Negative circuits for flows and submodular flows

AbstractFor solving minimum cost flow problems Goldberg and Tarjan [7] prove strongly polynomial bounds on the negative circuit method of Klein [9] which previously was not even known to be finite. Following the proposal of Goldberg and Tarjan, Cui and Fujishige [1] discuss the use of minimum mean circuits for solving the much more general minimum cost submodular flow problem and prove finiteness where the minimum mean circuit is chosen using a secondary criterium. We introduce certain additional positive weights on negative circuits and propose selecting a negative circuit with minimum ration of cost and weight. The resulting method for solving minimum cost submodular flow problems is pseudopolynomial. In fact, it terminates after at most m·U minimum ratio computations where m denotes the number of arcs and U the maximum capacity of an arc

Power packet transferability via symbol propagation matrix

Power packet is a unit of electric power transferred by a power pulse with an information tag. In Shannon's information theory, messages are represented by symbol sequences in a digitized manner. Referring to this formulation, we define symbols in power packetization as a minimum unit of power transferred by a tagged pulse. Here, power is digitized and quantized. In this paper, we consider packetized power in networks for a finite duration, giving symbols and their energies to the networks. A network structure is defined using a graph whose nodes represent routers, sources, and destinations. First, we introduce symbol propagation matrix (SPM) in which symbols are transferred at links during unit times. Packetized power is described as a network flow in a spatio-temporal structure. Then, we study the problem of selecting an SPM in terms of transferability, that is, the possibility to represent given energies at sources and destinations during the finite duration. To select an SPM, we consider a network flow problem of packetized power. The problem is formulated as an M-convex submodular flow problem which is known as generalization of the minimum cost flow problem and solvable. Finally, through examples, we verify that this formulation provides reasonable packetized power.Comment: Submitted to Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science

Destination-based Routing and Circuit Allocation for Future Traffic Growth

Internet traffic continues to grow relentlessly, driven largely by increasingly high- \\ resolution video streaming, the increasing adoption of cloud computing, the emergence of 5G networks, and the ever-growing reach of social media and social networks. Existing networks use packet switching to route packets on a hop-by-hop basis from the source to the destination. However, they suffer from two shortcomings. First, in existing networks, packets are routed along a fixed shortest path using the Open Shortest Path First (OSPF) protocol or obliviously load-balanced across equal-cost paths using the Equal-Cost Multi-Path (ECMP) protocol. These routing protocols do not fully utilize the network capacity because they do not adapt to network congestions in their routing decisions. Second, although studies have shown that the majority of packets processed by Internet routers are pass-through traffic, packets nonetheless have to be queued and routed at every hop in existing networks, which unnecessarily adds substantial delays and processing costs.In this thesis, we present two new approaches to overcome these shortcomings. First, we propose new backpressure-based routing algorithms which use only shortest-path routes when they are sufficient to accommodate the given traffic load, but will incrementally expand routing choices as needed to accommodate increasing traffic loads. This avoids the poor delay performance inherent in backpressure-based routing algorithms where packets may take long detours under light or moderate loads, and still retains the notable advantage, the network-wide optimal throughput, because packets are adaptively routed along less congested paths.Second, we propose a unified packet and circuit switched network in which the underlying optical transport is used to circuit-switch pass-through traffic by means of pre-established circuits. This avoids unnecessary packet queuing delays and processing costs at each hop. We propose a novel convex optimization framework based on a new destination-based multicommodity flow formulation for the allocation of circuits in such unified networks

Optimizing Opinions with Stubborn Agents

We consider the problem of optimizing the placement of stubborn agents in a social network in order to maximally influence the population. We assume the network contains stubborn users whose opinions do not change, and non-stubborn users who can be persuaded. We further assume the opinions in the network are in an equilibrium that is common to many opinion dynamics models, including the well-known DeGroot model. We develop a discrete optimization formulation for the problem of maximally shifting the equilibrium opinions in a network by targeting users with stubborn agents. The opinion objective functions we consider are the opinion mean, the opinion variance, and the number of individuals whose opinion exceeds a fixed threshold. We show that the mean opinion is a monotone submodular function, allowing us to find a good solution using a greedy algorithm. We find that on real social networks in Twitter consisting of tens of thousands of individuals, a small number of stubborn agents can non-trivially influence the equilibrium opinions. Furthermore, we show that our greedy algorithm outperforms several common benchmarks. We then propose an opinion dynamics model where users communicate noisy versions of their opinions, communications are random, users grow more stubborn with time, and there is heterogeneity is how users' stubbornness increases. We prove that under fairly general conditions on the stubbornness rates of the individuals, the opinions in this model converge to the same equilibrium as the DeGroot model, despite the randomness and user heterogeneity in the model.Comment: 40 pages, 11 figure