Nicholas J. A. Harvey

Non-Adaptive Fault Diagnosis for All-Optical Networks via Combinatorial Group Testing on Graphs

We consider the problem of detecting failures for all-optical networks, with the objective of keeping the diagnosis cost low. Compared to the passive paradigm based on parity check in SONET, optical probing signals are sent proactively along lightpaths to probe their state of health and failure pattern is identified through the set of test results (i.e., probe syndromes). As an alternative to our previous adaptive approach where all the probes are sent sequentially, we consider in this work a non-adaptive approach where all the probes are sent in parallel. The design objective is to minimize the number of parallel probes, so as to keep network cost low. The non-adaptive fault diagnosis approach motivates a new technical framework that we introduce: combinatorial group testing with graph-based constraints. Using this framework, we develop several new probing schemes to detect network faults for all-optical networks with different topologies. The efficiency of our schemes often depends on the network topology; in many cases we can show that our schemes are optimal in minimizing the number of probes.

On the complexity of reconfiguration problems

Reconfiguration problems arise when we wish to find a step-by-step transformation between two feasible solutions of a problem such that all intermediate results are also feasible. We demonstrate that a host of reconfiguration problems derived from NP-complete problems are PSPACE-complete, while some are also NP-hard to approximate. In contrast, several reconfiguration versions of problems in P are solvable in polynomial time.

On the capacity of information networks

We consider information networks in the absence of interference and noise, and present an upper bound on the rate at which information can be transmitted using network coding. Our upper bound is based on combining properties of entropy with a strong information inequality derived from the structure of the network.The undirected k-pairs conjecture states that the information capacity of an undirected network supporting k point-to-point connections is achievable by multicommodity flows. Our techniques prove the conjecture for a non-trivial class of graphs, and also yield the first known proof of a gap between the sparsity of an undirected graph and its capacity. We believe that these techniques may be instrumental in resolving the conjecture completely. We demonstrate the importance of the undirected k-pairs conjecture by connecting it with a long-standing open question in Input/Output (I/O) complexity. We also show that proving the conjecture would provide the strongest known lower bound for computation in the oblivious cell-probe model and give a non-trivial lower bound for two-tape oblivious Turing machines.Finally, we conclude by considering the capacity of directed information networks. We construct a family of directed graphs whose capacity is much larger than the rate achievable using only multicommodity flows. The gap that we exhibit is linear in the number of vertices, edges, and commodities of the graph, which is asymptotically optimal.

A general framework for graph sparsification

We present a general framework for constructing cut sparsifiers in undirected graphs --- weighted subgraphs for which every cut has the same weight as the original graph, up to a multiplicative factor of (1 ε). Using this framework, we simplify, unify and improve upon previous sparsification results. As simple instantiations of this framework, we show that sparsifiers can be constructed by sampling edges according to their strength (a result of Benczur and Karger), effective resistance (a result of Spielman and Srivastava), edge connectivity, or by sampling random spanning trees. Sampling according to edge connectivity is the most aggressive method, and the most challenging to analyze. Our proof that this method produces sparsifiers resolves an open question of Benczur and Karger. While the above results are interesting from a combinatorial standpoint, we also prove new algorithmic results. In particular, we develop techniques that give the first (optimal) O(m)-time sparsification algorithm for unweighted graphs. Our algorithm has a running time of O(m) + ~O(n/ε2) for weighted graphs, which is also linear unless the input graph is very sparse itself. In both cases, this improves upon the previous best running times (due to Benczur and Karger) of O(m log2 n) (for the unweighted case) and O(m log3 n) (for the weighted case) respectively. Our algorithm constructs sparsifiers that contain O(n log n/ε2) edges in expectation; the only known construction of sparsifiers with fewer edges is by a substantially slower algorithm running in O(n3 m / ε2) time. A key ingredient of our proofs is a natural generalization of Kargers bound on the number of small cuts in an undirected graph. Given the numerous applications of Kargers bound, we suspect that our generalization will also be of independent interest.

Semi-matchings for Bipartite Graphs and Load Balancing

We consider the problem of fairly matching the left-hand vertices of a bipartite graph to the right-hand vertices. We refer to this problem as the semi-matching problem; it is a relaxation of the known bipartite matching problem. We present a way to evaluate the quality of a given semi-matching and show that, under this measure, an optimal semi-matching balances the load on the right hand vertices with respect to any L p -norm. In particular, when modeling a job assignment system, an optimal semi-matching achieves the minimal makespan and the minimal flow time for the system.

The complexity of matrix completion

Given a matrix whose entries are a mixture of numeric values and symbolic variables, the matrix completion problem is to assign values to the variables so as to maximize the resulting matrix rank. This problem has deep connections to computational complexity and numerous important algorithmic applications. Determining the complexity of this problem is a fundamental open question in computational complexity. Under different settings of parameters, the problem is variously in P, in RP, or NP-hard. We shed new light on this landscape by demonstrating a new region of NP-hard scenarios. As a special case, we obtain the first known hardness result for matrices in which each variable appears only twice.Another particular scenario that we consider is the simultaneous matrix completion problem, where one must simultaneously maximize the rank for several matrices that share variables. This problem has important applications in the field of network coding. Recent work has given a simple, greedy, deterministic algorithm for this problem, assuming that the algorithm works over a sufficiently large field. We show an exact threshold for the field size required to find a simultaneous completion efficiently. This result implies that, surprisingly, the simple greedy algorithm is optimal: finding a simultaneous completion over any smaller field is NP-hard.

Algebraic Structures and Algorithms for Matching and Matroid Problems

We present new algebraic approaches for several well-known combinatorial problems, including non-bipartite matching, matroid intersection, and some of their generalizations. Our work yields new randomized algorithms that are the most efficient known. For non-bipartite matching, we obtain a simple, purely algebraic algorithm with running time O(nw) where n is the number of vertices and w is the matrix multiplication exponent. This resolves the central open problem of Mucha and Sankowski (2004). For matroid intersection, our algorithm has running time O(nrw - 1)for matroids with n elements and rank r that satisfy some natural conditions. This algorithm is based on new algebraic results characterizing the size of a maximum intersection in contracted matroids. Furthermore, the running time of this algorithm is essentially optimal

Learning submodular functions

There has been much interest in the machine learning and algorithmic game theory communities on understanding and using submodular functions. Despite this substantial interest, little is known about their learnability from data. Motivated by applications, such as pricing goods in economics, this paper considers PAC-style learning of submodular functions in a distributional setting. A problem instance consists of a distribution on {0,1}n and a real-valued function on {0,1}n that is non-negative, monotone, and submodular. We are given poly(n) samples from this distribution, along with the values of the function at those sample points. The task is to approximate the value of the function to within a multiplicative factor at subsequent sample points drawn from the same distribution, with sufficiently high probability. We develop the first theoretical analysis of this problem, proving a number of important and nearly tight results. For instance, if the underlying distribution is a product distribution then we give a learning algorithm that achieves a constant-factor approximation (under some assumptions). However, for general distributions we provide a surprising Omega(n1/3) lower bound based on a new interesting class of matroids and we also show a O(n1/2) upper bound. Our work combines central issues in optimization (submodular functions and matroids) with central topics in learning (distributional learning and PAC-style analyses) and with central concepts in pseudo-randomness (lossless expander graphs). Our analysis involves a twist on the usual learning theory models and uncovers some interesting structural and extremal properties of submodular functions, which we suspect are likely to be useful in other contexts. In particular, to prove our general lower bound, we use lossless expanders to construct a new family of matroids which can take wildly varying rank values on superpolynomially many sets; no such construction was previously known. This construction shows unexpected extremal properties of submodular functions.

Deterministic SkipNet

We present a deterministic scalable overlay network. In contrast, most previous overlay networks use randomness or hashing (pseudo-randomness) to achieve a uniform distribution of data and routing traffic.

An Algorithmic Proof of the Lovasz Local Lemma via Resampling Oracles

The Lovasz Local Lemma is a seminal result in probabilistic combinatorics. It gives a sufficient condition on a probability space and a collection of events for the existence of an outcome that simultaneously avoids all of those events. Finding such an outcome by an efficient algorithm has been an active research topic for decades. Breakthrough work of Moser and Tardos (2009) presented an efficient algorithm for a general setting primarily characterized by a product structure on the probability space. In this work we present an efficient algorithm for a much more general setting. Our main assumption is that there exist certain functions, called resampling oracles, that can be invoked to address the undesired occurrence of the events. We show that, in all scenarios to which the original Lovasz Local Lemma applies, there exist resampling oracles, although they are not necessarily efficient. Nevertheless, for essentially all known applications of the Lovasz Local Lemma and its generalizations, we have designed efficient resampling oracles. As applications of these techniques, we present new results for packings of Latin transversals, rainbow matchings and rainbow spanning trees.