Permutation Betting Markets: Singleton Betting with Extra Information

Save this PDF as:

Size: px
Start display at page:

Transcription

5 0.6 u u 2 u 3 u u 2 u 3 0. l l 2 l 3 l l 2 l 3 G Figure : definition of G is a special case of the revenue maximization problem. In the following, we give a simple combinatorial algorithm for the existence problem, and an LP-based algorithm for the revenue maximization problem. First, we give some notation that will be used throughout this section. Corresponding bipartite graph G I. Given an instance of the singleton betting problem with a set of bets I, we construct a bipartite graph G I. For every candidate, we place a vertex in the upper part of G I and for every position, we place a vertex in the lower part of G I. Let U G denote the set of vertices in the upper part and L G denote the set of vertices in the lower part. We denote the ith vertex of the upper part by u i and the jth vertex of the lower part by l j. Finally, for every triple (b i, x i, y i) I, we put an edge between u xi U G and l yi L G with weight b i. Note that it is possible to have multiple edges between two nodes in G I. Given a simple edge-weighted bipartite graph G(U G, L G, E), let w G be the weight of the edge between vertices u i U and l j L. Given a multigraph G(V, E), let G (V, E ) with edge weights w be a simple edge-weighted graph with the same set of vertices, i.e, V = V such that w ij, the weight of the edge between vertices i, j V in G, is equal to the number of edges between vertices i and j in G (as shown in Figure ). Note that w G is equal to the number of edges between vertices i and j in G. Each bet has a corresponding edge in G I. Therefore, when the auctioneer accepts a bet, we can also say that the auctioneer accepts the corresponding edge. For example, consider the singleton betting market depicted in Figure. There are nine bets corresponding to edges of the bipartite graph. As a result, the set of bets, I contains triples: 3 G 2 (0.6,, ),(0.,, ),(0.3,,),(0.02,,2),..., (0., 3,3),(0.7, 3,3) In order to prove the existence of Ĥ, we need the following Lemma. In this example, if the auctioneer accepts all bets, he/she gets \$3.32 before the outcome. If candidates, 2 and 3 stand in positions, 2 and 3 respectively, the auctioneers should pay = \$6 to traders. Accepted graph. If the auctioneer accepts a subset of bets, there is a subgraph in G which is formed by the edges corresponding to the accepted bets. We call this graph the accepted graph. We say an auctioneer will win with respect to an accepted graph H, if accepting the bets corresponding to the edges of H gives a positive revenue to the auctioneer in every possible outcome. Graph theoretic preliminaries. In every graph G, let the value of the maximum weighted matching be M G. Vector α = (α, α 2,..., α V (G) ) is called a weighted vertex cover of graph G if for every edge e = (i, j) we have w G e α i + α j. The minimum weighted vertex cover is a weighted vertex cover which minimizes the sum i V (G) αi. 4.2 The existence problem In this section, we give a necessary and sufficient condition for the existence problem which can be checked in linear time. Theorem 2. Given a set of bets I for the singleton betting problem, there exists a set of risk-free bets for the existence problem if and only if there exists a vertex i V (G I) and a set of edges A such that (i) edges in A are adjacent to vertex i, and each pair of edges in A has only one endpoint in common which is i; and (ii) the total weight of edges in A exceeds one, or equivalently e A wg I e >. Proof: Proof of sufficiency: Suppose that there is a vertex i with the desired properties. Without loss of generality, suppose i is in the upper part of G I. Assume that there exists a set A of edges which are adjacent to i and the sum of their weights is greater than. If the auctioneer accepts the bets corresponding to edges in A and rejects the other ones, the amount of money which is earned by him/her is greater than \$. On the other hand, the auctioneer must pay at most \$ in the worst case, because all the accepted bets have the candidate i in common and their positions (the third element in the triples of bets) are distinct, so in each outcome the candidate i stands at only one position, and we lose at most \$. Proof of necessity : Suppose there is no vertex i with the desired property of the theorem and there is a subgraph H of G I such that the auctioneer will win if he accepts the bets corresponding to edges of H. First, we find the subgraph Ĥ of G I such that the auctioneer will win if he accepts the corresponding bets of Ĥ and we have wĥ, for all i, j. This means that there exists at most one edge between every pair of vertices u i UĤ and l j LĤ in Ĥ. Finally, we prove that if a subgraph like Ĥ exists, we will reach a contradiction. Lemma 2. Let G be a weighted simple bipartite graph with integer weights. If the value of the maximum weighted matching in G is M G, then there exists a vertex i with the following property: If we decrease the weights of all edges adjacent to i by unit, the value of the maximum weighted matching in the remaining graph will be M G.

6 Proof: The dual of the maximum weighted matching problem in a bipartite graph G is the following problem: assign values α i and β j to the vertices of G (α i to vertex u i U and β j to vertex l j L) such that for every edge e = (u i, l j), we have that α i +β j, and we also want to minimize the objective function u i U αi+ l j L βj. Based on the weak duality theorem, we know that the minimum feasible value of u i U αi+ l j L βj is greater than or equal to MG in G. Consider the optimal dual solution α i and β j for u i U and l i L. Note that the weights of the edges in G are integer, therefore, all values of α i and β j are integers. We also know that M G, so at least one α i or one β j is greater than 0. Without loss of generality, suppose α k > 0, and because α k is an integer number, we conclude that α k. Now, we can decrease the weights of the edges adjacent to u k by unit and let G be the remaining graph. It is clear that (β j = β j for all l j L, α i = α i for all u i U, u i u k and α k = α k ) is a feasible solution for the dual problem in graph G with value u i U α G i + l j L β G j = M G. Therefore, the value of every weighted matching in G is not greater than M G. On the other hand, consider the maximum weighted matching in G with value M G. It is clear that the value of this matching in G is M G. So the value of the maximum weighted matching in G is M G. Now we return to the proof of Theorem 2. Consider a graph H and assume that the auctioneer accepts the bets corresponding to the edges of H. (Note that we assume that the auctioneer will win by accepting these bets). Let the value of the maximum weighted matching in H be M H. It is clear that for some permutations, the auctioneer must pay M H to the traders. On the other hand, the auctioneer gets e H wh e amount of money from traders at first. Since the auctioneer is seeking a risk-free subset, we should have: M H < e H w H e () If H has an edge with weight greater than, there are at least two edges in H between the endpoints of that edge with weight greater than one. We repeat the following procedure iteratively, until there is no edge with weight greater than in H. - We know that there exists a vertex i in H with the desired property of Lemma 2. For every vertex j, remove one of the edges between vertices i and j in H. Let Ḣ be the remaining graph. According to Lemma 2, if we decrease the weights of edges adjacent to i in H by unit, the value of the maximum weighted matching in H will decrease by exactly unit. Therefore, we have MḢ = M H. We assume that there is no vertex with the desired property of Theorem 2. Therefore, the sum of the weights of the removed edges from H is not greater than, and we have e Ḣ wḣe e H wh e. Using Equation we conclude: MḢ = M H < e H w H e e Ḣ wḣe (2) This proves that if the auctioneer accepts the bets corresponding to the edges of Ḣ, he wins. Therefore, we H U L t s 0 0 Figure 2: Constructing the network flow F using graph H can replace H by Ḣ and repeat this procedure iteratively until we reach a graph H with no edge weight greater than one. This proves that there exists a graph H such that the auctioneer wins with respect to H, and there is at most one edge between any pair of vertices in H. Now, we construct a network flow F using the graph H as follows.. Add two vertices s and t to H. Let s be the source of our network flow and t be its sink. 2. Put an edge between s and each vertex u i in the upper part of H with capacity of. 3. Put an edge between each vertex l j in the lower part of H and t with capacity of. 4. Let the capacity of each edge from vertex u i in the upper part to vertex l j in the lower part be equal to w H u i,l j. We know that there is no vertex with the desired property of Theorem 2. Therefore, for every vertex i, the sum of the weights of the edges adjacent to i in H is not greater than. So, we have a flow with value e H wh e in the network flow F. Construct a new network flow F by rounding up the capacities of edges in F. It is clear that the value of the maximum flow in F is not less than the value of the maximum flow in F, because we do not decrease the capacities. On the other hand, we can see that the value of the maximum flow in F is equal to the value of the maximum weighted matching in H (M H ). Knowing these facts, we can conclude: M H = max flow in F max flow in F e H w H e (3) which is a contradiction (see Equation.) Verifying the necessary and sufficient condition of Theorem 2 for all vertices in graph G I can be done in running time O( I +m+n) where n and m are the number of candidates and positions respectively. As a result, Theorem 2 gives a linear-time algorithm for the existence problem.

7 0.6 u l 0.02 u 2 u 3 u u 2 u l 2 l 3 l l 2 l 3 H Figure 3: A solution with revenue 0.2 H We can add one of the edges between i and j in G H to H. Note that such an edge exists because we assume that w H < w G. It implies that the value of w H increases by one. But, it is clear that the minimum weighted vertex cover of H is still a weighted vertex cover. Using the fact that the value of the maximum weighted matching is equal to the value of the minimum weighted vertex cover, we conclude that the value of e H wh e M H will increase by adding one of edges between vertices i and j to the optimal accepted graph H. This contradicts the optimality of H. 4.3 The revenue maximization problem In this section, we propose a polynomial-time algorithm for finding a subset of bets with the maximum guaranteed revenue to the auctioneer. The algorithm is based on a linear programming (LP) formulation. We first characterize an LP whose optimal integer solution is equal to our optimal solution in the singleton betting problem. We relax the linear program to a fractional linear program, and then prove that we can change any optimal fractional solution of our LP to an integer solution with the same objective function in polynomial-time. Note that in the revenue maximization problem, we have a weighted bipartite graph G with multiple edges each of which corresponds to one bet, and we want to find a subgraph H of G that maximizes: we H M H (4) e H In other words, if the auctioneer accepts the bets of the edges of graph H, he earns e H wh e amount of money from traders, and he should pay M H in the worst case outcome. For example, consider the singleton betting market shown in Figure. If the auctioneer accepts bets (0.7, 2, 2), (0.4, 2,3), (0.4, 3, 2) and (0.7, 3,3), the accepted graph is H that is shown in Figure 3. Note that e H wh e is equal to 2.2, and the maximum weighted matching in graph H, M H is 2. So the revenue is = 0.2 in this case. Now we characterize the structure of the maximum revenue solution. Let the subgraph H G be the maximum revenue solution. Consider the minimum weighted vertex cover of H. Assume that the value α i is assigned to the vertex u i U H, and β j is assigned to the vertex l j L H in this weighted vertex cover. Lemma 3. In the optimal accepted graph H, w H is equal to min(w G, α i + β j) for every pair of vertices i and j. Proof: We know that H is a subgraph of G, thus w H is not greater than w G. On the other hand, the values α, α 2,..., α V (H) and β, β 2,..., β V (H) form a weighted vertex cover of graph H, so we have that w H α i + β j. For the sake of contradiction, assume w H is not equal to min(w G, α i + β j) for some i and j, thus w H < w G and w H < α i + β j. The value w H is integer, for any pair of vertices i and j. Therefore, by definition, α i and β j are also integers, for every i and j. Thus, w H + α i + β j. See the definition of G in the preliminaries. Lemma 3 implies that using values of α i and β j, we can determine the value w H by setting it to min(w G, α i +β j). Now, we can write an integer linear program whose optimal solution determines our optimal solution for the singleton betting problem. In this ILP, we want to find the values of α i and β j. We also want to choose edges which should be added to the optimal subgraph H. For ease of notation, let w,t G be the weight of the t th edge between vertices i and j in G. Without loss of generality assume that the edges between vertices i and j are sorted in decreasing order with respect to their weight such that we have w,t G w G,t+. The following program is the ILP which helps us in computing the minimum weighted vertex cover: max ( w G,ty,t x i x j) (5) w G t= y,t = Y i U, j L Y x i + x j i U, j L x i, x j 0 i U, j L y,t {0, } i U, j L, t k ij The ILP variables x i, x j, y,t and Y are defined as follows: x i is the value of α i in the minimum weighted vertex cover of H. x j is the value of β j in the minimum weighted vertex cover of H. Y is the value of w H which is equal to number of edges between vertices i and j in H. y,t is a number which is equal to 0 or, and indicates whether the t th edge between i and j in G belongs to H. By strong duality, the value of the maximum weighted matching is equal to the value of the minimum weighted vertex cover. Since the ILP variables x and x correspond to the vectors of the minimum weighted vertex cover (α and β), the value x i + x j in the objective function of the ILP is equal to M H. As a result, the optimal solution of the integer linear program 5 characterizes the maximum revenue of the singleton betting problem. In order to solve the integer linear program 5, we relax the integer constraints y,t {0, } to linear fractional constraints 0 y,t. As a result, we get the following linear programming relax-

8 ation: max ( w G,ty,t x i x j) (6) w G t= y,t = Y i U, j L Y x i + x j i U, j L x i, x j 0 i U, j L 0 y,t i U,j L, t k ij We can solve the linear program 6. Now, the question is how to round the solution of 6 and construct an integer solution for program 5. The following Lemma 4 shows that the integrality gap of this linear program is and any solution of this LP can be rounded to an integer solution in polynomial time without changing the value of the objective function. Here, we prove this fact by showing that LP 6 is totally unimodular. For completeness, we give an explicit polynomial-time rounding method for rounding fractional solutions of this LP to optimal integer solutions in the appendix. Lemma 4. The integrality gap of LP 6 is and an optimal integer solution of ILP 5 can be found in polynomial-time by solving the LP relaxation 6. Proof: Here, we prove this lemma by showing that the LP is totally unimodular. For a constructive proof of this lemma, see the appendix. There are four types of variables in LP 6, i.e. y,t, Y, x i, x j. Let v be a vector that contains all types of variables. We can write the constraints of LP 6 as a matrix inequality Av b where A and b are defined as follows: A is a matrix whose number of rows and columns are equal to the number of constraints and variables in the LP respectively, and entries of A correspond to coefficients in the linear constraints of this LP. The vector b contains the right hand side values of the constraints. It is well known that in order to prove that the integrality gap of LP 6 is, it suffices to show that A is totally unimodular [8, 23]. This fact also implies a polynomial-time algorithm for rounding any fractional solution of LP 6 to an optimal integer solution to ILP 5. For contradiction, assume A is not totally unimodular. In that case, A should have a square submatrix with determinant not equal to 0, or. Suppose K is the smallest square submatrix of A with this property. It is not hard to see that each row or column of K has at least two non-zero entries. Since, if there is a row (or column) with only one non-zero entry a (a is either or ), we can say that the absolute value of determinant of K is equal to the absolute value of determinant of K where K is the matrix that is obtained from K by removing the row and column of entry a. So the determinant of K is also not equal to 0, or. This contradicts with the assumption that K is the smallest square submatrix with determinant not equal to 0, or. According to the fact that each row of K has at least two non-zero entries, we conclude that rows corresponding to constraints like y,t or y,t 0 are not selected as rows of K. Therefore, without loss of generality, we can delete these rows from A, and assume that K is a submatrix of the remaining matrix. Since each column of K also has at least two non-zero entries, we can say that the columns corresponding to variables y,t are not selected as columns of K, because each of these columns in the remaining matrix contains exactly one non-zero entry. Similarly we can remove these columns, and assume that K is a submatrix of the remaining matrix. Now consider a row corresponding to a constraint of the form y,t = Y. Since the columns of variables of the form y,t are removed later, the row corresponding to this constraint has only one non-zero entry. Again we can remove these rows from our matrix. In the remaining matrix, the columns of variables of form Y has only one non-zero entry, therefore we remove these columns too. The remaining matrix has only rows of constraints of form Y x i + x j and columns of variables of form x i or x j. Note that each row of this matrix has exactly two non-zero variable with the same sign. Partition the columns into two sets B = {x, x 2,..., x n} and C = {x, x 2,..., x n}. One of those two non-zero entries belongs to a column in set B, and the other one belongs to a column in set C. According to [2], the determinant of any square submatrix of this matrix, including K is equal to 0, or which is again a contradiction. Therefore we conclude A is totally unimodular, and thus, the integrality gap of the corresponding integer linear program is equal to. Using ILP 5 and the result of Lemma 4, it follows that the revenue maximization problem for singleton betting is polynomial-time solvable. We conclude this section by the following theorem. Theorem 3. The revenue maximization problem for the auctioneer in singleton betting can be solved in polynomial time. Note that linear program 6 is a small polynomial-size linear program that can be solved very efficiently in practice as well. This is different from the exponential-size linear programming formulation of Chen et.al [3] for divisible variant of the subset betting. 4.4 The revenue maximization problem with extra information In this section, we study the revenue maximization problem for singleton betting when we are given a set of pieces of extra information about the outcome of the betting market. Each piece of the extra information is a forbidden pair (x,y) which means that candidate x never ends up in position y. The auctioneer may gain this type of information from various sources, or can predict such forbidden pairs with such a high confidence that he/she does not bear any risk by assuming these forbidden pairs. Let F be the set of these pairs. Given a set of forbidden pairs, a possible outcome we can illustrated by a perfect matching among candidates and positions in which no forbidden pair appears. In that case, the auctioneer can use this information in his/her favor and in order to solve the revenue maximization problem, he should take into account such extra information. In this section, we show that the LP-based revenue maximization

9 u u 2 u 3 u u 2 u l l 2 l 3 l l 2 l 3 H H 2 In the above LP, α i is the variable corresponding to the constraint of candidate i U, β j is the variable corresponding to the constraint of position j L, and δ is the variable corresponding to the constraint of the forbidden edge (i, j). Note that all variables including δ can get arbitrarily large positive or negative values. Variables δ does not contribute in the cost function, so by setting δ = +, we can eliminate the constraints of the form α i+β j+δ w H. Therefore, we can rewrite the dual program as follows: Figure 4: Best solution with extra information algorithm from the previous section can be extended to solve the revenue maximization problem with extra information. Before stating the algorithm, we discuss an example. Consider the singleton betting market in Figure. Suppose we know that candidates and 2 do not stand at position in any possible outcome (see Figure 4). In other words, the edges (,) and (2,) are forbidden pairs. Using this extra information, we can say that the candidate 3 necessarily stands at position. Therefor, the auctioneer gains the maximum revenue by accepting all bets except bet (0.02,,2). The auctioneer gets \$3.3 before the outcome, and will pay at most \$ to the traders after the outcome. First, we show how to calculate the minimum revenue over all possible outcomes with respect to a given accepted graph H. Then we propose a linear programming method to find, an accepted graph which maximizes this minimum revenue over the possible outcomes. Note that a possible outcome can be shown by a perfect matching M among candidates and positions that does not use the forbidden pairs. The sum of weights of edges that are in both M and H (See definition of G in Subsection 4.) is the value that we should pay to the traders in this outcome. Therefore, in order to find the minimum revenue over all possible outcomes, we should find the maximum weighted perfect matching in H without using forbidden edges. We can solve this by finding the integer solution of the following LP: max ( w H x ) (7) n x = i U j= n x = j L i= x = 0 (i, j) F x 0 i U, j L We consider the dual of the above program: min ( α i + β j) (8) α i + β j w H α i + β j + δ w H (i, j) F (i, j) F min ( α i + β j) (9) α i + β j w H (i, j) F This program finds the values α i and β j such that for each non-forbidden edge (i, j), we have α i+β j w H. Since LP 7 is the similar to the LP of the maximum weighted matching problem, the integrality gap of LP 7 is [23]. Therefore the best fractional solution of the dual program 9 is equal to the maximum integer solution of LP 7. Now we propose an algorithm to find an accepted graph maximizing this minimum revenue over the possible outcomes. With the same argument as Lemma 3, we can use values of α i and β j of dual program 9 to determine the value w H by setting it to min(w G, α i + β j). Now, we can write an integer linear program whose optimal solution determines our optimal solution for the singleton betting problem with extra information. The ILP is very similar to ILP 5. Again we want to find the values of α i and β j and we also want to choose edges which should be added to the optimal subgraph H. The following program is the ILP for computing the minimum weighted vertex cover in this setting: max ( w G,ty,t x i x j) (0) w G t= y,t = Y Y x i + x j i U, j L (i, j) F y,t {0, } i U, j L, t k ij Theorem 4. The revenue maximization problem for the auctioneer in singleton betting with extra information can be solved in polynomial time. Proof: Similar to the proof of Lemma 4, we can show that the constraint matrix of ILP 0 is totally unimodular. Therefore if we relax the integer constraints y,t {0, } to linear fractional constraints 0 y,t, we can solve this linear programming relaxation and round it to optimal integer solution of ILP 0. The optimal solution of ILP 0, corresponds to the maximum revenue of singleton betting problem with extra information. 5. THE PROBABILISTIC SETTING In this section, we study the betting problem in the probabilistic setting. We first define the problem formally. Assume that the auctioneer has a probability distribution q over the possible outcome permutations, i.e., σ is a permutation q(σ) =. Given a probability distribution q, a desired revenue x,

11 design. Information Systems Frontiers, 5():07-9, [2] I. Heller, C.B. Tompkins. An extension of a theorem of Dantzig s, in: H.W. Kuhn, A.W. Tucker (Eds.), Linear Inequalities and Related Systems. Princeton University Press, Princeton, NJ, 956, pp [3] N. Immorlica, D. Karger, M. Minkoff, and V. S. Mirrokni. On the costs and benefits of procrastination: Approximation algorithms for stochastic combinatorial optimization problems. In SODA, [4] B. Lehmann, D. Lehmann, and N. Nisan. Combinatorial Auctions with Decreasing Marginal Utilities, ACM EC 200. [5] Y. Nikulin. Robustness in combinatorial optimization and scheduling theory: An annotated bibliography. Technical Report SOR-9-3, Statistics and Operation Research, [6] N. Nisan, T. Roughgarden, E. Tardos and V. Vazirani. Algorithmic Game Theory. Cambridge University Press, [7] D. M. Pennock, S. Lawrence, C. L. Giles, and F. A. Nielsen. The real power of artiŕcial markets. Science, 29: , February [8] C. Papadimitriou, and K. Steiglitz. Combinatorial Optimization: Algorithms and Complexity N.Y. : Dover Publications, Inc., 998. [9] C. Plott and S. Sunder. Efficiency of experimental security markets with insider information: An application of rational expectations models. Journal of Political Economy, 90:663 98, 982 [20] C. Plott and S. Sunder. Rational expectations and the aggregation of diverse information in laboratory security markets. Econometrica, 56:085 8, 988. [2] T. Sandholm. Algorithm for optimal winner determination in combinatorial auctions. Artificial Intelligence, 35:-54, [22] D. Shmoys and S. Swamy. Stochastic optimization is (almost) as easy as deterministic optimization. In FOCS, [23] A. Schrijver, Combinatorial Optimization - Polyhedra and Efficiency, Springer-Verlag, Berlin, APPENDIX A. A CONSTRUCTIVE PROOF OF LEMMA 4 Proof: In this LP, we know that all values w G,t are nonnegative. So, in any of its optimal solutions, we have Y = min(w G, x i + x j), y,t = for t Y, y,t = 0 for t > Y +, and y,t = Y Y for t = Y +. Note that the sequence w G,t is sorted in decreasing order with respect to their values. Consider an optimal fractional solution. There exists at least one x i or one x j which is not integer. The reason is that if all values of x i and x j were integer, all the other variables would be integer too. Without loss of generality, assume that x s is not an integer number. Define critical, empty and full edges in an optimal solution of LP as follows: e = (i, j) is a critical edge if we have Y = x i + x j = w G, e = (i, j) is an empty edge if we have Y = x i + x j < w G, and e = (i, j) is a full edge if we have Y = w G < x i +x j. Consider the set of vertices in the upper (lower) part of G which have a path to u s through the critical edges. Name this set of vertices C U(C L). For any critical edge e = (i, j), we know that x i + x j = w G which is an integer number, so if one of the numbers x i and x j is not integer, the other one is not an integer number either. Since x s is not an integer number, numbers x i and x j are not integer where u i C U and l j C L. This means that x i and x j are greater than zero, and if we change them by a sufficiently small ε in the optimal solution, the constraints x i, x j 0 still hold. Now, we modify the optimal solution as follows: x i = x i + ε, i C U x j = x j ε, j C L We study in detail the changes that occur in the optimal solution by these changes of the values of these variables. First, we define ε to be zero, and then we slowly increase it and observe the changes in the solution. In the following, we study the situation of edges in different categories. In some cases, we say that we stop increasing ε if some certain events occur. Note that we keep increasing ε until at least one of these events happen, or one of the variables x i or x j becomes integer. all edges between C U and C L: For edge e = (i, j), the value of x i + x j does not change in this case, so nothing changes for this edge. all full edges between C U and (L G C L): A full edge e = (i, j) which is adjacent to C U remains full by increasing ε. The reason is that Y = w G < x i + x j + ε. all full edges between (U G C U) and C L: We keep increasing ε until the equality Y = w G = x i +x j ε holds for some full edges such as e = (i, j) which is adjacent to C L. In fact we stop increasing ε whenever some full edges change to critical edges. all empty edges between C U and (L G C L): Consider an empty edge e = (i, j) which is adjacent to C U. Since the value of x i increases by ε, the value of Y = x i + x j also increases. We stop increasing ε whenever some empty edges convert to critical edges. We may stop increasing ε in another situation. While we are increasing ε, the value of Y also increases for any empty edge e = (i, j) which is adjacent to C U. By increasing Y, the value of y,t will increase too, where t is the first index with y,t <. When a variable y,t < reaches value, we stop increasing ε. Note that increasing ε can be stopped by this kind of edge in two different situations: when an empty edge changes to a critical one, or a non-integer variable y,t becomes. One can see that these increments increase the objective function as well. Assume that the rate of increasing objective function by this type of change is R.

12 all empty edges between (U G C U) and C L: Consider an empty edge e = (i, j) which is adjacent to C L. The value of Y decreases by increasing ε. Since we k y,k = Y, by decreasing Y the value of y,t also decreases, where t is the maximum index with a nonzero value. We stop increasing ε whenever a variable y,t > 0 reaches the value 0. These changes decrease the objective function too. Assume that the rate of decreasing the objective function by this type of changes is R 2. all edges between (U G C U) and (L G C L): For edge e = (i, j), the value of x i + x j does not change in this case, so nothing changes for this edge. reason is that the number of critical edges increases in these scenarios, and this number is upper bounded by the number of edges. The number of steps which end with scenarios 3 or 4 is also polynomial, since at each step one of the y variables become integral. In addition, we know that the above increasing process stops for a value ε < because before ε reaches, scenario 5 occurs, so each variable y,t triggers the conditions of scenarios 3 or 4 at most one time. Therefore, the number of steps between two consecutive gold steps is also polynomial. We can thus conclude that our algorithm runs in polynomial time. Since we do not lose any value in the modification steps, it is clear that at the end of the algorithm, we have an integer solution with the same value of the objective function. One can verify that when we increase ε slowly the amount of change in the objective function is exactly: εr = ε C U ε C L + εr εr 2 If we have R > 0, we can increase the objective function by increasing ε, which contradicts our optimality assumption. If we have R < 0, we also reach a contradiction, because if we start with ε = 0, and decrease its value instead, we can prove similarly, and everything is similar to the case ε > 0. Therefore, R should be zero. In this case, we will increase ε from zero until one of these scenarios occurs:. One of the full edges adjacent to C L changes to a critical edge, 2. One of the empty edges adjacent to C U changes to a critical edge, 3. One variable y,t < reaches the value, 4. One variable y,t > 0 reaches the value 0, or 5. One of the non-integer variables x i or x j reaches an integer value. Note that we only modify some noninteger variables x i and x j in our algorithm. If scenarios or 2 occur, the number of critical edges will increase, and we can continue the algorithm iteratively with the new sets C U and C L. If scenarios 3 or 4 occur, the only change is the sets C U and C L, and we can continue our algorithm with the new sets C U and C L. If scenario 5 occurs, the number of integer variables x i and x j will increase, and we can continue our algorithm if any non-integer variable remains. It is clear that during the changes, we have the invariant R = 0 in all cases. Otherwise, we can increase the objective function which is a contradiction. During these changes, we stop the algorithm if we reach an integer solution. We now prove that our algorithm runs in polynomial time. One can see that increasing ε in our algorithm is done step by step. A step starts by increasing ε, and ends when one of scenarios -5 occur. We say a step is a gold step if it ends with scenario 5. In other words, the number of integer variables x i and x j increases at the end of a gold step. It is easy to see that the number of gold steps is at most U G + L G. So, we only need to prove that the number of steps between two consecutive gold steps is a polynomial function of the input size. First, we observe that the number of steps which end with scenario or 2 is polynomial. The

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

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

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

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

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

! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. #-approximation algorithm.

Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of three

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

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

! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. !-approximation algorithm.

Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of

Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should I do? A. Theory says you're unlikely to find a poly-time algorithm. Must sacrifice one

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

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

5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1

5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition

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

2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

Online Adwords Allocation Shoshana Neuburger May 6, 2009 1 Overview Many search engines auction the advertising space alongside search results. When Google interviewed Amin Saberi in 2004, their advertisement

Algorithm Design and Analysis

Algorithm Design and Analysis LECTURE 27 Approximation Algorithms Load Balancing Weighted Vertex Cover Reminder: Fill out SRTEs online Don t forget to click submit Sofya Raskhodnikova 12/6/2011 S. Raskhodnikova;

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

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

1 Polyhedra and Linear Programming

CS 598CSC: Combinatorial Optimization Lecture date: January 21, 2009 Instructor: Chandra Chekuri Scribe: Sungjin Im 1 Polyhedra and Linear Programming In this lecture, we will cover some basic material

The multi-integer set cover and the facility terminal cover problem

The multi-integer set cover and the facility teral cover problem Dorit S. Hochbaum Asaf Levin December 5, 2007 Abstract The facility teral cover problem is a generalization of the vertex cover problem.

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

Betting Boolean-Style: A Framework for Trading in Securities Based on Logical Formulas

Betting Boolean-Style: A Framework for Trading in Securities Based on Logical Formulas Lance Fortnow 1 Department of Computer Science, University of Chicago, 1100 E. 58th St., Chicago, IL 60637 Joe Kilian

Min-cost flow problems and network simplex algorithm

Min-cost flow problems and network simplex algorithm The particular structure of some LP problems can be sometimes used for the design of solution techniques more efficient than the simplex algorithm.

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,

Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS

Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS 1.1.25 Prove that the Petersen graph has no cycle of length 7. Solution: There are 10 vertices in the Petersen graph G. Assume there is a cycle C

max cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x

Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study

Good luck, veel succes!

Final exam Advanced Linear Programming, May 7, 13.00-16.00 Switch off your mobile phone, PDA and any other mobile device and put it far away. No books or other reading materials are allowed. This exam

Scheduling to Minimize Power Consumption using Submodular Functions

Scheduling to Minimize Power Consumption using Submodular Functions Erik D. Demaine MIT edemaine@mit.edu Morteza Zadimoghaddam MIT morteza@mit.edu ABSTRACT We develop logarithmic approximation algorithms

P versus NP, and More

1 P versus NP, and More Great Ideas in Theoretical Computer Science Saarland University, Summer 2014 If you have tried to solve a crossword puzzle, you know that it is much harder to solve it than to verify

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

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

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

CSC2420 Fall 2012: Algorithm Design, Analysis and Theory

CSC2420 Fall 2012: Algorithm Design, Analysis and Theory Allan Borodin November 15, 2012; Lecture 10 1 / 27 Randomized online bipartite matching and the adwords problem. We briefly return to online algorithms

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

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

Prediction Markets. 1 Introduction. 1.1 What is a Prediction Market? Algorithms Seminar 12.02.2014

Algorithms Seminar 12.02.2014 Tawatchai Siripanya Prediction Markets Wolfgang Mulzer, Yannik Stein 1 Introduction 1.1 What is a Prediction Market? Prediction markets are defined as analytical markets that

10. Graph Matrices Incidence Matrix

10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a

Why? A central concept in Computer Science. Algorithms are ubiquitous.

Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online

Complexity and Completeness of Finding Another Solution and Its Application to Puzzles

yato@is.s.u-tokyo.ac.jp seta@is.s.u-tokyo.ac.jp Π (ASP) Π x s x s ASP Ueda Nagao n n-asp parsimonious ASP ASP NP Complexity and Completeness of Finding Another Solution and Its Application to Puzzles Takayuki

11. APPROXIMATION ALGORITHMS

11. APPROXIMATION ALGORITHMS load balancing center selection pricing method: vertex cover LP rounding: vertex cover generalized load balancing knapsack problem Lecture slides by Kevin Wayne Copyright 2005

Analysis of Approximation Algorithms for k-set Cover using Factor-Revealing Linear Programs

Analysis of Approximation Algorithms for k-set Cover using Factor-Revealing Linear Programs Stavros Athanassopoulos, Ioannis Caragiannis, and Christos Kaklamanis Research Academic Computer Technology Institute

GRAPH THEORY and APPLICATIONS. Trees

GRAPH THEORY and APPLICATIONS Trees Properties Tree: a connected graph with no cycle (acyclic) Forest: a graph with no cycle Paths are trees. Star: A tree consisting of one vertex adjacent to all the others.

Discuss the size of the instance for the minimum spanning tree problem.

3.1 Algorithm complexity The algorithms A, B are given. The former has complexity O(n 2 ), the latter O(2 n ), where n is the size of the instance. Let n A 0 be the size of the largest instance that can

OPRE 6201 : 2. Simplex Method

OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2

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.

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.,

The Conference Call Search Problem in Wireless Networks

The Conference Call Search Problem in Wireless Networks Leah Epstein 1, and Asaf Levin 2 1 Department of Mathematics, University of Haifa, 31905 Haifa, Israel. lea@math.haifa.ac.il 2 Department of Statistics,

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

On Integer Additive Set-Indexers of Graphs

On Integer Additive Set-Indexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A set-indexer of a graph G is an injective set-valued function f : V (G) 2 X such that

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

Lecture 7: Approximation via Randomized Rounding

Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

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,

Exponential time algorithms for graph coloring

Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A k-labeling of vertices of a graph G(V, E) is a function V [k].

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

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

JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004

Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February

A Simple Characterization for Truth-Revealing Single-Item Auctions

A Simple Characterization for Truth-Revealing Single-Item Auctions Kamal Jain 1, Aranyak Mehta 2, Kunal Talwar 3, and Vijay Vazirani 2 1 Microsoft Research, Redmond, WA 2 College of Computing, Georgia

Partitioning edge-coloured complete graphs into monochromatic cycles and paths

arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political

A Working Knowledge of Computational Complexity for an Optimizer

A Working Knowledge of Computational Complexity for an Optimizer ORF 363/COS 323 Instructor: Amir Ali Ahmadi TAs: Y. Chen, G. Hall, J. Ye Fall 2014 1 Why computational complexity? What is computational

Online Primal-Dual Algorithms for Maximizing Ad-Auctions Revenue

Online Primal-Dual Algorithms for Maximizing Ad-Auctions Revenue Niv Buchbinder 1, Kamal Jain 2, and Joseph (Seffi) Naor 1 1 Computer Science Department, Technion, Haifa, Israel. 2 Microsoft Research,

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

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

COLORED GRAPHS AND THEIR PROPERTIES

COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloring

Minimum Makespan Scheduling

Minimum Makespan Scheduling Minimum makespan scheduling: Definition and variants Factor 2 algorithm for identical machines PTAS for identical machines Factor 2 algorithm for unrelated machines Martin Zachariasen,

Theory of Linear Programming

Theory of Linear Programming Debasis Mishra April 6, 2011 1 Introduction Optimization of a function f over a set S involves finding the maximum (minimum) value of f (objective function) in the set S (feasible

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

Two General Methods to Reduce Delay and Change of Enumeration Algorithms

ISSN 1346-5597 NII Technical Report Two General Methods to Reduce Delay and Change of Enumeration Algorithms Takeaki Uno NII-2003-004E Apr.2003 Two General Methods to Reduce Delay and Change of Enumeration

Introduction to Logic in Computer Science: Autumn 2006

Introduction to Logic in Computer Science: Autumn 2006 Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss 1 Plan for Today Now that we have a basic understanding

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,

Outline. NP-completeness. When is a problem easy? When is a problem hard? Today. Euler Circuits

Outline NP-completeness Examples of Easy vs. Hard problems Euler circuit vs. Hamiltonian circuit Shortest Path vs. Longest Path 2-pairs sum vs. general Subset Sum Reducing one problem to another Clique

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

Computational Learning Theory Spring Semester, 2009/10 Lecture 11: Sponsored search Lecturer: Yishay Mansour Scribe: Ben Pere, Jonathan Heimann, Alon Levin 11.1 Sponsored Search 11.1.1 Introduction Search

One last point: we started off this book by introducing another famously hard search problem:

S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani 261 Factoring One last point: we started off this book by introducing another famously hard search problem: FACTORING, the task of finding all prime factors

Chapter 7. Sealed-bid Auctions

Chapter 7 Sealed-bid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)

On Competitiveness in Uniform Utility Allocation Markets

On Competitiveness in Uniform Utility Allocation Markets Deeparnab Chakrabarty Department of Combinatorics and Optimization University of Waterloo Nikhil Devanur Microsoft Research, Redmond, Washington,

LINEAR PROGRAMMING P V Ram B. Sc., ACA, ACMA Hyderabad

LINEAR PROGRAMMING P V Ram B. Sc., ACA, ACMA 98481 85073 Hyderabad Page 1 of 19 Question: Explain LPP. Answer: Linear programming is a mathematical technique for determining the optimal allocation of resources

The Trip Scheduling Problem

The Trip Scheduling Problem Claudia Archetti Department of Quantitative Methods, University of Brescia Contrada Santa Chiara 50, 25122 Brescia, Italy Martin Savelsbergh School of Industrial and Systems

Graphs without proper subgraphs of minimum degree 3 and short cycles

Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract

Pricing of Limit Orders in the Electronic Security Trading System Xetra

Pricing of Limit Orders in the Electronic Security Trading System Xetra Li Xihao Bielefeld Graduate School of Economics and Management Bielefeld University, P.O. Box 100 131 D-33501 Bielefeld, Germany

Lecture 3: Finding integer solutions to systems of linear equations

Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture

Introduction to Flocking {Stochastic Matrices}

Supelec EECI Graduate School in Control Introduction to Flocking {Stochastic Matrices} A. S. Morse Yale University Gif sur - Yvette May 21, 2012 CRAIG REYNOLDS - 1987 BOIDS The Lion King CRAIG REYNOLDS

THE SCHEDULING OF MAINTENANCE SERVICE

THE SCHEDULING OF MAINTENANCE SERVICE Shoshana Anily Celia A. Glass Refael Hassin Abstract We study a discrete problem of scheduling activities of several types under the constraint that at most a single

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

arxiv:1112.0829v1 [math.pr] 5 Dec 2011

How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

Notes on Determinant

ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

Discrete Applied Mathematics. The firefighter problem with more than one firefighter on trees

Discrete Applied Mathematics 161 (2013) 899 908 Contents lists available at SciVerse ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam The firefighter problem with

Approximating Minimum Bounded Degree Spanning Trees to within One of Optimal

Approximating Minimum Bounded Degree Spanning Trees to within One of Optimal ABSTACT Mohit Singh Tepper School of Business Carnegie Mellon University Pittsburgh, PA USA mohits@andrew.cmu.edu In the MINIMUM

School Timetabling in Theory and Practice

School Timetabling in Theory and Practice Irving van Heuven van Staereling VU University, Amsterdam Faculty of Sciences December 24, 2012 Preface At almost every secondary school and university, some

MATHEMATICAL ENGINEERING TECHNICAL REPORTS. The Independent Even Factor Problem

MATHEMATICAL ENGINEERING TECHNICAL REPORTS The Independent Even Factor Problem Satoru IWATA and Kenjiro TAKAZAWA METR 2006 24 April 2006 DEPARTMENT OF MATHEMATICAL INFORMATICS GRADUATE SCHOOL OF INFORMATION

Linear Codes. Chapter 3. 3.1 Basics

Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length

Scheduling Shop Scheduling. Tim Nieberg

Scheduling Shop Scheduling Tim Nieberg Shop models: General Introduction Remark: Consider non preemptive problems with regular objectives Notation Shop Problems: m machines, n jobs 1,..., n operations

Solutions to Exercises 8

Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 8 (1) Suppose that G is a graph in which every vertex has degree at least k, where k 1, and in which every cycle contains at least 4 vertices.

Sharing Online Advertising Revenue with Consumers

Sharing Online Advertising Revenue with Consumers Yiling Chen 2,, Arpita Ghosh 1, Preston McAfee 1, and David Pennock 1 1 Yahoo! Research. Email: arpita, mcafee, pennockd@yahoo-inc.com 2 Harvard University.

6. Mixed Integer Linear Programming

6. Mixed Integer Linear Programming Javier Larrosa Albert Oliveras Enric Rodríguez-Carbonell Problem Solving and Constraint Programming (RPAR) Session 6 p.1/40 Mixed Integer Linear Programming A mixed

A 2-factor in which each cycle has long length in claw-free graphs

A -factor in which each cycle has long length in claw-free graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science

Planar Tree Transformation: Results and Counterexample

Planar Tree Transformation: Results and Counterexample Selim G Akl, Kamrul Islam, and Henk Meijer School of Computing, Queen s University Kingston, Ontario, Canada K7L 3N6 Abstract We consider the problem