Fast Distributed Approximation for Max-Cut

Finding a maximum cut is a fundamental task in many computational settings. Surprisingly, it has been insufficiently studied in the classic distributed settings, where vertices communicate by synchronously sending messages to their neighbors according to the underlying graph, known as the $\mathcal{LOCAL}$ or $\mathcal{CONGEST}$ models. We amend this by obtaining almost optimal algorithms for Max-Cut on a wide class of graphs in these models. In particular, for any $\epsilon > 0$, we develop randomized approximation algorithms achieving a ratio of $(1-\epsilon)$ to the optimum for Max-Cut on bipartite graphs in the $\mathcal{CONGEST}$ model, and on general graphs in the $\mathcal{LOCAL}$ model. We further present efficient deterministic algorithms, including a $1/3$-approximation for Max-Dicut in our models, thus improving the best known (randomized) ratio of $1/4$. Our algorithms make non-trivial use of the greedy approach of Buchbinder et al. (SIAM Journal on Computing, 2015) for maximizing an unconstrained (non-monotone) submodular function, which may be of independent interest

Streaming complexity of CSPs with randomly ordered constraints

We initiate a study of the streaming complexity of constraint satisfaction problems (CSPs) when the constraints arrive in a random order. We show that there exists a CSP, namely $\textsf{Max-DICUT}$, for which random ordering makes a provable difference. Whereas a $4/9 \approx 0.445$ approximation of $\textsf{DICUT}$ requires $\Omega(\sqrt{n})$ space with adversarial ordering, we show that with random ordering of constraints there exists a $0.48$-approximation algorithm that only needs $O(\log n)$ space. We also give new algorithms for $\textsf{Max-DICUT}$ in variants of the adversarial ordering setting. Specifically, we give a two-pass $O(\log n)$ space $0.48$-approximation algorithm for general graphs and a single-pass $\tilde{O}(\sqrt{n})$ space $0.48$-approximation algorithm for bounded degree graphs. On the negative side, we prove that CSPs where the satisfying assignments of the constraints support a one-wise independent distribution require $\Omega(\sqrt{n})$-space for any non-trivial approximation, even when the constraints are randomly ordered. This was previously known only for adversarially ordered constraints. Extending the results to randomly ordered constraints requires switching the hard instances from a union of random matchings to simple Erd\"os-Renyi random (hyper)graphs and extending tools that can perform Fourier analysis on such instances. The only CSP to have been considered previously with random ordering is $\textsf{Max-CUT}$ where the ordering is not known to change the approximability. Specifically it is known to be as hard to approximate with random ordering as with adversarial ordering, for $o(\sqrt{n})$ space algorithms. Our results show a richer variety of possibilities and motivate further study of CSPs with randomly ordered constraints

Mixed integer programming formulations and heuristics for joint production and transportation problems.

In this thesis we consider different joint production and transportation problems. We first study the simplest two-level problem, the uncapacitated two-level production-in-series lot-sizing problem (2L-S/LS-U). We give a new polynomial dynamic programming algorithm and a new compact extended formulation for the problem and for an extension with sales. Some computational tests are performed comparing several reformulations on a NP-Hard problem containing the 2L-S/LS-U as a relaxation. We also investigate the one-warehouse multi-retailer problem (OWMR), another NP-Hard extension of the 2L-S/LS-U. We study possible ways to tackle the problem effectively using mixed integer programming (MIP) techniques. We analyze the projection of a multi-commodity reformulation onto the space of the original variables for two special cases and characterize valid inequalities for the 2L-S/LS-U. Limited computational experiments are performed to compare several approaches. We then analyze a more general two-level production and transportation problem with multiple production sites. Relaxations for the problem for which reformulations are known are identified in order to improve the linear relaxation bounds. We show that some uncapacitated instances of the basic problem of reasonable size can often be solved to optimality. We also show that a hybrid MIP heuristic based on two different MIP formulations permits us to find solutions guaranteed to be within 10% of optimality for harder instances with limited transportation capacity and/or with additional sales. For instances with big bucket production or aggregate storage capacity constraints the gaps can be larger. In addition, we study a different type of production and transportation problem in which cllients place orders with different sizes and delivery dates and the transportation is performed by a third company. We develop a MIP formulation and an algorithm with a local search procedure that allows us to solve large instances effectively.

Fundamentals

Volume 1 establishes the foundations of this new field. It goes through all the steps from data collection, their summary and clustering, to different aspects of resource-aware learning, i.e., hardware, memory, energy, and communication awareness. Machine learning methods are inspected with respect to resource requirements and how to enhance scalability on diverse computing architectures ranging from embedded systems to large computing clusters

Fundamentals

Volume 1 establishes the foundations of this new field. It goes through all the steps from data collection, their summary and clustering, to different aspects of resource-aware learning, i.e., hardware, memory, energy, and communication awareness. Machine learning methods are inspected with respect to resource requirements and how to enhance scalability on diverse computing architectures ranging from embedded systems to large computing clusters

A General Stabilization Bound for Influence Propagation in Graphs

We study the stabilization time of a wide class of processes on graphs, in which each node can only switch its state if it is motivated to do so by at least a $\frac{1+\lambda}{2}$ fraction of its neighbors, for some $0 < \lambda < 1$. Two examples of such processes are well-studied dynamically changing colorings in graphs: in majority processes, nodes switch to the most frequent color in their neighborhood, while in minority processes, nodes switch to the least frequent color in their neighborhood. We describe a non-elementary function $f(\lambda)$, and we show that in the sequential model, the worst-case stabilization time of these processes can completely be characterized by $f(\lambda)$. More precisely, we prove that for any $\epsilon>0$, $O(n^{1+f(\lambda)+\epsilon})$ is an upper bound on the stabilization time of any proportional majority/minority process, and we also show that there are graph constructions where stabilization indeed takes $\Omega(n^{1+f(\lambda)-\epsilon})$ steps