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

Distributed coloring in sparse graphs with fewer colors

This paper is concerned with efficiently coloring sparse graphs in the distributed setting with as few colors as possible. According to the celebrated Four Color Theorem, planar graphs can be colored with at most 4 colors, and the proof gives a (sequential) quadratic algorithm finding such a coloring. A natural problem is to improve this complexity in the distributed setting. Using the fact that planar graphs contain linearly many vertices of degree at most 6, Goldberg, Plotkin, and Shannon obtained a deterministic distributed algorithm coloring $n$-vertex planar graphs with 7 colors in $O(\log n)$ rounds. Here, we show how to color planar graphs with 6 colors in \mbox{polylog}(n) rounds. Our algorithm indeed works more generally in the list-coloring setting and for sparse graphs (for such graphs we improve by at least one the number of colors resulting from an efficient algorithm of Barenboim and Elkin, at the expense of a slightly worst complexity). Our bounds on the number of colors turn out to be quite sharp in general. Among other results, we show that no distributed algorithm can color every $n$-vertex planar graph with 4 colors in $o(n)$ rounds.Comment: 16 pages, 4 figures - An extended abstract of this work was presented at PODC'18 (ACM Symposium on Principles of Distributed Computing

Topics in Graph Algorithms: Structural Results and Algorithmic Techniques, with Applications

Coping with computational intractability has inspired the development of a variety of algorithmic techniques. The main challenge has usually been the design of polynomial time algorithms for NP-complete problems in a way that guarantees some, often worst-case, satisfactory performance when compared to exact (optimal) solutions. We mainly study some emergent techniques that help to bridge the gap between computational intractability and practicality. We present results that lead to better exact and approximation algorithms and better implementations. The problems considered in this dissertation share much in common structurally, and have applications in several scientific domains, including circuit design, network reliability, and bioinformatics. We begin by considering the relationship between graph coloring and the immersion order, a well-quasi-order defined on the set of finite graphs. We establish several (structural) results and discuss their potential algorithmic consequences. We discuss graph metrics such as treewidth and pathwidth. Treewidth is well studied, mainly because many problems that are NP-hard in general have polynomial time algorithms when restricted to graphs of bounded treewidth. Pathwidth has many applications ranging from circuit layout to natural language processing. We present a linear time algorithm to approximate the pathwidth of planar graphs that have a fixed disk dimension. We consider the face cover problem, which has potential applications in facilities location and logistics. Being fixed-parameter tractable, we develop an algorithm that solves it in time O(5k + n2) where k is the input parameter. This is a notable improvement over the previous best known algorithm, which runs in O(8kn). In addition to the structural and algorithmic results, this text tries to illustrate the practicality of fixed-parameter algorithms. This is achieved by implementing some algorithms for the vertex cover problem, and conducting experiments on real data sets. Our experiments advocate the viewpoint that, for many practical purposes, exact solutions of some NP-complete problems are affordable

Some Optimally Adaptive Parallel Graph Algorithms on EREW PRAM Model

The study of graph algorithms is an important area of research in computer science, since graphs offer useful tools to model many real-world situations. The commercial availability of parallel computers have led to the development of efficient parallel graph algorithms. Using an exclusive-read and exclusive-write (EREW) parallel random access machine (PRAM) as the computation model with a fixed number of processors, we design and analyze parallel algorithms for seven undirected graph problems, such as, connected components, spanning forest, fundamental cycle set, bridges, bipartiteness, assignment problems, and approximate vertex coloring. For all but the last two problems, the input data structure is an unordered list of edges, and divide-and-conquer is the paradigm for designing algorithms. One of the algorithms to solve the assignment problem makes use of an appropriate variant of dynamic programming strategy. An elegant data structure, called the adjacency list matrix, used in a vertex-coloring algorithm avoids the sequential nature of linked adjacency lists. Each of the proposed algorithms achieves optimal speedup, choosing an optimal granularity (thus exploiting maximum parallelism) which depends on the density or the number of vertices of the given graph. The processor-(time)2 product has been identified as a useful parameter to measure the cost-effectiveness of a parallel algorithm. We derive a lower bound on this measure for each of our algorithms

Solving Hard Computational Problems Efficiently: Asymptotic Parametric Complexity 3-Coloring Algorithm

Many practical problems in almost all scientific and technological disciplines have been classified as computationally hard (NP-hard or even NP-complete). In life sciences, combinatorial optimization problems frequently arise in molecular biology, e.g., genome sequencing; global alignment of multiple genomes; identifying siblings or discovery of dysregulated pathways.In almost all of these problems, there is the need for proving a hypothesis about certain property of an object that can be present only when it adopts some particular admissible structure (an NP-certificate) or be absent (no admissible structure), however, none of the standard approaches can discard the hypothesis when no solution can be found, since none can provide a proof that there is no admissible structure. This article presents an algorithm that introduces a novel type of solution method to "efficiently" solve the graph 3-coloring problem; an NP-complete problem. The proposed method provides certificates (proofs) in both cases: present or absent, so it is possible to accept or reject the hypothesis on the basis of a rigorous proof. It provides exact solutions and is polynomial-time (i.e., efficient) however parametric. The only requirement is sufficient computational power, which is controlled by the parameter $\alpha\in\mathbb{N}$. Nevertheless, here it is proved that the probability of requiring a value of $\alpha>k$ to obtain a solution for a random graph decreases exponentially: $P(\alpha>k) \leq 2^{-(k+1)}$, making tractable almost all problem instances. Thorough experimental analyses were performed. The algorithm was tested on random graphs, planar graphs and 4-regular planar graphs. The obtained experimental results are in accordance with the theoretical expected results.Comment: Working pape

Distributed Approximation of Maximum Independent Set and Maximum Matching

We present a simple distributed $\Delta$-approximation algorithm for maximum weight independent set (MaxIS) in the $\mathsf{CONGEST}$ model which completes in $O(\texttt{MIS}(G)\cdot \log W)$ rounds, where $\Delta$ is the maximum degree, $\texttt{MIS}(G)$ is the number of rounds needed to compute a maximal independent set (MIS) on $G$, and $W$ is the maximum weight of a node. %Whether our algorithm is randomized or deterministic depends on the \texttt{MIS} algorithm used as a black-box. Plugging in the best known algorithm for MIS gives a randomized solution in $O(\log n \log W)$ rounds, where $n$ is the number of nodes. We also present a deterministic $O(\Delta +\log^* n)$-round algorithm based on coloring. We then show how to use our MaxIS approximation algorithms to compute a $2$-approximation for maximum weight matching without incurring any additional round penalty in the $\mathsf{CONGEST}$ model. We use a known reduction for simulating algorithms on the line graph while incurring congestion, but we show our algorithm is part of a broad family of \emph{local aggregation algorithms} for which we describe a mechanism that allows the simulation to run in the $\mathsf{CONGEST}$ model without an additional overhead. Next, we show that for maximum weight matching, relaxing the approximation factor to ($2+\varepsilon$) allows us to devise a distributed algorithm requiring $O(\frac{\log \Delta}{\log\log\Delta})$ rounds for any constant $\varepsilon>0$. For the unweighted case, we can even obtain a $(1+\varepsilon)$-approximation in this number of rounds. These algorithms are the first to achieve the provably optimal round complexity with respect to dependency on $\Delta$

Best of Two Local Models: Local Centralized and Local Distributed Algorithms

We consider two models of computation: centralized local algorithms and local distributed algorithms. Algorithms in one model are adapted to the other model to obtain improved algorithms. Distributed vertex coloring is employed to design improved centralized local algorithms for: maximal independent set, maximal matching, and an approximation scheme for maximum (weighted) matching over bounded degree graphs. The improvement is threefold: the algorithms are deterministic, stateless, and the number of probes grows polynomially in $\log^* n$, where $n$ is the number of vertices of the input graph. The recursive centralized local improvement technique by Nguyen and Onak~\cite{onak2008} is employed to obtain an improved distributed approximation scheme for maximum (weighted) matching. The improvement is twofold: we reduce the number of rounds from $O(\log n)$ to $O(\log^*n)$ for a wide range of instances and, our algorithms are deterministic rather than randomized

Solving constraint-satisfaction problems with distributed neocortical-like neuronal networks

Finding actions that satisfy the constraints imposed by both external inputs and internal representations is central to decision making. We demonstrate that some important classes of constraint satisfaction problems (CSPs) can be solved by networks composed of homogeneous cooperative-competitive modules that have connectivity similar to motifs observed in the superficial layers of neocortex. The winner-take-all modules are sparsely coupled by programming neurons that embed the constraints onto the otherwise homogeneous modular computational substrate. We show rules that embed any instance of the CSPs planar four-color graph coloring, maximum independent set, and Sudoku on this substrate, and provide mathematical proofs that guarantee these graph coloring problems will convergence to a solution. The network is composed of non-saturating linear threshold neurons. Their lack of right saturation allows the overall network to explore the problem space driven through the unstable dynamics generated by recurrent excitation. The direction of exploration is steered by the constraint neurons. While many problems can be solved using only linear inhibitory constraints, network performance on hard problems benefits significantly when these negative constraints are implemented by non-linear multiplicative inhibition. Overall, our results demonstrate the importance of instability rather than stability in network computation, and also offer insight into the computational role of dual inhibitory mechanisms in neural circuits.Comment: Accepted manuscript, in press, Neural Computation (2018