NP-Hardness Results Related to PPAD

Transcription

1 NP-Hardness Results Related to PPAD Chuangyin Dang Dept. of Manufacturing Engineering & Engineering Management City University of Hong Kong Kowloon, Hong Kong SAR, China Yinyu Ye Dept. of Management Science & Engineering Stanford University Stanford, CA September, 009 Abstract Let P = {x R n Ax b} be a polytope satisfying that each row of A has at most one positive entry. The problem we consider is to determine whether there is an integer point in P, which is known to be an NP-complete problem. Applying an integer labeling rule and a triangulation of an augmented integral set in R n+, we show This work was partially supported by GRF: CityU 08 of the Government of Hong Kong SAR, China and by NSF DMS-06045, USA.

2 in this paper that determining whether there is an integer point in P can be reduced, in polynomial time, to certain decision problems related to PPAD (polynomial parity argument for directed graphs). Consequently, we prove that these decision problems are all NP-hard. Key Words: Integer Point, Polytope, Integer Programming, Integer Labeling, Triangulation, Pivoting Procedure, Simplicial Method, PPAD, NPhard. Introduction The problem we consider is as follows: Determine whether there is an integer point in a polytope given by P = {x R n Ax b}, where a a a n a a a n A = a m a m a mn satisfies that each row of A has at most one positive entry and b = (b, b,..., b m ). It has been shown: Theorem (Lagarias, 985). Determining whether there is an integer point in P is an NP-complete problem. Applying an integer labeling rule and a triangulation of an augmented integral set in R n+, we show in this paper that determining whether there is an integer point in P can be reduced, in polynomial time, to certain decision problems related to PPAD. Consequently, we prove these decision problems

3 are all NP-hard. The idea of this paper is stimulated from the work in Dang (009), Dang and Maaren (998, 999, 00) and Dang and Ye (009) and has its foundations in simplicial methods for computing fixed points of a continuous mapping that were originated in Scarf (967) and substantially developed in the literature (e.g., Allgower and Georg, 000; Dang, 99, 995; Eaves, 97; Eaves and Saigal, 97; Kojima and Yamamoto, 984; Kuhn, 968; van der Laan and Talman, 979, 98; Merrill, 97; Scarf, 97; Todd, 976; Wright, 98). The rest of this paper is organized as follows. In Section, we introduce an integer labeling rule and a triangulation, and analyze their properties and structures. In Section, we show that the problem can be reduced in polynomial time to certain decision problems related to PPAD, and then draw our main conclusions. Integer Labeling Rule and Triangulation Let M = {,,..., m}, N = {,,..., n}, and N 0 = {,,..., n + }. For i M, let a i denote the ith row of A. Thus, A = (a, a,..., a m ). Let e = (,,..., ) R n. Without loss of generality, we assume throughout this paper that P is bounded and full dimensional. As a result of the property of A, one can easily obtain that, for any x = (x, x,..., x n) P and x = (x, x,..., x n) P, x = max(x, x ) = (max{x, x }, max{x, x },..., max{x n, x n}) P.

4 This implies that max x P e x has a unique solution, which is denoted by x max = (x max, x max,..., x max n ). Let x min = (x min, x min,..., x min n ), where x min j = min x P x j, j =,,..., n. Clearly, x min x x max for all x P. For any real number α, let α denote the greatest integer less than or equal to α and α the smallest integer greater than or equal to α. Let D(P ) = {x R n x l x x u }, where x u = x max = ( x max, x max,..., x max n ) and x l = x min = ( x min, x min,..., x min n ). Thus, x D(P ) for all integer points x P since x min x x max for all x P. Without loss of generality, we assume that x l x u. The sizes of both x l and x u are bounded by polynomials of the size of A and b if they are rational, since x l and x u are obtained from the solutions of linear programs with rational data A and b. For x R n, let 0 R n if x P, f(x) = a i x b i i I(x) a a i a i i if x / P, where I(x) = {i M a i x b i > 0}. Then, we have Lemma. For any x R n, f(x) = 0 if and only if x P. Proof. We only need prove the only part. Suppose that there is some x R n with f(x) = 0 and x / P. Then, I(x). For any given y P 4

5 (that is, b i a i y 0 for all i M), 0 = (x y) f(x) = (x y) i I(x) = i I(x) = i I(x) i I(x) = i I(x) > 0. a i x b i a a i a i i a i x b i a a i a i i (x y) a i x b i (a a i a i i x b i + b i a i y) a i x b i (a a i a i i x b i ) (a i x b i) a i a i Thus, a contradiction occurs. This completes the proof. Let x 0 = (x 0, x 0,..., x 0 n) be any given integer point of R n and, for y R n, C R n and d {, }, let the augmented set Γ(y, C, d) = {t(x, 0) + ( t)(y, d) x C and 0 t } R n+. Then, we define the following integer labeling rule: Definition (An Integer Labeling Rule). For each integer point (x, γ) Γ(x 0, R n, ) Γ(x 0, R n, ), we assign to (x, γ) an integer label l(x, γ) {0} N 0 as follows.. l(x 0, ) = and l(x 0, ) = n +. 5

6 . For (x, 0) with x D(P ), 0 if f(x) = 0 or x P, l(x, 0) = max{k f k (x) = max j N f j (x)} if f j (x) > 0 for some j N, n + if f(x) 0 and f(x) 0.. For (x, 0) with x j > x u j for some j N, l(x, 0) = max{k x k x u k = max j N x j x u j }. 4. For (x, 0) with x x u and x j < x l j for some j N, n + if x < x l, l(x, 0) = max{k x k x l k = max j N x j x l j} otherwise. Example. Consider P = x = (x, x ) x x, 7 6 x + x, x x 9 5, which has an integer point. Figure illustrates the integer labeling rule of Definition on R {0} for this polytope. Example. Consider P = x = (x, x ) x x, x + x 5, 5 x x 8 5, which has no integer point. Figure illustrates the integer labeling rule of Definition on R {0} for this polytope. 6

7 D(P) x l x u x max x min Figure : An Illustration of the Integer Labeling Rule on R {0} for Example 7

8 D(P) x l x u x max x min Figure : An Illustration of the Integer Labeling Rule on R {0} for Example 8

9 For our further development, we need a triangulation of Γ(x 0, R n, ) Γ(x 0, R n, ) that subdivides each of Γ(x 0, R n, d), d {, }, into simplices in such a way that every integer point of Γ(x 0, R n, ) Γ(x 0, R n, ) is a vertex of some simplex of the triangulation and every vertex of a simplex of the triangulation is an integer point of Γ(x 0, R n, ) Γ(x 0, R n, ). Any cubic triangulation of R n is suitable for the purpose. For simplicity, we choose the K -triangulation in Freudenthal (94), which is as follows. A simplex of the K -triangulation of Γ(x 0, R n, ) Γ(x 0, R n, ) is the convex hull of n + vectors, y 0, y,..., y n+, given by y 0 = y, y k = y k + u π(k), k =,,..., n, and y n+ = (x 0, d), where y = (y, y,..., y n+ ) is an integer point in R n {0}, d {, }, and π = (π(), π(),..., π(n), π(n+)) is a permutation of elements of N 0 with π(n + ) = n +. Let K be the set of all such simplices. Since a simplex of the K -triangulation is uniquely determined by y, d, and π, we use K (y, d, π) to denote it. Two simplices of K are adjacent if they share a common facet. For a given simplex σ = K (y, d, π) with vertices y 0, y,..., y n+, its adjacent simplex opposite to a vertex, say y i, is given by K (ȳ, d, π), where ȳ, d, and π are generated according to the pivot rules given in the following table. Pivot Rules of the K -Triangulation of Γ(x 0, R n, ) Γ(x 0, R n, ) i ȳ d π 0 y + u π() d (π(),..., π(n), π(), π(n + )) i < n y d (π(),..., π(i + ), π(i),..., π(n + )) n y u π(n) d (π(n), π(),..., π(n ), π(n + )) n + y d π 9

10 Let K be the set of faces of simplices of K. A q-dimensional simplex of K with vertices y 0, y,..., y q is denoted by y 0, y,..., y q. For σ K with σ R n {0}, let grid(σ) = max{ x y (x, 0) σ and (y, 0) σ}, where denotes the infinity norm. We define mesh(k ) = max{grid(σ) σ K and σ R n {0}}. Then, mesh(k ) =. Definition. A q-dimensional simplex σ = y 0, y,..., y q of K is complete if l(y i ) l(y j ) for any i j, A q-dimensional simplex σ = y 0, y,..., y q of K is almost complete if labels of q + vertices of σ consist of q different integers. From Definition, it is easy to see that an almost complete simplex has exactly two complete facets that carry the same set of integer labels. Lemma. If f(x) 0 and f(x) 0, then, for any y P, there is some k N satisfying that x k y k < 0. Proof. Since f(x) 0, I(x). Let y be a point in P, that is 0

11 b i a i y 0 for all i M. Suppose that x y 0. Then, 0 (x y) f(x) = (x y) i I(x) = i I(x) a i x b i a a i a i i a i x b i (a a i a i i x b i + b i a i y) i I(x) a i x b i (a a i a i i x b i ) = i I(x) > 0. (a i x b i) a i a i Thus, a contradiction occurs. The lemma follows immediately. For y R n and K N, let the higher level cone originated at y along certain directions given in K H(y, K) = {y + h R n 0 h j, j K, and h j = 0, j / K}. Lemma. If z 0 is an integer point of P, then, for any K N, each integer point of H(z 0, K) {0} carries an integer label of either 0 or an integer in K. Proof. Let (x, 0) be an integer point of H(z 0, K) {0}. Consider that x D(P ). From Lemma, we know that l(x, 0) n+, since x z 0, which is equivalent to that either f(x) = 0 or f j (x) > 0 for some j N. Let λ = x z 0. Then, λ j 0, j K, and λ j = 0, j / K. Thus, for any

12 constraint i with a ij 0 for all j K, a i x = a i z 0 + a i λ b i + a i λ = b i + j K a ijλ j b i. This implies that every violated constraint i I(x), if it exists, must have a ij > 0 for a j K. Since at most one coefficient a ij > 0 for every i, we have a ij 0 for all j / K and for every possible violated constraint. Therefore, if f j (x) > 0, one must have j K, that is, either l(x, 0) = 0 or l(x, 0) K from the labeling rule. Consider x / D(P ). Since x l z 0 x, hence, x j > x u j for some j N. From x H(z 0, K), we know that x j = zj 0 x u j for all j / K. Therefore, according to the labeling rule, l(x, 0) K. This completes the proof. This lemma plays an essential role in our development. As a corollary of Lemma, we obtain that Corollary. If P contains an integer point z 0, then there is no complete n-dimensional simplex in the higher level cone H(z 0, N) {0} carrying all integer labels in N 0, and, for any j N and k N 0, there is no complete (n )-dimensional simplex in H(z 0, N\{j}) {0} carrying all integer labels in N 0 \{k}. Let Ω = {x R n x l e x x u + e}

13 and Ω denote the boundary of Ω. Clearly, Ω strictly contains D(P ). Figure illustrates Γ(x 0, Ω, ) Γ(x 0, Ω, ) with integer labels for Example. Let the cube C(x u ) = {x R n x u x x u + e}. Then, C(x u ) = H(x u, N) Ω. Lemma 4. Γ(x 0, C(x u ), ) contains all the complete n-dimensional simplices in Γ(x 0, Ω, ) carrying all integer labels in N 0. Proof. Let σ be a complete n-dimensional simplex in Γ(x 0, Ω, ) carrying all integer labels in N 0. Then, (x 0, ) must be a vertex of σ. Since l(x 0, ) = n +, hence, the facet of σ opposite to (x 0, ) must be a complete (n )-dimensional simplex in Ω {0} carrying all integer labels in N. Let τ = (y, 0), (y, 0),..., (y n, 0) be a complete (n )-dimensional simplex in Ω {0} carrying all integer labels in N. Without loss of generality, we assume that l(y i, 0) = i, i =,,..., n. Since τ is an (n )-dimensional simplex on the boundary of Ω {0}, there must be an index h N such that yh = y h =... = yn h. Suppose τ {x Ω x h = x l h } {0}, that is, this common entry hits the lower bound side of Ω. But for any integer point (x, 0) R n {0}, from (4) of the labeling rule, we have l(x, 0) = n + if x < x l. Hence, y i x l for every vertex (y i, 0) of τ. In particular, y h h x l h = max j N yh j x l j 0.

14 (x 0, ) 0 (x 0, ) Γ(x 0,Ω, ) Γ(x 0,Ω,) 0 Figure : An Illustration of Γ(x 0, Ω, ) Γ(x 0, Ω, ) with Integer Labels for Example 4

15 This contradicts with yh h = xl h < xl h. Thus, the common entry of τ must hits the upper bound of Ω, or τ {x Ω x h = x u h + } {0}. For all i =,,..., n, from l(y i, 0) = i = max{k y i k xu k = max j N y i j x u j } and yh i = xu h +, we derive that y i i x u i = max j N yi j x u j =. Therefore, as a result of mesh(k ) =, we obtain that τ C(x u ) {0}. This completes the proof. Lemma 4 says that each of possible complete n-dimensional simplices in Γ(x 0, Ω, ) carrying all integer labels in N 0 must be contained by Γ(x 0, C(x u ), ) formed from (x 0, ) and the cube C(x u ). Next, we will prove that such a complete n-dimensional simplex exists and it is unique. Let τ = y, y,..., y n+ and σ = y 0, y,..., y n+ with y 0 = (x u, 0), y k = y k + u k, k =,,..., n, and y n+ = (x 0, ). Then, l(y i ) = i, i =,,..., n +. Thus, τ is a complete n-dimensional simplex in Γ(x 0, Ω, ) carrying all integer labels in N 0. Lemma 5. τ is a unique complete n-dimensional simplex in Γ(x 0, Ω, ) carrying all integer labels in N 0. Proof. Let τ = v, v,..., v n be a complete (n )-dimensional simplex in Ω {0} with l(v i ) = i, i =,,..., n. From Lemma 4, we obtain that 5

16 τ C(x u ) {0}. Thus, from () of the labeling rule and the definition of the K -triangulation, we drive that v = (x u, 0) + u and v i = v i + u i, i =,,..., n. Therefore, v, v,..., v n, (x 0, ) = τ. This completes the proof. Let τ = y 0, y,..., y n, y n+ and σ = y 0, y,..., y n+ with y 0 = (x l e, 0), y k = y k + u k+, k =,,..., n, y n = y n + u, and y n+ = (x 0, ). Then, l(y 0 ) = n+, l(y k ) = k +, k =,,..., n, and l(y n+ ) =. Thus, τ is a complete n-dimensional simplex in Γ(x 0, Ω, ) carrying all integer labels in N 0. Lemma 6. τ is a unique complete n-dimensional simplex in Γ(x 0, Ω, ) carrying all integer labels in N 0. Proof. Let τ = v 0, v, v,..., v n be a complete (n )-dimensional simplex in Ω {0} with l(v 0 ) = n + and l(v i ) = i +, i =,,..., n. From the labeling rule, we know that (x l e, 0) is a unique integer point in Ω {0} carrying integer label n +. Thus, v 0 = (x l e, 0). Then, from (4) of the labeling rule and the definition of the K -triangulation, we drive that v i = v i + u i+, i =,,..., n. Therefore, v 0, v,..., v n, (x 0, ) = τ. This completes the proof. 6

17 Polynomial-Time Reduction to Certain Decision Problems Related to PPAD Recently, as a subclass of total search problems, a PPAD class was proposed by Papadimitriou (994) (see also Daskalakis et al. (009)), which leads us to the following work. We show in this section that determining whether there is an integer point in P can be reduced in polynomial time to certain decision problems related to PPAD. In the following discussions,. a C(n)S stands for a complete n-dimensional simplex in Γ(x 0, Ω, ) Γ(x 0, Ω, ) carrying all integer labels in N 0 ;. an AC(n + )S stands for an almost complete (n + )-dimensional simplex in Γ(x 0, Ω, ) Γ(x 0, Ω, ) carrying only integer labels in N 0 ; and. a C(n + )S stands for a complete (n + )-dimensional simplex in Γ(x 0, Ω, ) Γ(x 0, Ω, ) carrying all integer labels in {0} N 0. Let Φ be a graph defined as follows. Nodes of Φ consist of. all C(n)Ss,. all AC(n + )Ss, and. all C(n + )Ss. There is an edge between two nodes of graph Φ if one is a complete facet of the other or they have a common complete facet carrying all integer labels in N 0. 7

18 Let us first determine the degree of each node in graph Φ.. Consider node σ of a C(n)S. From the definition of Φ, one can obtain that node σ is only adjacent to a node given by one of the following: (a) an AC(n + )S or (b) a C(n + )S. Thus, node σ has degree one (which is called an unbalanced node).. Consider node σ of an AC(n + )S. From the definition of Φ, one can obtain that node σ is only adjacent to a pair of nodes given by one of the following pairs: (a) two C(n)Ss, (b) a C(n)S and an AC(n + )S, (c) a C(n)S and a C(n + )S, (d) two AC(n + )Ss, (e) an AC(n + )S and a C(n + )S, or (f) two C(n + )Ss. Thus, node σ has degree two (which is called a balanced node).. Consider node σ of a C(n+)S. From the definition of Φ, one can obtain that node σ is only adjacent to a node given by one of the following: (a) a C(n)S, 8

19 (b) an AC(n + )S, or (c) a C(n + )S. Thus, node σ has degree one (an unbalanced node). From the construction of graph Φ, one can see that each node of Φ belongs uniquely to one of these three categories. The above results show that the degree of each node of Φ is at most two and that C(n + )Ss and C(n)Ss are only nodes that have degree one. Therefore, we come to Lemma 7. Each connected component of graph Φ has one of the following forms: A simple circuit, in which each of nodes has degree two. A simple path, in which each of end nodes has degree one and is given by one of. a C(n)S, or. a C(n + )S. We show next how to determine the direction of an edge in a path of Φ using the orientation of simplices given in Eaves and Scarf (976) and Todd (976a). Let τ = y 0,..., y i, y i+,..., y n+ be a complete facet of σ = y 0, y,..., y n+ K carrying all integer labels in N 0. Let p = (p, p,..., p n+ ) be the permutation of elements of {0,..., i, i+,..., n+ } such that l(y p i ) = i, i =,,..., n +. 9

20 Definition. The orientation of τ with respect to σ is given by (( ) ( ) ( ) ( or σ (τ) = sign(det y p, y p,, y p, n+ y i )) ). It is easy to see that if σ is an almost complete (n+)-dimensional simplex of K carrying only integer labels in N 0 and τ and τ are two complete facets of σ, then or σ (τ ) = or σ (τ ). It is well known that adding a multiple of a column to another column does not change the determinant of a matrix and that exchanging two columns just changes the sign of the determinant of a matrix. Thus, applying these two column operations and the pivoting rules of the K -triangulation, one can derive that Theorem. If τ is a common complete n-dimensional facet of σ and σ in K carrying all integer labels in N 0, then or σ (τ) = or σ (τ). From Lemma 5 and Lemma 6, we know that there are only two C(n)Ss, which are given by τ and τ. We use τ to determine the direction of an edge of graph Φ. Definition 4. Let σ be a node of Φ given by either an A(n + )S or a C(n + )S and τ a complete facet of σ carrying all integer labels in N 0. If or σ (τ) = or eσ ( τ ), the edge leaves σ from τ. If or σ (τ) = or eσ ( τ ), the edge enters σ from τ. A direct calculation yields that or eσ ( τ ) = or eσ ( τ ). Given this definition, Corollary, Lemma 5, Lemma 6 and Lemma 7 together imply that Theorem. There are two given unbalanced nodes τ and τ for the directed graph Φ, and one inward and one outward. If P contains no integer point, 0

21 then the graph has no other unbalanced nodes, which implies that it has a unique path from τ to τ. On the other hand, if P contains an integer point, then other unbalanced nodes exist (should be at least two) and each of them should be a C(n + )S that has an integer point of P as its vertex. Furthermore, the path from τ will end at a C(n + )S. Proof. An unbalanced node is either a C(n)S or a C(n + )S. Lemma 5 and Lemma 6 together show that there are only two C(n)Ss, which are τ and τ. Lemma 7 implies that the graph has one path starting from τ, denoted by P ( τ ), and one path ending at τ, denoted by P ( τ ). Consider the case that P contains no integer point. Then, there is no C(n+)S. Thus, P ( τ ) must end at the other C(n)S ( τ ) since it is contained in the bounded set Γ(x 0, Ω, ) Γ(x 0, Ω, ). Therefore, the graph has a unique path. Consider the case that P contains an integer point. Let z 0 be an integer point of P. From Corollary, we derive that there is no complete n- dimensional simplex in the boundary of Γ(x 0, H(z 0, N), ) carrying all integer labels in N 0. This together with Lemma 5 imply that P ( τ ) is contained in the bounded set Γ(x 0, H(z 0, N) Ω, ). Thus, P ( τ ) must end at a C(n+)S. From Corollary, we derive that there is no complete n-dimensional simplex in the boundary of Γ(x 0, H(z 0, N), ) Γ(x 0, H(z 0, N), ) carrying all integer labels in N 0. This together with Lemma 5 and Lemma 6 imply that P ( τ ) is contained in the bounded set Γ(x 0, L(z 0, Ω), ) Γ(x 0, L(z 0, Ω), ), where L(z 0, Ω) = {x Ω x j zj 0 for some j N}. Therefore, P ( τ ) must start from a C(n + )S. This completes the proof.

22 This theorem reduces the problem of determining whether there is an integer point in P to certain decision problems related to the PPAD graph, since graph Φ has only two alternatives: either the unbalanced node on the other end of the path P ( τ ) is a C(n + )S that implies P containing an integer point, or it is the only other known unbalanced node C(n)S. Figure 4 and Figure 5 illustrate the paths projected on R {0} for Example and the path projected on R {0} for Example, respectively. In Example, the paths, either starting from unbalanced node τ or ending at unbalanced node τ, either ends at or starts from a C(n + )S, respectively. In Example, the path starting from the unbalanced node τ ends at the only other unbalanced node τ, where (x 0, ) and (x 0, ) are switched in the middle of the path. Finally, the following method shows how the unique successor node of a predecessor node in a simple path can be computed in polynomial time in graph Φ. Initialization: Choose τ 0 { τ, τ }. Let σ 0 be the unique simplex in Γ(x 0, Ω, ) Γ(x 0, Ω, ) with τ 0 being one of its facets, y + the vertex of σ 0 opposite to τ 0, and k = 0, and go to Step. Step : Compute l(y + ). If l(y + ) = 0, the method terminates and an integer point of P has been found. Otherwise, let y be the vertex of σ k other than y + and carrying integer label l(y + ), and τ k+ the facet of σ k opposite to y, and go to Step. Step : If τ k+ Γ(x 0, Ω, ) Γ(x 0, Ω, ), the method terminates and

23 D(P) x l x u τ τ Figure 4: An Illustration of the Paths Projected on R {0} for Example

24 D(P) x l x u τ τ Figure 5: An Illustration of the Path Projected on R {0} for Example 4

25 P has no integer point. Otherwise, let σ k+ be the unique (n + )- dimensional simplex that is adjacent to σ k and has τ k+ as a facet, y + the vertex of σ k+ opposite to τ k+, and k = k +, and go to Step. It is easy to see that the computational complexity of an iteration of the method is bounded by O(n ) arithmetic operations over integers. Thus, the unique successor of a predecessor in a simple path can be generated in polynomial time. Therefore, we come to our major conclusions: Theorem 4. Given a directed graph G and TWO unbalanced nodes (with degree one), one inward and one outward, it is NP-hard to decide Are the two given unbalanced nodes connected? Is the path from the given inward unbalanced node unique in G? Are the given two unbalanced nodes the only unbalanced nodes in G? Or its complement: are there other unbalanced nodes in G besides the given two? It is also NP-hard to find any pair of connected unbalanced nodes in G; the unbalanced node connected to a known unbalanced node. We need to mention that a similar result for decising Nash Matrix Game equilibria has been developed in Gilboa and Zemel [4]: given a matrix game, does there exists a unique Nash equilibrium in the game? Their result does not imply our results presented in Theorem 4. 5

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