}$, $l$ is larger than the number of $z$ symbols in $x^{\prime}$, all $z$ symbols are be replaced with 1; this ensures the map is defined for all $l\geq 0$.