Scheduling jobs with hard deadlines over Multiple Access and Degraded Broadcast Channels

We consider the problem of scheduling jobs with given start and finish times over two classes of multi-user channels, namely Multiple Access Channels and Degraded Broadcast Channels, and derive necessary and sufficient conditions for feasible scheduling of the jobs

Curvature and Optimal Algorithms for Learning and Minimizing Submodular Functions

We investigate three related and important problems connected to machine learning: approximating a submodular function everywhere, learning a submodular function (in a PAC-like setting [53]), and constrained minimization of submodular functions. We show that the complexity of all three problems depends on the 'curvature' of the submodular function, and provide lower and upper bounds that refine and improve previous results [3, 16, 18, 52]. Our proof techniques are fairly generic. We either use a black-box transformation of the function (for approximation and learning), or a transformation of algorithms to use an appropriate surrogate function (for minimization). Curiously, curvature has been known to influence approximations for submodular maximization [7, 55], but its effect on minimization, approximation and learning has hitherto been open. We complete this picture, and also support our theoretical claims by empirical results.Comment: 21 pages. A shorter version appeared in Advances of NIPS-201

New Algorithmic Results in Clustering and Partitioning

Clustering and partitioning tasks have found widespread applications across computing. In machine learning, clustering represents the quintessential unsupervised learning task: grouping similar data points to discover structure in data. In operations research and combinatorial optimization, one is often interested in finding bottlenecks in a network, to identify possible weakness and points of failure. In this work, we discuss recent progress in better understanding computational aspects of clustering and partitioning. Our primary goal is establishing formal mathematical guarantees on the performance of clustering algorithms, as well as proving impossibility results to determine the inherent hardness of the problems we consider. In the first part of the thesis, we discuss graph partitioning tasks, focusing on the theory behind finding small vertex separators: few vertices which, when removed, disconnect the graph into large pieces. We design approximation algorithms for this problem, based on rounding natural convex relaxations. We also outline a recently uncovered connection between this problem and the fastest mixing random walk process on a graph with a target stationary distribution. In the second part of this work we discuss some algorithmic results in partitioning hypergraphs. We introduce a new, expressive class of hypergraph cut functions. We then design approximation algorithms for hypergraph generalizations of the minimum conductance cut problem by leveraging and extending techniques from spectral graph theory to the hypergraph regime. We prove our results for all the cut functions in our newly-defined class. In the process, we also improve on a popular primal-dual algorithmic framework for graph partitioning algorithms. Finally, we address the problem of learning partitions in an interactive way, by querying a same-cluster oracle, which determines whether two points belong to the same cluster. In this context we develop and analyze novel error-resistant algorithms, and provide complementary lower bounds, showing that our algorithms achieve optimal query complexity. To this end, we develop a new analytic framework based on modeling this task as a Rényi-Ulam liar game

"Graph Entropy, Network Coding and Guessing games"

We introduce the (private) entropy of a directed graph (in a new network coding sense) as well as a number of related concepts. We show that the entropy of a directed graph is identical to its guessing number and can be bounded from below with the number of vertices minus the size of the graph’s shortest index code. We show that the Network Coding solvability of each specific multiple unicast network is completely determined by the entropy (as well as by the shortest index code) of the directed graph that occur by identifying each source node with each corresponding target node. Shannon’s information inequalities can be used to calculate up- per bounds on a graph’s entropy as well as calculating the size of the minimal index code. Recently, a number of new families of so-called non-shannon-type information inequalities have been discovered. It has been shown that there exist communication networks with a ca- pacity strictly ess than required for solvability, but where this fact cannot be derived using Shannon’s classical information inequalities. Based on this result we show that there exist graphs with an entropy that cannot be calculated using only Shannon’s classical information inequalities, and show that better estimate can be obtained by use of certain non-shannon-type information inequalities

Congestion-Free Rerouting of Flows on DAGs

Changing a given configuration in a graph into another one is known as a reconfiguration problem. Such problems have recently received much interest in the context of algorithmic graph theory. We initiate the theoretical study of the following reconfiguration problem: How to reroute k unsplittable flows of a certain demand in a capacitated network from their current paths to their respective new paths, in a congestion-free manner? This problem finds immediate applications, e.g., in traffic engineering in computer networks. We show that the problem is generally NP-hard already for k=2 flows, which motivates us to study rerouting on a most basic class of flow graphs, namely DAGs. Interestingly, we find that for general k, deciding whether an unsplittable multi-commodity flow rerouting schedule exists, is NP-hard even on DAGs. Our main contribution is a polynomial-time (fixed parameter tractable) algorithm to solve the route update problem for a bounded number of flows on DAGs. At the heart of our algorithm lies a novel decomposition of the flow network that allows us to express and resolve reconfiguration dependencies among flows

Optimization of Multiclass Queueing Networks: Polyhedral and Nonlinear Characterizations of Achievable Performance

We consider open and closed multiclass queueing networks with Poisson arrivals (in open networks), exponentially distributed class dependent service times, and with class dependent deterministic or probabilistic routing. For open networks, the performance objective is to minimize, over all sequencing and routing policies, a weighted sum of the expected response times of different classes. Using a powerful technique involving quadratic or higher order potential functions, we propose variants of a method to derive polyhedral and nonlinear spaces which contain the entire set of achievable response times under stable and preemptive scheduling policies. By optimizing over these spaces, we obtain lower bounds on achievable performance. In particular, we obtain a sequence of progressively more complicated nonlinear approximations (relaxations) which are progressively closer to the exact achievable space. In the special case of single station networks (multiclass queues and Klimov's model) and homogenous multiclass networks, our characterization gives exactly the achievable region. Consequently, the proposed method can be viewed as the natural extension of conservation laws to multiclass queueing networks. For closed networks, the performance objective is to maximize throughput. We similarly find polyhedral and nonlinear spaces that include the performance space and by maximizing over these spaces we obtain an upper bound on the optimal throughput. We check the tightness of our bounds by simulating heuristic scheduling policies for simple open networks and we find that the first order approximation of our method is at least as good as simulation-based existing methods. In terms of computational complexity and in contrast to simulation-based existing methods, the calculation of our first order bounds consists of solving a linear programming problem with both the number of variables and constraints being polynomial (quadratic) in the number of classes in the network. The i-th order approximation involves solving a convex programming problem in dimension O(Ri+l), where R is the number of classes in the network, which can be solved efficiently using techniques from semi-definite programming