NP-Hardness Results Related to PPAD

Size: px
Start display at page:

Download "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

26 Acknowledgement We would like to thank Xiaotie Deng and Nimrod Megiddo for their helpful comments on a preliminary version of the paper. References [] E.L. Allgower and K. Georg (000). Piecewise linear methods for nonlinear equations and optimization, Journal of Computational and Applied Mathematics 4: [] C. Dang (99). The D -triangulation of R n for simplicial algorithms for computing solutions of nonlinear equations, Mathematics of Operations Research 6: [] C. Dang (995). Triangulations and Simplicial Methods, Lecture Notes in Mathematical Systems and Economics 4. [4] C. Dang (009). An arbitrary starting homotopy-like simplicial algorithm for computing an integer point in a class of polytopes, SIAM Journal on Discrete Mathematics : [5] C. Dang and H. van Maaren (998). A simplicial approach to the determination of an integral point of a simplex, Mathematics of Operations Research : [6] C. Dang and H. van Maaren (999). An arbitrary starting variable dimension algorithm for computing an integer point of a simplex, Computational Optimization and Applications 4:

27 [7] C. Dang and H. van Maaren (00). Computing an integer point of a simplex with an arbitrary starting homotopy-like simplicial algorithm, Journal of Computational and Applied Mathematics 9: [8] C. Dang and Y. Ye (009). Computing an integer point in a class of polytopes, Preprint. [9] C. Daskalakis, P.W. Coldberg, and C.H. Papadimitriou (009). The complexity of computing a Nash equilibrium, Communications of the ACM 5: [0] B.C. Eaves (97). Homotopies for the computation of fixed points, Mathematical Programming : -. [] B.C. Eaves and R. Saigal (97). Homotopies for the computation of fixed points on unbounded regions, Mathematical Programming : 5-7. [] B.C. Eaves and H.E. Scarf (976). The solution of systems of piecewise linear equations, Mathematics of Operations Research : -7. [] H. Freudenthal (94). Simplizialzerlegungen von beschrankter flachheit, Annals of Mathematics 4: [4] I. Gilboa and E. Zemmel (989). Nash and Correlated Equilibria: Some Complexity Considerations, Games and Economic Behavior :

28 [5] M. Kojima and Y. Yamamoto (984). A unified approach to the implementation of several restart fixed point algorithms and a new variable dimension algorithm, Mathematical Programming 8: [6] H.W. Kuhn (968). Simplicial approximation of fixed points, Proceedings of National Academy of Science 6: 8-4. [7] G. van der Laan and A.J.J. Talman (979). A restart algorithm for computing fixed points without an extra dimension, Mathematical Programming 7: [8] G. van der Laan and A.J.J. Talman (98). A class of simplicial restart fixed point algorithms without an extra dimension, Mathematical Programming 0: -48. [9] J.C. Lagarias (985). The computational complexity of simultaneous Diophantine approximation problems, SIAM Journal on Computing 4: [0] N. Megiddo (988). A note on the complexity of P-matrix LCP and computing an equilibrium, Research Report RJ 649, IBM Almaden Research Center, San Jose, CA. [] O.H. Merrill (97). Applications and Extensions of an Algorithm that Computes Fixed Points of Certain Upper Semi-Continuous Point to Set Mappings, PhD Thesis, Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI. 8

29 [] C.H. Papadimitriou (994). On the complexity of the parity argument and other inefficient proofs of existence, Journal of Computer and System Science 48: [] H.E. Scarf (967). The approximation of fixed points of a continuous mapping, SIAM Journal on Applied Mathematics 5: 8-4. [4] H.E. Scarf (Collaboration with T. Hansen) (97). The Computation of Economic Equilibria. Yale University Press, New Haven. [5] M.J. Todd (976). The Computation of Fixed Points and Applications, Lecture Notes in Economics and Mathematical Systems 4, Springer- Verlag, Berlin. [6] M.J. Todd (976a). Orientation in complementary pivoting algorithms, Mathematics of Operations Research : [7] A.H. Wright (98). The octahedral algorithm, a new simplicial fixed point algorithm, Mathematical Programming :

Approximation Algorithms

Approximation Algorithms Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

More information

2.3 Convex Constrained Optimization Problems

2.3 Convex Constrained Optimization Problems 42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

More information

Transportation Polytopes: a Twenty year Update

Transportation Polytopes: a Twenty year Update Transportation Polytopes: a Twenty year Update Jesús Antonio De Loera University of California, Davis Based on various papers joint with R. Hemmecke, E.Kim, F. Liu, U. Rothblum, F. Santos, S. Onn, R. Yoshida,

More information

Applied Algorithm Design Lecture 5

Applied Algorithm Design Lecture 5 Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design

More information

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

More information

3. Linear Programming and Polyhedral Combinatorics

3. Linear Programming and Polyhedral Combinatorics Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

More information

Absolute Value Programming

Absolute Value Programming Computational Optimization and Aplications,, 1 11 (2006) c 2006 Springer Verlag, Boston. Manufactured in The Netherlands. Absolute Value Programming O. L. MANGASARIAN olvi@cs.wisc.edu Computer Sciences

More information

4.6 Linear Programming duality

4.6 Linear Programming duality 4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal

More information

The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

More information

Fairness in Routing and Load Balancing

Fairness in Routing and Load Balancing Fairness in Routing and Load Balancing Jon Kleinberg Yuval Rabani Éva Tardos Abstract We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria

More information

CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

More information

24. The Branch and Bound Method

24. The Branch and Bound Method 24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that no

More information

Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

More information

8.1 Min Degree Spanning Tree

8.1 Min Degree Spanning Tree CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree

More information

Optimization in R n Introduction

Optimization in R n Introduction Optimization in R n Introduction Rudi Pendavingh Eindhoven Technical University Optimization in R n, lecture Rudi Pendavingh (TUE) Optimization in R n Introduction ORN / 4 Some optimization problems designing

More information

5.1 Bipartite Matching

5.1 Bipartite Matching CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the Ford-Fulkerson

More information

An improved on-line algorithm for scheduling on two unrestrictive parallel batch processing machines

An improved on-line algorithm for scheduling on two unrestrictive parallel batch processing machines This is the Pre-Published Version. An improved on-line algorithm for scheduling on two unrestrictive parallel batch processing machines Q.Q. Nong, T.C.E. Cheng, C.T. Ng Department of Mathematics, Ocean

More information

1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.

1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where. Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S

More information

Linear Programming I

Linear Programming I Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins

More information

Definition 11.1. Given a graph G on n vertices, we define the following quantities:

Definition 11.1. Given a graph G on n vertices, we define the following quantities: Lecture 11 The Lovász ϑ Function 11.1 Perfect graphs We begin with some background on perfect graphs. graphs. First, we define some quantities on Definition 11.1. Given a graph G on n vertices, we define

More information

ON THE COMPLEXITY OF THE GAME OF SET. {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu

ON THE COMPLEXITY OF THE GAME OF SET. {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu ON THE COMPLEXITY OF THE GAME OF SET KAMALIKA CHAUDHURI, BRIGHTEN GODFREY, DAVID RATAJCZAK, AND HOETECK WEE {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu ABSTRACT. Set R is a card game played with a

More information

The Line Connectivity Problem

The Line Connectivity Problem Konrad-Zuse-Zentrum für Informationstechnik Berlin Takustraße 7 D-14195 Berlin-Dahlem Germany RALF BORNDÖRFER MARIKA NEUMANN MARC E. PFETSCH The Line Connectivity Problem Supported by the DFG Research

More information

On Stability Properties of Economic Solution Concepts

On Stability Properties of Economic Solution Concepts On Stability Properties of Economic Solution Concepts Richard J. Lipton Vangelis Markakis Aranyak Mehta Abstract In this note we investigate the stability of game theoretic and economic solution concepts

More information

Actually Doing It! 6. Prove that the regular unit cube (say 1cm=unit) of sufficiently high dimension can fit inside it the whole city of New York.

Actually Doing It! 6. Prove that the regular unit cube (say 1cm=unit) of sufficiently high dimension can fit inside it the whole city of New York. 1: 1. Compute a random 4-dimensional polytope P as the convex hull of 10 random points using rand sphere(4,10). Run VISUAL to see a Schlegel diagram. How many 3-dimensional polytopes do you see? How many

More information

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

More information

Single machine parallel batch scheduling with unbounded capacity

Single machine parallel batch scheduling with unbounded capacity Workshop on Combinatorics and Graph Theory 21th, April, 2006 Nankai University Single machine parallel batch scheduling with unbounded capacity Yuan Jinjiang Department of mathematics, Zhengzhou University

More information

Practical Guide to the Simplex Method of Linear Programming

Practical Guide to the Simplex Method of Linear Programming Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

Definition of a Linear Program

Definition of a Linear Program Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1

More information

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem

More information

The Equivalence of Linear Programs and Zero-Sum Games

The Equivalence of Linear Programs and Zero-Sum Games The Equivalence of Linear Programs and Zero-Sum Games Ilan Adler, IEOR Dep, UC Berkeley adler@ieor.berkeley.edu Abstract In 1951, Dantzig showed the equivalence of linear programming problems and two-person

More information

Equilibrium computation: Part 1

Equilibrium computation: Part 1 Equilibrium computation: Part 1 Nicola Gatti 1 Troels Bjerre Sorensen 2 1 Politecnico di Milano, Italy 2 Duke University, USA Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium

More information

Finding and counting given length cycles

Finding and counting given length cycles Finding and counting given length cycles Noga Alon Raphael Yuster Uri Zwick Abstract We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected

More information

Lecture 4: BK inequality 27th August and 6th September, 2007

Lecture 4: BK inequality 27th August and 6th September, 2007 CSL866: Percolation and Random Graphs IIT Delhi Amitabha Bagchi Scribe: Arindam Pal Lecture 4: BK inequality 27th August and 6th September, 2007 4. Preliminaries The FKG inequality allows us to lower bound

More information

8.1 Examples, definitions, and basic properties

8.1 Examples, definitions, and basic properties 8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A k-form ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)-form σ Ω k 1 (M) such that dσ = ω.

More information

SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH 31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

More information

Solving polynomial least squares problems via semidefinite programming relaxations

Solving polynomial least squares problems via semidefinite programming relaxations Solving polynomial least squares problems via semidefinite programming relaxations Sunyoung Kim and Masakazu Kojima August 2007, revised in November, 2007 Abstract. A polynomial optimization problem whose

More information

Permutation Betting Markets: Singleton Betting with Extra Information

Permutation Betting Markets: Singleton Betting with Extra Information Permutation Betting Markets: Singleton Betting with Extra Information Mohammad Ghodsi Sharif University of Technology ghodsi@sharif.edu Hamid Mahini Sharif University of Technology mahini@ce.sharif.edu

More information

On the Unique Games Conjecture

On the Unique Games Conjecture On the Unique Games Conjecture Antonios Angelakis National Technical University of Athens June 16, 2015 Antonios Angelakis (NTUA) Theory of Computation June 16, 2015 1 / 20 Overview 1 Introduction 2 Preliminary

More information

Labeling outerplanar graphs with maximum degree three

Labeling outerplanar graphs with maximum degree three Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics

More information

Linear Programming. March 14, 2014

Linear Programming. March 14, 2014 Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1

More information

Linear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued.

Linear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued. Linear Programming Widget Factory Example Learning Goals. Introduce Linear Programming Problems. Widget Example, Graphical Solution. Basic Theory:, Vertices, Existence of Solutions. Equivalent formulations.

More information

Bargaining Solutions in a Social Network

Bargaining Solutions in a Social Network Bargaining Solutions in a Social Network Tanmoy Chakraborty and Michael Kearns Department of Computer and Information Science University of Pennsylvania Abstract. We study the concept of bargaining solutions,

More information

Market Equilibrium under Piecewise Leontief Concave Utilities

Market Equilibrium under Piecewise Leontief Concave Utilities Market Equilibrium under Piecewise Leontief Concave Utilities Jugal Garg Max-Planck-Institut für Informatik, Germany jugal.garg@gmail.com Abstract. Leontief function is one of the most widely used function

More information

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various

More information

Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai

Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques My T. Thai 1 Overview An overview of LP relaxation and rounding method is as follows: 1. Formulate an optimization

More information

Proximal mapping via network optimization

Proximal mapping via network optimization L. Vandenberghe EE236C (Spring 23-4) Proximal mapping via network optimization minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping Introduction this lecture:

More information

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of

More information

Solving Curved Linear Programs via the Shadow Simplex Method

Solving Curved Linear Programs via the Shadow Simplex Method Solving Curved Linear rograms via the Shadow Simplex Method Daniel Dadush 1 Nicolai Hähnle 2 1 Centrum Wiskunde & Informatica (CWI) 2 Bonn Universität CWI Scientific Meeting 01/15 Outline 1 Introduction

More information

Week 5 Integral Polyhedra

Week 5 Integral Polyhedra Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory

More information

Minimize subject to. x S R

Minimize subject to. x S R Chapter 12 Lagrangian Relaxation This chapter is mostly inspired by Chapter 16 of [1]. In the previous chapters, we have succeeded to find efficient algorithms to solve several important problems such

More information

Max-Min Representation of Piecewise Linear Functions

Max-Min Representation of Piecewise Linear Functions Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 1, 297-302. Max-Min Representation of Piecewise Linear Functions Sergei Ovchinnikov Mathematics Department,

More information

LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS. 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method

LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS. 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method LECTURE 5: DUALITY AND SENSITIVITY ANALYSIS 1. Dual linear program 2. Duality theory 3. Sensitivity analysis 4. Dual simplex method Introduction to dual linear program Given a constraint matrix A, right

More information

Analysis of an Artificial Hormone System (Extended abstract)

Analysis of an Artificial Hormone System (Extended abstract) c 2013. This is the author s version of the work. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purpose or for creating

More information

Lecture 2: August 29. Linear Programming (part I)

Lecture 2: August 29. Linear Programming (part I) 10-725: Convex Optimization Fall 2013 Lecture 2: August 29 Lecturer: Barnabás Póczos Scribes: Samrachana Adhikari, Mattia Ciollaro, Fabrizio Lecci Note: LaTeX template courtesy of UC Berkeley EECS dept.

More information

The Heat Equation. Lectures INF2320 p. 1/88

The Heat Equation. Lectures INF2320 p. 1/88 The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)

More information

Collinear Points in Permutations

Collinear Points in Permutations Collinear Points in Permutations Joshua N. Cooper Courant Institute of Mathematics New York University, New York, NY József Solymosi Department of Mathematics University of British Columbia, Vancouver,

More information

arxiv:1203.1525v1 [math.co] 7 Mar 2012

arxiv:1203.1525v1 [math.co] 7 Mar 2012 Constructing subset partition graphs with strong adjacency and end-point count properties Nicolai Hähnle haehnle@math.tu-berlin.de arxiv:1203.1525v1 [math.co] 7 Mar 2012 March 8, 2012 Abstract Kim defined

More information

T ( a i x i ) = a i T (x i ).

T ( a i x i ) = a i T (x i ). Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)

More information

Duplicating and its Applications in Batch Scheduling

Duplicating and its Applications in Batch Scheduling Duplicating and its Applications in Batch Scheduling Yuzhong Zhang 1 Chunsong Bai 1 Shouyang Wang 2 1 College of Operations Research and Management Sciences Qufu Normal University, Shandong 276826, China

More information

Lattice Point Geometry: Pick s Theorem and Minkowski s Theorem. Senior Exercise in Mathematics. Jennifer Garbett Kenyon College

Lattice Point Geometry: Pick s Theorem and Minkowski s Theorem. Senior Exercise in Mathematics. Jennifer Garbett Kenyon College Lattice Point Geometry: Pick s Theorem and Minkowski s Theorem Senior Exercise in Mathematics Jennifer Garbett Kenyon College November 18, 010 Contents 1 Introduction 1 Primitive Lattice Triangles 5.1

More information

Minimally Infeasible Set Partitioning Problems with Balanced Constraints

Minimally Infeasible Set Partitioning Problems with Balanced Constraints Minimally Infeasible Set Partitioning Problems with alanced Constraints Michele Conforti, Marco Di Summa, Giacomo Zambelli January, 2005 Revised February, 2006 Abstract We study properties of systems of

More information

1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form 1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

More information

University of Lille I PC first year list of exercises n 7. Review

University of Lille I PC first year list of exercises n 7. Review University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients

More information

Systems of Linear Equations

Systems of Linear Equations Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

More information

Numerical methods for American options

Numerical methods for American options Lecture 9 Numerical methods for American options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 American options The holder of an American option has the right to exercise it at any moment

More information

Discrete Optimization

Discrete Optimization Discrete Optimization [Chen, Batson, Dang: Applied integer Programming] Chapter 3 and 4.1-4.3 by Johan Högdahl and Victoria Svedberg Seminar 2, 2015-03-31 Todays presentation Chapter 3 Transforms using

More information

Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can

More information

Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.

Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i. Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(

More information

ALMOST COMMON PRIORS 1. INTRODUCTION

ALMOST COMMON PRIORS 1. INTRODUCTION ALMOST COMMON PRIORS ZIV HELLMAN ABSTRACT. What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type

More information

International Doctoral School Algorithmic Decision Theory: MCDA and MOO

International Doctoral School Algorithmic Decision Theory: MCDA and MOO International Doctoral School Algorithmic Decision Theory: MCDA and MOO Lecture 2: Multiobjective Linear Programming Department of Engineering Science, The University of Auckland, New Zealand Laboratoire

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

Math 215 HW #6 Solutions

Math 215 HW #6 Solutions Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T

More information

Optimal Index Codes for a Class of Multicast Networks with Receiver Side Information

Optimal Index Codes for a Class of Multicast Networks with Receiver Side Information Optimal Index Codes for a Class of Multicast Networks with Receiver Side Information Lawrence Ong School of Electrical Engineering and Computer Science, The University of Newcastle, Australia Email: lawrence.ong@cantab.net

More information

Facility Location: Discrete Models and Local Search Methods

Facility Location: Discrete Models and Local Search Methods Facility Location: Discrete Models and Local Search Methods Yury KOCHETOV Sobolev Institute of Mathematics, Novosibirsk, Russia Abstract. Discrete location theory is one of the most dynamic areas of operations

More information

1 Solving LPs: The Simplex Algorithm of George Dantzig

1 Solving LPs: The Simplex Algorithm of George Dantzig Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.

More information

Nonlinear Programming Methods.S2 Quadratic Programming

Nonlinear Programming Methods.S2 Quadratic Programming Nonlinear Programming Methods.S2 Quadratic Programming Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard A linearly constrained optimization problem with a quadratic objective

More information

Introduction to Scheduling Theory

Introduction to Scheduling Theory Introduction to Scheduling Theory Arnaud Legrand Laboratoire Informatique et Distribution IMAG CNRS, France arnaud.legrand@imag.fr November 8, 2004 1/ 26 Outline 1 Task graphs from outer space 2 Scheduling

More information

POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS

POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS RUSS WOODROOFE 1. Unique Factorization Domains Throughout the following, we think of R as sitting inside R[x] as the constant polynomials (of degree 0).

More information

Integer factorization is in P

Integer factorization is in P Integer factorization is in P Yuly Shipilevsky Toronto, Ontario, Canada E-mail address: yulysh2000@yahoo.ca Abstract A polynomial-time algorithm for integer factorization, wherein integer factorization

More information

Compact Representations and Approximations for Compuation in Games

Compact Representations and Approximations for Compuation in Games Compact Representations and Approximations for Compuation in Games Kevin Swersky April 23, 2008 Abstract Compact representations have recently been developed as a way of both encoding the strategic interactions

More information

This exposition of linear programming

This exposition of linear programming Linear Programming and the Simplex Method David Gale This exposition of linear programming and the simplex method is intended as a companion piece to the article in this issue on the life and work of George

More information

On the k-path cover problem for cacti

On the k-path cover problem for cacti On the k-path cover problem for cacti Zemin Jin and Xueliang Li Center for Combinatorics and LPMC Nankai University Tianjin 300071, P.R. China zeminjin@eyou.com, x.li@eyou.com Abstract In this paper we

More information

Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible:

Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Cramer s Rule and the Adjugate Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Theorem [Cramer s Rule] If A is an invertible

More information

4. Expanding dynamical systems

4. Expanding dynamical systems 4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,

More information

Lecture 7: NP-Complete Problems

Lecture 7: NP-Complete Problems IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 7: NP-Complete Problems David Mix Barrington and Alexis Maciel July 25, 2000 1. Circuit

More information

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

More information

Co-Clustering under the Maximum Norm

Co-Clustering under the Maximum Norm algorithms Article Co-Clustering under the Maximum Norm Laurent Bulteau 1, Vincent Froese 2, *, Sepp Hartung 2 and Rolf Niedermeier 2 1 IGM-LabInfo, CNRS UMR 8049, Université Paris-Est Marne-la-Vallée,

More information

Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs

Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Yong Zhang 1.2, Francis Y.L. Chin 2, and Hing-Fung Ting 2 1 College of Mathematics and Computer Science, Hebei University,

More information

Lecture 1: Systems of Linear Equations

Lecture 1: Systems of Linear Equations MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables

More information

Approximating the Minimum Chain Completion problem

Approximating the Minimum Chain Completion problem Approximating the Minimum Chain Completion problem Tomás Feder Heikki Mannila Evimaria Terzi Abstract A bipartite graph G = (U, V, E) is a chain graph [9] if there is a bijection π : {1,..., U } U such

More information

6.2 Permutations continued

6.2 Permutations continued 6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of

More information

Offline sorting buffers on Line

Offline sorting buffers on Line Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: rkhandekar@gmail.com 2 IBM India Research Lab, New Delhi. email: pvinayak@in.ibm.com

More information

Nan Kong, Andrew J. Schaefer. Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA

Nan Kong, Andrew J. Schaefer. Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA A Factor 1 2 Approximation Algorithm for Two-Stage Stochastic Matching Problems Nan Kong, Andrew J. Schaefer Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA Abstract We introduce

More information

Lecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method

Lecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method Lecture 3 3B1B Optimization Michaelmas 2015 A. Zisserman Linear Programming Extreme solutions Simplex method Interior point method Integer programming and relaxation The Optimization Tree Linear Programming

More information

We shall turn our attention to solving linear systems of equations. Ax = b

We shall turn our attention to solving linear systems of equations. Ax = b 59 Linear Algebra We shall turn our attention to solving linear systems of equations Ax = b where A R m n, x R n, and b R m. We already saw examples of methods that required the solution of a linear system

More information

Date: April 12, 2001. Contents

Date: April 12, 2001. Contents 2 Lagrange Multipliers Date: April 12, 2001 Contents 2.1. Introduction to Lagrange Multipliers......... p. 2 2.2. Enhanced Fritz John Optimality Conditions...... p. 12 2.3. Informative Lagrange Multipliers...........

More information

Pacific Journal of Mathematics

Pacific Journal of Mathematics Pacific Journal of Mathematics GLOBAL EXISTENCE AND DECREASING PROPERTY OF BOUNDARY VALUES OF SOLUTIONS TO PARABOLIC EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS Sangwon Seo Volume 193 No. 1 March 2000

More information

Duality of linear conic problems

Duality of linear conic problems Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

More information

Adaptive Online Gradient Descent

Adaptive Online Gradient Descent Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

More information