Let f(z) = 1 z+2z+2 and note that on the real axis f corresponds to the integrand above. From the equation x2 + 2x + 2 = 0 and the quadratic formula, we find that the singularities of f are −1± i and both are clearly simple poles. Let R > √ 2, c1 : z = x (−R ≤ x ≤ R), and c2 : z = Re (0 ≤ θ ≤ π). Then c = c1+c2 is a simple closed contour that contains −1+i in its interior (see figure). By the R...

(b) In the relaxation we can get many fractional paths between each demand pair of total unit value. These paths can be read out of the solution in polynomial time in the following way. We find an edge e of the smallest positive flow value c, and subtract flow of this value from a cycle or a path going through e over edges of positive flow value. Repeating this procedure we will decompose the f...

There are a large number of identical purchasers of boat rentals, none of whom own a boat, and whose utility from a boat ride depends on the weather. The state of the weather can take one of four values s = +2,+1,−1,−2. In state s, the utility of a renter is βs log t− p.t where β is a positive number, t is the amount of time that the boat is rented for, and p is the price paid per unit time of ...

Privacy-preserving data splitting is a technique that aims to protect data privacy by storing different fragments of data in different locations. In this work we give a new combinatorial formulation to the data splitting problem. We see the data splitting problem as a purely combinatorial problem, in which we have to split data attributes into different fragments in a way that satisfies certain...

If we insert a set of n items into a binary search tree using TREE-INSERT, the resulting tree may be horribly unbalanced. As we saw in class, however, we expect randomly built binary search trees to be balanced. (Precisely, a randomly built binary search tree has expected height O(lg n).) Therefore, if we want to build an expected balanced tree for a fixed set of items, we could randomly permut...

We study the approximability of the weighted edge-dominating set problem. Although even the unweighted case is NP-Complete, in this case a solution of size at most twice the minimum can be efficiently computed due to its close relationship with minimum maximal matching; however, in the weighted case such a nice relationship is not known to exist. In this paper, after showing that weighted edge ...

We consider the image denoising problem using total variation (TV) regularization. This problem can be computationally challenging to solve due to the non-differentiability and non-linearity of the regularization term. We propose an alternating direction augmented Lagrangian (ADAL) method, based on a new variable splitting approach that results in subproblems that can be solved efficiently and ...

Starting from her home, a service provider visits several customers, following a predetermined route, and returns home after all customers are visited. The problem is to …nd a fair allocation of the total cost of this tour among the customers served. A transferableutility cooperative game can be associated with this cost allocation problem. We introduce a new class of games, which we refer as t...

As the number of simultaneously active groups increases, IP multicast suffers from scalability problem. In backbone networks, the state scalability problem is even more serious. In this paper, we propose a novel scheme, called AMBTS, which uses the concept of aggregated multicast, and employs tree splitting before aggregating. We design an algorithm to divide the leaf nodes of backbone into dif...

Bonald et al. have studied insensitivity in data networks assuming a fixed route for each flow class. If capacity allocation and routing are balanced and the capacity of a given class is shared equally between the flows, the network state distribution and flow level performance are insensitive to any detailed traffic characteristics except the traffic loads. In this paper, we consider optimal i...