Maximum Weight Matching via Max-Product Belief Propagation

Max-product "belief propagation" is an iterative, local, message-passing algorithm for finding the maximum a posteriori (MAP) assignment of a discrete probability distribution specified by a graphical model. Despite the spectacular success of the algorithm in many application areas such as iterative decoding, computer vision and combinatorial optimization which involve graphs with many cycles, theoretical results about both correctness and convergence of the algorithm are known in few cases (Weiss-Freeman Wainwright, Yeddidia-Weiss-Freeman, Richardson-Urbanke}. In this paper we consider the problem of finding the Maximum Weight Matching (MWM) in a weighted complete bipartite graph. We define a probability distribution on the bipartite graph whose MAP assignment corresponds to the MWM. We use the max-product algorithm for finding the MAP of this distribution or equivalently, the MWM on the bipartite graph. Even though the underlying bipartite graph has many short cycles, we find that surprisingly, the max-product algorithm always converges to the correct MAP assignment as long as the MAP assignment is unique. We provide a bound on the number of iterations required by the algorithm and evaluate the computational cost of the algorithm. We find that for a graph of size $n$, the computational cost of the algorithm scales as $O(n^3)$, which is the same as the computational cost of the best known algorithm. Finally, we establish the precise relation between the max-product algorithm and the celebrated {\em auction} algorithm proposed by Bertsekas. This suggests possible connections between dual algorithm and max-product algorithm for discrete optimization problems.Comment: In the proceedings of the 2005 IEEE International Symposium on Information Theor

Algorithms for Fast Aggregated Convergecast in Sensor Networks

Fast and periodic collection of aggregated data is of considerable interest for mission-critical and continuous monitoring applications in sensor networks. In the many-to-one communication paradigm, referred to as convergecast, we focus on applications wherein data packets are aggregated at each hop en-route to the sink along a tree-based routing topology, and address the problem of minimizing the convergecast schedule length by utilizing multiple frequency channels. The primary hindrance in minimizing the schedule length is the presence of interfering links. We prove that it is NP-complete to determine whether all the interfering links in an arbitrary network can be removed using at most a constant number of frequencies. We give a sufficient condition on the number of frequencies for which all the interfering links can be removed, and propose a polynomial time algorithm that minimizes the schedule length in this case. We also prove that minimizing the schedule length for a given number of frequencies on an arbitrary network is NP-complete, and describe a greedy scheme that gives a constant factor approximation on unit disk graphs. When the routing tree is not given as an input to the problem, we prove that a constant factor approximation is still achievable for degree-bounded trees. Finally, we evaluate our algorithms through simulations and compare their performance under different network parameters

Multi-agent pathfinding for unmanned aerial vehicles

Unmanned aerial vehicles (UAVs), commonly known as drones, have become more and more prevalent in recent years. In particular, governmental organizations and companies around the world are starting to research how UAVs can be used to perform tasks such as package deliver, disaster investigation and surveillance of key assets such as pipelines, railroads and bridges. NASA is currently in the early stages of developing an air traffic control system specifically designed to manage UAV operations in low-altitude airspace. Companies such as Amazon and Rakuten are testing large-scale drone deliver services in the USA and Japan. To perform these tasks, safe and conflict-free routes for concurrently operating UAVs must be found. This can be done using multi-agent pathfinding (mapf) algorithms, although the correct choice of algorithms is not clear. This is because many state of the art mapf algorithms have only been tested in 2D space in maps with many obstacles, while UAVs operate in 3D space in open maps with few obstacles. In addition, when an unexpected event occurs in the airspace and UAVs are forced to deviate from their original routes while inflight, new conflict-free routes must be found. Planning for these unexpected events is commonly known as contingency planning. With manned aircraft, contingency plans can be created in advance or on a case-by-case basis while inflight. The scale at which UAVs operate, combined with the fact that unexpected events may occur anywhere at any time make both advanced planning and planning on a case-by-case basis impossible. Thus, a new approach is needed. Online multi-agent pathfinding (online mapf) looks to be a promising solution. Online mapf utilizes traditional mapf algorithms to perform path planning in real-time. That is, new routes for UAVs are found while inflight. The primary contribution of this thesis is to present one possible approach to UAV contingency planning using online multi-agent pathfinding algorithms, which can be used as a baseline for future research and development. It also provides an in-depth overview and analysis of offline mapf algorithms with the goal of determining which ones are likely to perform best when applied to UAVs. Finally, to further this same goal, a few different mapf algorithms are experimentally tested and analyzed

Fast Algorithms for Parameterized Problems with Relaxed Disjointness Constraints

In parameterized complexity, it is a natural idea to consider different generalizations of classic problems. Usually, such generalization are obtained by introducing a "relaxation" variable, where the original problem corresponds to setting this variable to a constant value. For instance, the problem of packing sets of size at most $p$ into a given universe generalizes the Maximum Matching problem, which is recovered by taking $p=2$. Most often, the complexity of the problem increases with the relaxation variable, but very recently Abasi et al. have given a surprising example of a problem --- $r$-Simple $k$-Path --- that can be solved by a randomized algorithm with running time $O^*(2^{O(k \frac{\log r}{r})})$. That is, the complexity of the problem decreases with $r$. In this paper we pursue further the direction sketched by Abasi et al. Our main contribution is a derandomization tool that provides a deterministic counterpart of the main technical result of Abasi et al.: the $O^*(2^{O(k \frac{\log r}{r})})$ algorithm for $(r,k)$-Monomial Detection, which is the problem of finding a monomial of total degree $k$ and individual degrees at most $r$ in a polynomial given as an arithmetic circuit. Our technique works for a large class of circuits, and in particular it can be used to derandomize the result of Abasi et al. for $r$-Simple $k$-Path. On our way to this result we introduce the notion of representative sets for multisets, which may be of independent interest. Finally, we give two more examples of problems that were already studied in the literature, where the same relaxation phenomenon happens. The first one is a natural relaxation of the Set Packing problem, where we allow the packed sets to overlap at each element at most $r$ times. The second one is Degree Bounded Spanning Tree, where we seek for a spanning tree of the graph with a small maximum degree

Hierarchical and High-Girth QC LDPC Codes

We present a general approach to designing capacity-approaching high-girth low-density parity-check (LDPC) codes that are friendly to hardware implementation. Our methodology starts by defining a new class of "hierarchical" quasi-cyclic (HQC) LDPC codes that generalizes the structure of quasi-cyclic (QC) LDPC codes. Whereas the parity check matrices of QC LDPC codes are composed of circulant sub-matrices, those of HQC LDPC codes are composed of a hierarchy of circulant sub-matrices that are in turn constructed from circulant sub-matrices, and so on, through some number of levels. We show how to map any class of codes defined using a protograph into a family of HQC LDPC codes. Next, we present a girth-maximizing algorithm that optimizes the degrees of freedom within the family of codes to yield a high-girth HQC LDPC code. Finally, we discuss how certain characteristics of a code protograph will lead to inevitable short cycles, and show that these short cycles can be eliminated using a "squashing" procedure that results in a high-girth QC LDPC code, although not a hierarchical one. We illustrate our approach with designed examples of girth-10 QC LDPC codes obtained from protographs of one-sided spatially-coupled codes.Comment: Submitted to IEEE Transactions on Information THeor