A Simplicial Approach for Discrete Fixed Point Theorems (Extended Abstract)

A Simplicial Approach for DiscreteFixed Point Theorems(Extended Abstract)Xi Chen1and Xiaotie Deng2⋆1Department of Computer Science,Tsinghua Universityxichen00@2Department of Computer Science,City University of Hong Kongdeng@.hkAbstract.We present a new discretefixed point theorem based on anovel definition of direction-preserving maps over simplicial structures.We show that the result is more general and simpler than the two re-cent discretefixed point theorems by deriving both of them from ours.The simplicial approach applied in the development of the new theoremreveals a clear structural comparison with the classical approach for thecontinuous case.1IntroductionThere has recently been a sequence of works related tofixed point theorems in a discrete disguise,started with the seminal work of Iimura[12]which introduced a crucial concept of direction-preserving maps.Iimura,Murota and Tamura[13] corrected the proof of Iimura for the definition domains of the maps.With a different technique,Chen and Deng introduced another discretefixed point the-orem in order to achieve the optimal algorithmic bound forfinding a discrete fixed point for allfinite dimensions[1].In[14],Laan,Talman and Yang designed an iterative algorithm for the discrete zero point problem.Based on Sperner’s lemma which is fundamental for deriving Brouwer’sfixed point theorem,Friedl, Ivanyosy,Santha and Verhoeven defined the black-box Sperner problems.They √also obtained aon the simplicial structure in the study of the discrete version will help us gain a full and clear understanding of the mathematical structures and properties related to discretefixed point theorems.Furthermore,even for continuousfixed point theorems,discrete structural propositions are needed to derive them.Our study would provide a unified view of thefixed point theorem,both discrete and continuous,instead of treating them with ad hoc techniques.Our simplicial ap-proach unveils the mystery behind the recent results on discretefixed points and settles them under the same mathematical foundation as the classical continuous fixed point theorems.The discrete nature of thefixed point theorem has been noticed previously, mainly due to the proof techniques of Sperner’s lemma[15].The recent effort in direct formulation of the discrete version of thefixed point theorem would be especially useful in the complexity analysis of related problems.The recent work in characterizing the complexity of Nash Equilibria,by Daskalakis,Goldberg, Papadimitriou[8],Chen and Deng[2],Daskalakis and Papadimitriou[9],Chen and Deng[3],has been based on another innovative formulation of the2D(or 3D)discretefixed point problem,where afixed point is a collection of four[6] (or eight[8])corners of a unit square(or cube).It’s difficult to generalize such a formulation to high dimensional spaces,since a hypercube has an exponential number of corners,which is computationally infeasible.Instead,a simplicial de-finition has been necessary in extending those results to a non-approximability work obtained recently[9].Wefirst introduce notations and definitions with a review of previous works of Murota,Iimura and Tamura[13],as well as Chen and Deng[1].The simplicial model is then introduced in section3and the fundamental discretefixed point theorem is proved in section4.In section5,we present the discrete Brouwer’s fixed point theorem for simplicial direction-preserving maps,with the theorem of Murota,Iimura and Tamura[13]derived as a simple corollary.In Section6, we give an explicit explanation for the definition of bad cubes in[1]and show that,the theorem of Chen and Deng is a special case of the fundamentalfixed point theorem.Finally,we conclude in section7.2Preliminaries2.1Notations and DefinitionsInformally speaking,map F(or function f)is hypercubic direction-preserving on afinite set X⊂Z d if for every two neighboring points in X,their directions given by F(or f)are not opposite.The neighborhood relation considered here is defined by the infinity norm.Definition1(Hypercubic Direction-Preserving Maps)Let X be afinite subset of Z d.Map F from X to R d is said to be hypercubic direction-preserving on X if for every two points r1,r2∈X with||r1−r2||∞≤1,we have(F i(r1)−r1i)(F i(r2)−r2i)≥0,for all i:1≤i≤d.2Definition2(Hypercubic Direction-Preserving Functions)Let X be a finite subset of Z d.Function f from set X to{0,±e1,±e2...±e d−1,±e d}is said to be hypercubic direction-preserving if for every two points r1,r2∈X such that ||r1−r2||∞≤1,we have||f(r1)−f(r2)||∞≤1.Point r∈X is called afixed point of F(or f)if F(r)=r(or f(r)=0). 2.2The Fixed Point Theorem of Murota,Iimura and TamuraMurota,Iimura and Tamura proved in[13]that every hypercubic direction-pre-serving map from an integrally convex set X toX to denote the convex hull offinite set X⊂Z d.Definition3(Integrally Convex Sets)Finite set X⊂Z d is integrally con-vex if for all x∈X∩N(x)where N(x)={r∈Z d|||r−x||∞<1}.Theorem1([13])Let X be an integrally convex set in Z d,then every hyper-cubic direction-preserving map F from X to3Simplicial Direction-Preserving Maps and FunctionsIn this section,we introduce simplicial direction-preserving maps and functions based on simplicial structures.Let X be afinite set in R d.Here we only con-sider nondegenerate cases whereX such that for every S∈S,V S⊂X,where V S is the vertex set of simplex S.Given a simplicial decomposition S of X,we use F S to denote the set of (d−1)-simplices on the boundary ofX:F S={(d−1)-simplex S∈S|S⊂F and F is a facet ofX}.Definition8(Simplicial Direction-Preserving Maps)A simplicial direc-tion-preserving map is a triple M=(F,X,S).Here X is afinite set in R d and S is a simplicial decomposition of X.Map F from X to R d satisfies for every two points r1,r2∈X,if there exists a simplex S∈S such that r1,r2∈V S,then (F i(r1)−r1i)(F i(r2)−r2i)≥0,for all i:1≤i≤d.Definition9(Simplicial Direction-Preserving Functions)A triple G= (f,X,S)is said to be a simplicial direction-preserving function if X is afinite set in R d,S is a simplicial decomposition of X,and function f from set X to {0,±e1,...±e d}satisfies for every two points r1,r2∈X,if there exists S∈S such that r1,r2∈V S,then||f(r1)−f(r2)||∞≤1.In other words,for every two neighboring points in X,their directions given by map F(or function f)can’t be opposite.The only difference with the hy-percubic model is that the neighborhood relation is now defined by simplices in the simplicial decomposition S instead of unit d-cubes in Z d.4The Fundamental Discrete Fixed Point TheoremIn this section,we present the fundamental discretefixed point theorem which is both simple and powerful.Any simplicial direction-preserving function which satisfies the boundary condition of the theorem must have afixed point.Definition10(Bad Simplices)Let G=(f,X,S)be a simplicial direction-preserving function,where X⊂Z d.A t-simplex S∈S where0≤t≤d is said to be bad(relative to function G)if f(V S)={e1,e2,...e t+1},where V S is the vertex set of S.We use N G to denote the number of bad(d−1)-simplices in F S.4Lemma1For any simplicial direction-preserving function G=(f,X,S),if there exists nofixed point in X,then N G is even.Proof.Firstly,one can show that for every(d−1)-simplex S∈S,if S∈F S, then there exists exactly one d-simplex in S containing S.Otherwise,there are exactly two such ing this property,the parity of N G is same as the one of the following summation:d-simplex S d∈Sbad(d−1)-simplices in S d.As G is direction-preserving and has nofixed point,the number of bad(d−1)-simplices in S d is either0or2.Therefore,the summation above must be even.We now get the fundamental theorem as a simple corollary of Lemma1. Theorem3(The Fundamental Discrete Fixed Point Theorem)Let G=(f,X,S)be a simplicial direction-preserving function.If N G,i.e.the number of bad(d−1)-simplices on the boundary is odd,then G must have afixed point r∈X such that f(r)=0.5The Discrete Brouwer’s Fixed Point TheoremIn this section,the fundamental discretefixed point theorem will be employed to prove afixed point theorem concerning simplicial direction-preserving maps. It can be recognized as a discrete version of Brouwer’sfixed point theorem.It states that for any simplicial direction-preserving map from somefinite set to its convex hull,there must exist afixed point in the definition domain.We will also derive the theorem of Murota,Iimura and Tamura as a simple corollary.Actually,the one derived here is much stronger than theirs.5.1PreliminariesWe use e k to denote the k th unit vector of Z d where e k k=1and e k i=0for all i:1≤i=k≤d.Definition11For every(d−1)-simplex S∈F S,we let e S be the unit vector which is outgoing and perpendicular to S.For all r∈X if e S·(r−r S)>0for some r S∈S. Construction1(Extension of Simplicial Decomposition)Let X⊂R d be afinite set and S be a simplicial decomposition of X.For every point r/∈Given a simplicial direction-preserving map M=(F,X,S),we can convert it into a direction-preserving function G=(f,X,S)as follows.Construction2Given a simplicial direction-preserving map M=(F,X,S), we can build a simplicial direction-preserving function G=(f,X,S)as follows. For every r∈X,if F(r)=r,then f(r)=0.Otherwise,let i:1≤i≤d be the smallest integer such that F i(r)−r i=0,then f(r)=sign(F i(r)−r i)e i.5.2The Key LemmaLemma2Let M=(F,X,S)be a simplicial direction-preserving map where F is from X toX′where F′(r)=F((n+1)r) n+1.Let G′be the function constructed from map M′=(F′,X′,S′),then it is easy to check that N G=N G′.Therefore,we only need to prove the lemma for maps M=(F,X,S)with X⊂(−1,1)d.From now on,we always assume that X⊂(−1,1)d.If f has afixed point in set B X,then the lemma is proven.Otherwise,we extend(by applying Construction1for d times)G=(f,X,S)to be a new function G∗=(f∗,X∗,S∗)such that X⊂X∗,Lemma3guarantees the existence of G′.Lemma3For every triple G=(f,X,S)which satisfies all the conditions inDefinition12,there exists a triple G′=(f′,X′,S′)which is an extension of G along the k th coordinate.Proof.We only need to prove that there exists a simplicial decomposition S′ofX′satisfies S⊂S′.Actually,it can be constructed as follows.We list all the points in X′−X using the lexicographic order,add them into X one by one,and extend the original decomposition S step by step using the method described inDefinition1.Definition13G∗is said to be an extension of G if there exists a sequence G d+1...G2,G1such that G d+1=G,G∗=G1and G i is an extension of G i+1along the i th coordinate.The existence of G∗is guaranteed by Lemma3.Let G k=(f k,X k,S k).The following lemma is easy to check.Lemma4If r∈X k,r′∈[−1,1]d and r i=r′i for all1≤i<k,then r′∈X k.The following two corollaries can be derived from Lemma4.Corollary1X. Corollary2If v=e k−1or−e k−1is an outgoing vector at r∈X k,thenr+ǫv+di=kc i e i/∈X.If G is the function constructed from M using the method in Definition2,then any extension G∗of G is also simplicial direction-preserving.7Proof.We will use induction on k to prove that triple G k=(f k,X k,S k)is a simplicial direction-preserving function.The case for k=d+1is trivial.For case k≤d,if G k is not direction-preserving,then there must exist S∈S k−S k+1 and r1,r2∈V S such that f k(r1)and f k(r2)are opposite.As G k+1is direction-preserving,one of these two points belongs to X k+1,and the other is added into X k when G k+1is extended along the k th coordinate.Without loss of generality,we assume r1∈X k+1and r2k=1.As a result, f k(r2)=−e k and f k(r1)=+e k,as f k(r1)is opposite to f k(r2).The value of f k(r1)implies r1∈X and f(r1)=+e k.The way we construct G from map M guarantees F i(r1)=r1i for any1≤i<k and F k(r1)>ing Corollary2, we see that+e k is not an outgoing vector at r1∈X k+1,as F(r1)∈X k+1. Hence lemma5asserts that no pair(r2,S)where r2∈X k−X k+1and S∈S k−S k+1can satisfy that r2k=+1and r1,r2∈V S,which contradicts with our assumption in thefirst paragraph.Property1is a simple corollary of Lemma7.Lemma7For every triple G=(f,X,S)wherefinite set X⊂(−1,1)d,S is a simplicial decomposition of X and f is a direction function on X,if G∗= (f∗,X∗,S∗)is an extension of G,then N G∗is odd.Proof.We will use induction on d.The base case for d=1is trivial.For case d≥2,wefirst prove that every bad(d−1)-simplex S∈S∗on the boundary of [−1,1]d satisfies r d=−1for any r∈V S.Let S∈F S∗be such a simplex,then there exists r∈V S satisfies f∗(r)=+e d according to the definition of badness. Thus r is added into X when it is extended along the d th coordinate.We have r d=−1and−1<r i<1for any1≤i<d.This shows that S is covered by the hyperplane H which passes0[d→−1]and is perpendicular to+e d.Assume N is the number of bad(d−1)-simplices in H,then we only need to prove N is odd.Let X′=P(X)and X′′=P(H∩X∗)where projection P(r)=(r1,r2,...r d−1)∈R d−1,then S′and S′′are simplicial decompositions of X′and X′′respectively whereS′= P(S)S∈S∗and V S⊂X[d→−1] ,S′′= P(S)S∈S∗and S⊂H .Let f′be an arbitrary direction function on X′,then we extend it to be direc-tion function f′′on X′′as follows.For any r∈X′,f′′(r)=f′(r).Otherwise, f′′(r)=f∗((r1,r2,...r d−1,−1)).It’s easy to check that G′′=(f′′,X′′,S′′)is an extension of triple G′=(f′,X′,S′)in the(d−1)-dimensional space.Using arguments which are similar to those in the proof of Lemma1,the parity of N is same as the number of bad(d−2)-simplices on the boundary of8G′′.We know that the latter is odd according to the induction hypothesis,and Property1is proven.Property2.Let M=(F,X,S)be a simplicial direction-preserving map where F maps X toX, there must exist afixed point in X.Now we prove thefixed point theorem of Murota,Iimura and Tamura[13] as a direct corollary of Theorem4.Lemma8(Property of Integrally Convex Sets[13])For every integrally convex set X,there exists a simplicial decomposition S ofX,letting S x∈S be the smallest simplex containing x,then all of its vertices belong to N(x)={r∈Z d|||r−x||∞<1}.Let F be a hypercubic direction-preserving map from integrally convex set X⊂Z d todirection-preserving map from X toC a,b=X=∪P∈P P,and for every polytope P∈P, all of its vertices are drawn from X;2).For every two polytopes P1,P2∈P,if P1∩P2=∅,then P1∩P2is a face of both P1and P2.Definition16Let P be a convex t-polytope in R d and V P be its vertex set.The center point c P of polytope P is defined as c P= r∈V P(r/|V P|).Obviously,we have c P∈P and c P/∈V P.For example,let C⊂Z d be a t-cube centered at r∈Z d and perpendicular to T,then the center point c ofX into X and construct a simplicial decomposition S′for the new set X′. Details of the construction are described by the algorithm in Figure1. Every lattice set C a,b⊂R d has a natural convex subdivision P where P= {Fig.1.The Construction of S′and X′6.2Extension of Hypercubic Direction-Preserving FunctionsLet f be a hypercubic direction-preserving function on C a,b,we now extend it onto set D a,b as follows.For every r∈D a,b−C a,b,assume it is the center point of t-cube C⊂C a,b.If0∈f(C),then f(r)=0.Otherwise,let1≤t≤d be the largest integer such that f(C)∩{±e t}=∅,then f(r)=e t if e t∈f(C)and f(r)=−e t if−e t∈f(C).One can prove the following two properties.Property1Let f be a hypercubic direction-preserving function on C a,b⊂Z d, then G=(f,D a,b,S)is a simplicial direction-preserving function.Property2If the extended function G=(f,D a,b,S)has afixed point in D a,b, then the original function must have afixed point in C a,b.6.3The Nature of Bad CubesWe are ready to give an explicit explanation for the definition of bad cubes. Lemma9Let f be a hypercubic direction-preserving function on C a,b⊂Z d and G=(f,D a,b,S)be the extend function.For every t-cube C t in C a,b where 0≤t≤d−1,it is bad relative to f iffthe cardinality of the following set is odd:S C t= t-simplex S∈S is bad relative to G S⊂C t,then the way we build simplicial decomposition S implies that t-simplex S t⊂C t where S t−1is used to denote(t−1)-simplices in S.Firstly,we prove that,if t-cube C t⊂Z d is not bad,then|S C t|is even.If 0∈f(C t),then f(c)=0.As each t-simplex inC t,e k/∈f(V S),and thus,S C t=∅.Otherwise,we have f(C t)= {e1,...,e t+1},and thus f(c)=e t+1.Because C t is not bad,the number of bad11(t−1)-cubes on the boundary of C t is ing the induction hypothesis on t−1,a(t−1)-cube is bad iffthe number of bad(t−1)-simplices in it is odd. As a result,the number of bad(t−1)-simplices on the boundary of C t is even. Using the equation in thefirst paragraph,we know|S C t|is even too.On the other hand,we prove if C t is bad,then|S C t|is odd.Since f(C t)= {e1,...,e t+1},we have f(c)=e t+1.As the number of bad(t−1)-cubes on the boundary of C t is odd,the number of bad(t−1)-simplices on the boundary of C t is also odd,according to the induction hypothesis on case t−ing the equation in thefirst paragraph again,we know|S C t|is odd.We now get Lemma10as a direct corollary of Lemma9.Lemma10The parity of N B(the number of bad(d−1)-cubes on the boundary of C a,b)is same as the one of N G(the number of bad(d−1)-simplices on the boundary of D a,b).With Property1,2and Lemma10above,Chen and Deng’s theorem can be immediately derived from the fundamental discretefixed point theorem.7Concluding RemarksIn this paper,we generalize the concept of direction-preserving maps and char-acterize a new class of discrete maps over simplicial structures.The fundamental discretefixed point theorem is then proposed,which is based on the counting of bad(d−1)-simplices on the boundary.The power of this theorem is demon-strated in two ways.First,it is applied to prove the discrete Brouwer’sfixed point theorem which is much more general than the one of Murota,Iimura and Tamura.Second,we resolve the puzzle of bad cubes,and show that the bound-ary condition of Chen and Deng’s theorem is exactly equivalent to the one of the fundamental theorem.Our work would immediately imply the corresponding discrete concept of degree.It would be an especially interesting problem to study the case when the fixed point is defined in the recent model of a set of points.An immediate follow-up research direction is to understand other concepts and theorems related to degree.A clear understanding would definitely advance the state of art of the numerical computation of related problems,such as the case of discretefixed points versus approximatefixed points[1].References1.X.Chen and X.Deng.On Algorithms for Discrete and Approximate BrouwerFixed Points.In STOC2005,pages323–330.2.X.Chen and X.Deng.3-Nash is PPAD-complete.ECCC,TR05-134,2005.3.X.Chen and X.Deng.Settling the Complexity of Two-Player Nash-Equilibrium.FOCS,2006.124.X.Chen,X.Deng,and S-H puting Nash Equilibria:Approximationand Smoothed Complexity.FOCS,2006.5.X.Chen and ttice Embedding of Direction-Preserving CorrespondenceOver Integrally Convex Set.AAIM2006,2006,pp.53–63.6.X.Chen and X.Deng.On the Complexity of2D Discrete Fixed Point Problem.ICALP2006,pp.489–5007.P.Crescenzi and R.Silvestri.Sperner’s lemma and robust put.Complexity,7(2):163–173,1998.8. 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