Understanding PPA-Completeness

We consider the problem of finding a fully colored base triangle on the 2-dimensional Möbius band under the standard boundary condition, proving it to be PPA-complete. The proof is based on a construction for the DPZP problem, that of finding a zero point under a discrete version of continuity condition. It further derives PPA-completeness for versions on the Möbius band of other related discrete fixed point type problems, and a special version of the Tucker problem, finding an edge such that if the value of one end vertex is x, the other is −x, given a special anti-symmetry boundary condition. More generally, this applies to other non-orientable spaces, including the projective plane and the Klein bottle. However, since those models have a closed boundary, we rely on a version of the PPA that states it as to find another fixed point giving a fixed point. This model also makes it presentationally simple for an extension to a high dimensional discrete fixed point problem on a non-orientable (nearly) hyper-grid with a constant side length. 1998 ACM Subject Classification F.1.3 Complexity Measures and Classes


Introduction
In his seminal work on understanding the time complexity of the parity argument, Papadimitriou introduced the now well known class PPAD [27] that has influenced a generation of algorithmic game theorists in their study of economic computations.In the same paper, Papadimitriou also defined a more inclusive complexity class PPA (Polynomial Parity Argument) of search problems whose solution is guaranteed to exist through a proof based on the fact that "Any undirected graph with an odd-degree vertex must have another one".In contrast to PPA, PPAD is based on another straightforward principle: "Any directed graph that has an unbalanced node must have another".
The class PPA is a superset of PPAD, and the intuitive reason is that directions are helpful: Finding another node of the appropriate kind is harder to solve when there are no directions; in fact, oracle separation is known [3].This difference has also reflected in our understanding in the two classes, especially with regard to their complete problems.The class PPAD has now many problems that have been shown complete for PPAD such as in the incomplete list of 25 of them [22] gathered by Kintali.The class PPA-complete, however, did not fare as well.
On the one hand, there are many interesting existence theorems in Graph Theory, Combinatorics and Number Theory for which the computational problems are in PPA [27]: Smith's theorem [30] and related existentially polytime (graph) theorems [5], Chevalley's theorem [10] and Alon's Combinatorial Nullstellensatz [2], among others.Remarkably, the problem of factoring an integer has been recently proved to belong to PPA (via randomized reductions) [21], and the inclusion of this fundamental and critical problem gives the class a new significance.
On the other hand, we know few PPA-complete problems besides the generic one, unfortunately.The only exceptions are certain versions of Sperner's problem for rather esoteric non-orientable bodies.About ten years after the introduction of the class, Grigni [17] had the important idea that the right geometric context for PPA are non-orientable bodies, and showed that a version of the Sperner problem in the non-orientable three dimensional space is complete in the class.Soon after, Friedl et al. [15] strengthened it to a non-orientable and locally two-dimensional orientable space.
In general, it would be nice to have a growing strong collection of PPA-complete problems (like we have for PPAD), which with luck could eventually include factoring.The progress has been slow: another ten years passed without any progress in our understanding of the class PPA-complete for this problem many scientists are interested in.

Contributions
Our main results first end the quest for a complete fixed point characterization of the PPA-complete class.It provides a sharp division on what can be done and what cannot be done in computing different versions of the fixed point problem on the Möbius band.In particular, it does so by completing the task started by Friedl et al. [15], to reduce the next dimension demanded by the seminal result of Grigni [17], with the help of a technique developed by Chen and Deng [7], on the 2D Möbius version of a zero point problem, referred to as DPZP and conceptualised in [20,8,7,11].Together with the results of Grigni and Friedl, et al., they raise a theoretical connection of computational complexity to topology.The comparison between the 2D versions makes a strong case for this distinction.
Next, as the past works of Chen and Deng [7] as well as Deng, Qi, Saberi and Zhang [11] unify the complexity of the various discrete fixed point concepts in principle the above result implies that the same result holds for all the related discrete fixed points on the Möbius band.However, this may not always hold in general.We develop a new reduction approach to derive those results on the Möbius band.In particular, the 2D Tucker on orientable space were proven PPAD-hard, originally in the first principle by Pálvölgyi [26] and then by reduction to another discrete fixed point [11].Both approaches are complicated where applied to the Möbius version.Our new reduction approach makes it easy to be shown in both ways of containing and contained in the PPA class.The same holds for the other discrete fixed point problems.
Third, the simplicity of our 2D version has been handy to make further applications.On the higher constant dimension non-orientable space, all the discrete fixed point problems follow from the 2D results to become PPA-complete.Those cannot be easily obtained from

Relevance of the Möbius Band
The stories of the Möbius band have been a curiosity out of the Mind, such as a brain's toy of German mathematicians August Ferdinand Möbius (and Johann Benedict Listing), and the fascination art in the parade of ants by a Dutch artist M.C.Escher [13].In recent years, it becomes a possibility in scientific discoveries.Scientists made assembled object created by nano technology [18], proposed technical tool to develop negative refractive index materials [14], made experimental observation in electromagnetic metamaterial systems [6].In our work, it plays a role in understanding theoretical complexity of PPA-completeness.Hopefully, one day, they will become truly useful like other creatures of human imagination, if one so demands.

Related Literatures
The standard Sperner's problem, 3D-Sperner, is among the first natural problem proved to be PPAD-complete by Papadimitriou [27].The problem 2D-Sperner is proved to be C C C 2 0 1 6

23:4
Understanding PPA-Completeness PPAD-complete by Chen and Deng [7].In [17], Grigni proposed the brilliant idea using non-orientable space to model the 3D-Sperner as a PPA-complete problem.The only other known PPA-complete problem is the Sperner problem on a sophisticated locally 2D structure by Friedl, Ivanyos, Santha and Verhoeven [15].
Lemke-Howson's algorithm [24] for Nash equilibrium computation has started a path following paradigm.However, a worst case exponential lower bound was known for this algorithm by Savani and von Stengel [28].It was shown that the other PPAD-complete problems demand, under the oracle model, exponential time including the fixed point problem by Hirsch, Papadimitriou and Vavasis [19].It was further shown to have a tight exponential time by Chen and Deng [8], which was extended to include several discrete versions of the fixed point problem by Deng, Qi, Saberi and Zhang [11].
For the PPA class, the path following method was known to take an exponential time for the Smith problem by Krawczyk [23,4].It has been extended to related problems, such as an exponential time bound for finding the second perfect matching on Eulerian graphs by Edmonds and Sanita [12].An extensive discussion on related problems can be found in [5].Subsequently, Aisenberg et.al. [1] improved our result -proving that general version for 2D-Tucker is PPA-complete using an elegant trick.

Organization of Presentation
We prove that the natural Möbius band versions of the problems, Sperner, DPZP and Tucker to be PPA-complete.A neat reduction allows the problem of finding one fixed point be extended to given-one-find-another types of PPA problems.Along with several important technical details, a dicephalic snake lemma is crucial for the padding and folding to create a higher dimensional fixed point on a non-orientable grid in order to reduce the problem to one of constant side lengths.
The paper is laid out as follows: In Section 2, we will show some necessary definitions and notations.In Section 3, we show a key result, the proof of PPA-completeness of the problem mn-DPZP and its applications.In Section 4, we extend our work to prove a high-dimensional non-orientable version of fixed point.In Section 5, we applies our main result to other non-orientable spaces, including the projective plane and the Klein bottle.We prove the PPA-completeness of the problem of finding another fixed point on the projective plane and on the Klein bottle.In Section 6, we discuss the generality of the results obtained here in related settings.Finally we discuss potential future works.Because of the space limitation, we put most of our proofs in the Appendix.

Preliminaries and Definitions
PPA, (in its complete form, the Polynomial Parity Argument class), is a class of search problems based on an exponential size graph consisting of nodes of maximum degree two, with a given node of degree one.The problem asks for an output of another node of degree one, which is guaranteed to exist by the parity argument.More formally, we define it by a complete problem, named AEUL as follows: Definition 2.1 (Another End of Undirected Lines).Given an input circuit T n of polynomial size in n which takes as input u in the configuration space C n = {0, 1} n , returns as output T n (u) in the form v, w , v , or where v > w and v, w ∈ C n \ {u}.0 n is a given configuration of one tuple, i.e., |T n (0 n )| = 1.The search problem is to find another configuration v, v = 0 n such that |T n (v)| = 1.We should write it as AEUL for short.

Möbius Band
It is obtained from a rectangle by merging its left and right sides after twisting it 180 degrees (counter)-clockwise to form a one-boundary and one-surface band.Therefore, it is non-orientable.More formally, A Möbius band is obtained by twisting V N,M 180 degrees clockwise and then joining every vertex (N, y) with (−N, −y) to form a loop.We denote it by B N,M .A function f is defined on the Möbius band B N,M iff ∀y : −M ≤ y ≤ M , we have f ((N, y)) = f ((−N, −y)) on V N,M .

Definition 2.3 (Standard Triangulation).
For each i, j ∈ Z : −N ≤ i < N, −M ≤ j < M , we link (i, j) with (i + 1, j + 1) on the grids V N,M and B N,M .
We call every unit square in the standard triangulated grid V N,M a base square, every unit side length triangle of it a base triangle, its every edge a base edge.

Index
We now define the index [29,31] but adopt it for the non-orientable space B N,M .
Consider a coloring by {0, 1, 2} of vertices in B N,M , one vertex is assigned by one color.If a base triangle δ has all three colors, we define its index as 1.Otherwise, the index is 0. Alternatively, we define an edge index to be 1 if it is colored by both 1 and 2. The index of a base triangle is the sum of indices of its three edges, mod 2. It prepares us to define the index on Möbius band.

Definition 2.4 (Index of a Non-orientable Triangulated Möbius
Immediately, one derive the following lemma about indices on the Möbius band.Lemma 2.5.

DPZP
We should introduce several concepts to prepare its definition as a numeric version of the original direction preserving zero point.

Definition 2.7 (Möbius Numeric Direction-preserving Function). A function
Definition 2.10 (Numeric Möbius DPZP).Given as input, a triangulated Möbius Grid B N,M , and a polynomial-time machine F , which generates a numeric direction-preserving feasible admissible function As the function F (•, •) for mn-DPZP has five values, the index defined above does not apply.We should introduce a new definition of index for mn-DPZP.Definition 2.11 (Index of a Base Edge and a Base Triangle in mn-DPZP).Given an mn-DPZP grid B N,M , a coloring F : B N,M → {0, ±1, ±2}, of its vertices.The index of an edge is 1 if the colors of its two end vertices are {1, 2}, 0 otherwise.The index of a base triangle is the sum of the indices of its three edges (mod 2).Definition 2.12 (Index of mn-DPZP).Given a mn-DPZP grid B N,M , a coloring F : We have the following lemma on the Möbius grid.

Lemma 2.13. index(δ, F ) = 1 if and only if
Using the index on non-orientable surfaces, it is immediately that:

Lemma 2.14. The Numeric Möbius DPZP with the admissible boundary always has a zero point. Finding a zero point is a PPA problem.
We should next list the results for other related discrete fixed point concepts.We call the problem of finding a fully colored base triangle on Möbius band B N,M the m-Sperner problem.

Definition 2.15 (m-Sperner). Consider a triangulated Möbius grid B N,M and a polynomialtime machine G, which generates a function g on B
Further, we require that g(•) satisfies the m-Sperner boundary condition, defined as follows.
The required output is a base triangle which contains all three colors.Lemma 2.16.On any admissible triangulated Möbius band B N,M for an m-Sperner instance, the number of Sperner base triangles is odd.Finding one of those is in PPA.
Proof.As m-Sperner has index 1, the oddness follows.The reduction to an AEUL is similar to the above for the mn-DPZP problem.
We define the simple Möbius version of Tucker as follows.
Definition 2.17 (sm-Tucker).Consider a triangulated Möbius grid B N,M and a polynomialtime machine G, which generates a function g on B N,M : g(p) = G(p) ∈ {±1, ±2}, ∀p ∈ B N,M .Further, we require that g(•) satisfies the special antipodal boundary condition, defined as follows: The required output is a complementary edge.Lemma 2.18.On sm-Tucker, there is always a complementary edge.Finding one is in PPA.
Proof.Changing the colors {−1, −2} of the vertices in sm-Tucker into 0, we reduce the problem to m-Sperner.As the boundary of the m-Sperner has index 1, there is always a fully colored base triangle δ.The vertex colored 0 in δ was originally either −1 or −2 in the sm-Tucker, we obtain a complementary edge in the sm-Tucker.The claims follow.

PPA-completeness of mn-DPZP and Its Applications
We have already proven that mn-DPZP is in PPA in the last section.We now prove the PPAhardness of the mn-DPZP.For any input to AEU L(T n , C n , 0 n ), we construct an mn-DPZP instance in polynomial time so that each zero point in the mn-DPZP instance maps back to an end vertex for some lines in the original instance of AEU L(T n , C n , 0 n ), and vice versa.
Our proof embeds the AEUL(T n , C n , 0 n ) graph on the Möbius band.The reduction is motivated by the original proof of 2D Sperner being PPAD-complete by Chen and Deng [7].
Given a simple undirected graph , and E * = {(p, p ) : p − p 1 = 1}, i.e., (p, p ) is an edge in G * if and only if their L 1 distance is 1.For every p ∈ V * , let K p = q : q i ∈ {p i , p i +1} , i = 1, 2 to be the vertex set containing all 4 vertices in the base square having p at the left bottom corner, and E 1 p = {{p, p+(0, 1)}, {p+(1, 0), p+(1, 1)}}, E 2 p = {{p, p+(1, 0)}, {p+(0, 1), p+(1, 1)}} to be its two subsets of edges of K p .For p, q ∈ V * , if p i = q i , i = 1 or 2, let u 1 , u 2 , . . ., u m ∈ Z 2 be all the integer internal points on segment pq which are labeled along pq, where u 1 = p and u m = q.We say K p and K q are connected iff edges set On the Möbius band, we also allow that don't share the same xcoordinate nor y-coordinate, hence the edges introduced need to make turns in its directions to connect u 1 to u m .We make a special note that, at a turn on K u toward the right-upper direction, the edges {{u, u + (0, 1)}, {u + (0, 1), u + (1, 1)}} will be removed to make the nodes along the paths be of degree no more than two.
Intuitively, G * is a plannar embedding of the graph G for AEUL with vertices {0, 1, . . ., N − 1}.The construction is motivated by and has some similar details to that of Chen and Deng [7].Making it work on the Möbius band requires new ideas.For every i : 0 ≤ i < N , vertex i of G maps to a vertex set S i = ∪ 24i+11 k=24i {K (0,k) }.That is, we create a fixed-length "tube" S i for the vertex i in G.We call it a "vertex tube".Every such tube has two ends, called up and down, dependent on their values of the second coordinates on V * , denoted by S up i = K (0,24i+11) and S down i = K (0,24i) .We make a change in the embedding of the starting node 0 n : for i = 0, S 0 = ∪ For each vertex tube, we connect its up end to its bigger neighbour (if the degree of the vertex is 1, we also take it as the bigger one), and its down end to the smaller neighbour (if any).
If (i, j) is an edge in G, let y i , y j be the y-coordinates of the ends of tube i and j where need to be linked together.Let t = 12(N • max{i, j} + min{i, j}).We consider two different connection cases: 1. S up i − S down j or S down i − S up j : we add edges K (0,yi) K (t,yi) , K (t,yi) K (t,yj ) , K (t,yj ) K (0,yj ) into E * .

S up
i −S up j or S down i −S down j : w.l.o.g., we assume that i < j, we add edges Case 1 is illustrated in Figure 1, which is a normal case.The crucial difference that would involve in the Möbius band structure B 12N 2 ,24N is case 2, illustrated in Figure 2.For example, if degree of i is 2, i.e.T (n, i) = j, k , k > i, j, also we assume that i > j and T (n, j) = k, i .Let t = 10(n • i + j), we will link S down i and S down j by adding edges The remaining difficulties of the reduction are how to color the vertices of G * according to the requirements for Möbius DPZP and how to handle crossing paths.We should present techniques to handle them in the proof.

Lemma 3.1. mn-DPZP is PPA-hard.
We conclude that Möbius DPZP and Möbius Tucker are PPA-complete.Proof.sm-Tucker is in PPA by Lemma 2.18.
For PPA-hardness, we use the same construction as the proof of Lemma 3.1, except that we change vertices colored 0 to color −2.Therefore, at each vertex of color 0 in Lemma 3.1, we have an edge of color +2 and −2; and vice versa.The reduction follows.
Therefore, the theorem holds.
Finally we show that m-Sperner is PPA-complete.
Proof.First, m-Sperner is in PPA by Lemma 2.16.
To prove it is PPA-hard, we simply replace vertices colored {−1, −2} to color 0 in the instance constructed in the PPA-hardness proof of mn-DPZP.Finding a fully colored triangle δ in the m-Sperner instance will imply a true zero point in the mn-DPZP instance because the direction preserving condition, Definition 2.7, for mn-DPZP will prevent another vertex in the same base triangle of color∈ {−1, −2}.
The claim follows.

High Dimensional Non-orientable Discrete Fixed Point
In the above, some 2D fixed point problems on the Möbius band are proven PPA-complete.
The generalized problem in higher dimension space with all constant side lengths is considered in this section.The proof is motivated by a construction in [9].To handle the non-orientable space, the key changes are on the snake lemma.We need a dicephalic snake version.Considerable changes and new ideas are required to make it through.To avoid tedious details, we should present a version of the construction and the proof.To observe the page limit, we place all the proofs and some lemmas at the appendices.

Uniform Boundary Discrete Fixed Points on Möbius Band
We introduce a version here for which the boundary of the 2D Möbius band consists vertices all of the same color.Every instance of the problem has index 0.This naturally leads to a version of the fixed point problem where one fixed point is given and another is sought after.We call such a case the uniform boundary coloring.More precisely, the coloring function f is of uniform boundary on Möbius band B N,M if it satisfies that: (1) f ((x, ±M )) = 0, ∀x ∈ Z, −N ≤ x ≤ N .(2) Möbius condition, i.e. f ((N, y)) = f ((−N, −y)), ∀y ∈ Z, −M ≤ y ≤ M .Then the Möbius Sperner problem can be defined as follows.

High Dimensional Möbius Sperner
We extend the 2-dimensional uniform boundary Möbius Sperner proven PPA-complete in the above to higher dimension.First we define the well-behaved function.

Definition 4.3 (Well-behaved Function [9]). A polynomial-time computable integer function
For a positive integer d and a vector r ∈ Z d + , let be the hyper grid with side length r (note that is 2(r i − 1) in the i-th dimension because of symmetry with respect to r i = 0).Note that its boundary is, in one dimension, intentionally left open, 3, . . ., d} are called reversing face.Even though they are not on the boundary, we include (2) here to make sure the consistency of function values on the non-orientable space.Fixing other variables, x 3 , x 4 , • • • , x d , we have a reversing plane for the variables x 1 and x 2 .
For any well-behaved function f , we define a corresponding Möbius-Sperner fixed point problem as follows.Definition 4.5 (Möbius Sperner f ).For a well-behaved function f and a parameter n, let m = f (n) and d = n/f (n) .An input instance of Möbius Sperner f is a pair (C, 0 n ) where C is a valid coloring function with parameter d and r where r i = 2 m , ∀i : 1 ≤ i ≤ d.Given a point p ∈ A d r where K p is of degree one, i.e., contains one panchromatic simplex in its triangulation, the output of this problem is another point q = p, such that K q contains another panchromatic simplex.
We have the following theorem.
Theorem 4.6.The problem Möbius Sperner f is PPA-complete for any well-behaved function f .One can show that this problem is in PPA.To prove the hardness, similar to the orientable space [9], we embed an instance of Möbius Sperner f2 , known in PPA-complete, into one dimensional higher space iteratively till Möbius Sperner f .We should show that the process can be done in a polynomial number of state transformations.In Subsection B, we show three crucial lemmas for our reduction.In Subsection C, we employ these three lemmas iteratively to build up our construction.Please see the Appendix for the detail proofs.

Discrete Fixed Points on Projective Space and Klein Bottle
The results we have discussed above extend to other non-orientable spaces.The general idea is to slice out a Möbius band from the more complicated non-orientable space and to color it properly, then to patch the rest of the space.Two of the most interesting ones are the projective space and Klein Bottle.While the Möbius band can be embedded into 3D Euclidean space, neither the projective space nor the Klein bottle can.In this section, we make a reduction from DPZP to both the Möbius band and the projective plane for the PPA-hardness.As usually, as both cases are two dimensional objects, it is easy to triangulate them and to develop a path following algorithm.
We have discussed two types of discrete fixed point problems in the above.1. finding one, and 2. (given one) finding another, dependent on the boundary conditions.As both the Projective space and the Klein bottle are closed without a boundary, we need to use the second version.
Our presentation will focus on the mn-DPZP version of the problems.The same applies to other types of discrete fixed point concepts discussed above.We omit them here as the results are similar.

Remarks and Discussion
We have discussed two types of discrete fixed point problems on the Möbius band: finding one, and (given one) finding another, dependent on the boundary conditions.We show both problems are PPA-complete for several versions of discrete fixed point models, including the Sperner's problem on the two dimensional Möbius band.

23:12
Understanding PPA-Completeness Our first step focuses on the 2D version.We start with mn-DPZP, which finds a zero point of a discrete version of the continuous functions.Based on this result, we derive PPA-completeness proof of several other related fixed point problems on the Möbius band.We discuss finding another for Möbius Sperner and Index1-Brouwer on Möbius Band.We discuss finding one for sm-Tucker and mn-DPZP.They are switchable into the other types.For example, we can change all negative colored vertices to color 0 in mn-DPZP to obtain a "finding one" version for Möbius Sperner.We leave those cases out in this version and only exemplify useful structures and techniques choosing the most typical cases.
In this work, the link between non-orientable topological space and undirected path following computational paradigm, started by Grigni in [17], is further ratified by the simple structure of 2D Möbius band.It deepens our understanding of the computational complexity difference between the two classes PPAD and PPA in terms of the underlying topological structures.
The simplicity of our construction allows itself to extend beyond the 2D Möbius band to more general cases.For example, the PPA completeness of the finding another fixed point version extends naturally to the Klein Bottle, the projective space, and to other non-orientable surfaces [32].Simplicity has played a role in raising further curiosities from the 2D Sperner work [7] in the orientable space, such as in [25,16].
Further the results extend to higher dimensions, even for the case where each side is of a constant length.One such high dimension non-orientable space case of finding-another fixed point is presented in Section 4. The result extends to different related solution concepts as in the previous related concepts.
Note that the discrete fixed point problems in our discussion has an exponential size configuration.Otherwise, we can enumerate the space to find a solution by brute force.To compute colors and function values, a polynomial size circuit is given as an input.Alternatively, an oracle model returns those values in a unit oracle time [19].It is known that there is an asymptotic matching bound for finding the Brouwer's fixed point in Euclidean space [8], which extends to other discrete fixed point models [11].The same holds for the nonorientable space we discuss here.The lower bound holds simply because the problem is harder in the non-orientable space.The upper bound follows by the standard divide-and-conquer on the index adopted for the non-orientable space.
We would like to see the natural 2D Möbius Sperner will encourage more constructive works to develop a better knowledge of the PPA-complete class.In particular, as had suggested by Grigni [17], we would like to see the computational complexity of the Smith's Theorem, known in the class of PPA, be eventually resolved.
is an odd number of zero point base triangles on the Möbius grid.Therefore, there is always a zero point inside the Möbius grid.
For the construction of the AEUL, we take the boundary edge (2, 1) as the origin vertex of AEUL.Two such edges of mn-DPZP are connected in AEUL if they are in the same base triangle.Any such edge in mn-DPZP is an leaf node in AEUL if it is the single {1, 2} edge in a base triangle.
Therefore, an end of lines of the AEUL instance is a base triangle of the mn-DPZP.Finding a zero point base triangle is an AEUL problem, and in PPA.
Proof of Lemma 3.1.Using the main structure presented above, we show how to color B 12N 2 ,24N , so that for any zero point of this mn-DPZP, we can get a corresponding solution for the search problem AEUL.
The circuit T n of AEUL generates an undirected graph G = (C n , E), where So given any G, we construct an instance (f, G * ) for mn-DPZP problem where f is a coloring function for the generated G * = B 12N 2 ,24N .We should also use T 12N 2 ,24N to refer to G * in case of no ambiguity, with the understanding that (−12N 2 , y) and (12N 2 , −y) are the same vertex.
In constructing the coloring function f for G * , we make use of the input circuit of T n , to identify edges connecting a node to another, and vice versa, and to identify the degree one node of the AEUL graph.
We define the coloring function as follows: 1. Color vertices on the boundary according to the admissible conditions, Definition 2.9.

Color the long vertex tube: ∀j
1, k = 0, 1, 2, . . ., 11.We need to make some modifications in the colors for the case k = 0 later.5. Coat vertex tubes (to protect positive colored 1 and 2 inside tube): ∀i : 0 < i < N : , . . ., 11. 6. Make feasible: fill in the the rest of the interior vertices by color −2.Some of those vertices will be re-colorred in the following steps.7. Direction preserving on end of lines: For a leaf vertex i : 0 < i < N , we have f (0, 24i) = f (1, 24i) = 0. 8. Build an edge path: Given an edge (i, j) ∈ E, w.l.o.g., assume that i < j, we construct a path between i and j in G * .Let (i , j) ∈ E and (i, j ) ∈ E. If j > j , then the upper end of tube for i is connected to that of j, else the lower end of the tube for i is connected to that of j.Therefore, there are four possibilities one end of the vertex tube is connected to another vertex tube.a. i > i and j < j : Lower end of vertex tube for i is connected to the upper end of the vertex tube for j.See Figure 1. b. i < i and j > j : Upper end of vertex tube for i is connected to the lower end of the vertex tube for j.See Figure 1.c. i < i and j < j : Lower end of vertex tube for i is connected to the lower end of the vertex tube for j.See We should make appropriate adjustments so that the colorings consistently link two vertex tubes.9. We need parallel paths of width 4, making the colors crossing it to be −1, 2, 1, −2 (or −2, 1, 2, −1 , dependent on the direction we are moving) to maintain the direction preserving conditions.The vertex tubes for i and j connected in the four ways specified above will maintain it, that their colorings are consistent.Note that if the path will pass through the direction-reversing line, it must satisfy the Möbius condition, that is, the four vertices crossing the path reverse their colors from from −1, 2, 1, −2 to −2, 1, 2, −1 (or vice versa) after crossing the reversing direction boundary.
The colorings along the parallel paths satisfy our condition of direction preserving, as well as feasibility and admissibility conditions, except the problem where two paths cross each other.We resolve it in the same way originated from [7], shown in Figure 3.All the changes are local and can be decided using the local information with a constant bounded number of uses of the circuit T .Now we have provided the admissible coloring function that, given any point in B 12N 2 ,24N , provides its coloring in polynomial time using the polynomial time circuit T .
Note that vertices of color 0 in G * only appear in the mapping from G to G * from a vertex of degree one in G.
Therefore, finding a vertex of color 0 in G * is equivalent to find the AEUL solution in G.
Hence we have proven that mn-DPZP is PPA-hard.
Proof of Lemma 4.2.Clearly, the degree of any instance is 0. Therefore, there is an even number of the fully colored base triangles.Given one fully colored Sperner base triangle, the existence of another follows by the above lemma.The problem is in PPA because the relationship of two edges on a base triangle of colors (1,2) still holds and the uniform color boundary condition prevents the paths in the underlying AEUL going out of boundary.
On the other direction, m-Sperner can be easily reduced to Uniform-Color-Boundary Möbius Sperner by coating an extra layer of vertices outside of the boundary and coloring them all zero.More specifically, for each instance of m-Sperner, we create a Uniform-Color-Boundary Möbius Sperner by adding new vertices {(i, ±(M + 1)) : −N ≤ i ≤ N } with all 23:17  color 0. After this construction, we have an instance of Uniform-Color-Boundary Möbius Sperner.There is a fully colored base triangle given ({(0, −M − 1), (0, −M ), (1, −M )}).Our goal is to find another which is also one for the original m-Sperner instance.
The Proof of Theorem 5.1.We make a reduction from mn-DPZP to the same problem on the projective space.We will define the DPZP on the projective plane.Then, we make a reduction of mn-DPZP to the DPZP on the projective plane.
First, a 2D projective plane can be obtained from the sphere of a 3D unit ball by identifying two points share the same diameter.In other words, (x, y, z) and (−x, −y, −z) in B = {(x, y, z) : x 2 + y 2 + z 2 = 1} are merged into one point.
Next, it can be decomposed into a Möbius band and a disc as follows.M = {(x, y, z) : |z| ≤ 1/2 : (x, y, z) ∈ B} and D + = {(x, y, z) ∈ B : z ≥ 1/2} D − = {(x, y, z) ∈ B : z ≤ −1/2}.Here D + and D − merge into one.We color the disc as in the central figure, at the center of the disk we place a zero point Figure 4.
Further, let M + = {(x, y, z) ∈ M : x ≥ 0 and M − = {(x, y, z) ∈ M : x ≤ 0. We have M + is a Möbius band and so is M − .Moreover, the interior of M − maps into that of M + in a 1-1 mapping.They have a shared boundary on x = 0 that corresponds to the direction reversing line discussed in the above.Using the standard mn-DPZP boundary condition 2.9, we can embed an mn-DPZP instance on M + as in Figure 5.
Connecting the Möbius band on M + with the disk D + along their boundaries, we construct a triangulated projective plane that has a zero point on D + with the task of finding another zero point, which can only be on M + .As M + is equivalent to an mn-DPZP instance, it follows that the task is a PPA-complete problem.The Proof of Theorem 5.2.Similarly, the PPA completeness of the finding another fixed point version extends naturally to the Klein Bottle [32].Here again, the Klein Bottle can only be embedded in the four dimensional space.It is rather awkward to present it in the 3D world we live.Here we present by a 2D view with some amendments for ease of discussion in Figure 6.All the three non-orientable 2D spaces are represented uniformly in the 2D grid, with their boundaries merged with the opposite sides as illustrated.This clear presentation allows a simple embedment of the DPZP grid on to the Klein bottle as presented in the following Figure 7.
Here we merge the top line and the bottom line.On it, there is a zero point in the middle.We are asked to find another zero point on the Klein bottle constructed from this grid.If we remove the top 4 lines and the bottom four lines, we obtain a mn-DPZP on the Möbius band where the only other zero points could be hidden.
Therefore, given the top zero point, finding another is to find a zero point in the mn-DPZP for the Möbius band in the middle.The embedding is carried out by a sequence of three polynomial-time transformations: L 1 (T, t, u), L 2 (T, u), and L 3 (T, t, a, b).L 1 (T, t, u) increases the t-th dimension size of the hyper grid from r t to u (requiring u > r t ).L 2 (T, u) extend the colouring into a space one dimension higher.L 3 (T, t, a, b) folds a Möbius grid T to T so that one more side length in a dimension is reduced to a constant size.At the same time, from every panchromatic simplex of T , one can find a panchromatic simplex of T efficiently.We should use e i as the vector for the i-coordinate.
{K (0,k) } in G * .Edge ij appears in G iff there is an undirected path between one of {S up i , S down i } and one of {S up j , S down j }.Let ij ∈ E and ik ∈ E be the two edges connected to j and k from i.If j > k we call j the bigger neighbour and k the smaller neighbour of the vertex i.

Definition 4 . 1 (Lemma 4 . 2 .
Möbius Sperner).The input is a polynomial-time machine F that generates a uniform boundary 3-coloring function f on B N,M : F (p) = f (p) ∈ {0, 1, 2}, ∀p ∈ B N,M , as well as a panchromatic base triangle.The required output is another panchromatic base triangle on B N,M .Note that index(B N,M , f ) is zero for a color function of uniform boundary on the Möbius band.According to Lemma 2.5, we have the following lemma: For any uniform boundary 3-coloring of the triangulated Möbius band B N,M , the number of panchromatic base triangles is even.Given one panchromatic base triangle, finding another is a PPA-complete problem.

Figure 2 .Figure 3
Figure 3 Connection Crossing: four cases are listed as Case 1, Case 2, Case 3, Case 4 as the normal order.

Figure 6
Figure 6 Grid Views: Möbius band, Klein bottle and Projective plane.

Figure 7
Figure 7 Embed DPZP on to Klein bottle.

A
triple T = (C, d, r) is a coloring triple if r ∈ Z d with r i ≥ 3 for all 1 ≤ i ≤ dand C is a valid coloring function with parameters d and r.Let Size [C] denote the number of gates plus the number of input and output variables in a function C.