Optimization bounds from the branching dual

We present a general method for obtaining strong bounds for discrete optimization problems that is based on a concept of branching duality. It can be applied when no useful integer programming model is available, and we illustrate this with the minimum bandwidth problem. The method strengthens a known bound for a given problem by formulating a dual problem whose feasible solutions are partial branching trees. It solves the dual problem with a “worst-bound” local search heuristic that explores neighboring partial trees. After proving some optimality properties of the heuristic, we show that it substantially improves known combinatorial bounds for the minimum bandwidth problem with a modest amount of computation. It also obtains significantly tighter bounds than depth-first and breadth-first branching, demonstrating that the dual perspective can lead to better branching strategies when the object is to find valid bounds.Accepted manuscrip

Decorous lower bounds for minimum linear arrangement

Minimum Linear Arrangement is a classical basic combinatorial optimization problem from the 1960s, which turns out to be extremely challenging in practice. In particular, for most of its benchmark instances, even the order of magnitude of the optimal solution value is unknown, as testified by the surveys on the problem that contain tables in which the best known solution value often has one more digit than the best known lower bound value. In this paper, we propose a linear-programming based approach to compute lower bounds on the optimum. This allows us, for the first time, to show that the best known solutions are indeed not far from optimal for most of the benchmark instances

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th

Hypothesis testing and causal inference with heterogeneous medical data

Learning from data which associations hold and are likely to hold in the future is a fundamental part of scientific discovery. With increasingly heterogeneous data collection practices, exemplified by passively collected electronic health records or high-dimensional genetic data with only few observed samples, biases and spurious correlations are prevalent. These are called spurious because they do not contribute to the effect being studied. In this context, the modelling assumptions of existing statistical tests and causal inference methods are often found inadequate and their practical utility diminished even though these models are increasingly used as decision-support tools in practice. This thesis investigates how modern computational techniques may broaden the fields of hypothesis testing and causal inference to handle the subtleties of large heterogeneous data sets, as well as simultaneously improve the robustness and theoretical understanding of machine learning algorithms using insights from causality and statistics. The first part of this thesis is concerned with hypothesis testing. We develop a framework for hypothesis testing on set-valued data, a representation that faithfully describes many real-world phenomena including patient biomarker trajectories in the hospital. Using similar techniques, we develop next a two-sample test for making inference on selection-biased data, in the sense that not all individuals are equally likely to be included in the study, a fact that biases tests if not accounted for and if the desideratum is to obtain conclusions that are generally applicable. We conclude this section with an investigation of conditional independence in high-dimensional data, such as found in gene expression data, and propose a test using generative adversarial networks. The second part of this thesis is concerned with causal inference and discovery, with a special focus on the influence of unobserved confounders that distort the observed associations between variables and yet may not be ruled out or adjusted for using data alone. We start by demonstrating that unobserved confounders may bias substantially the generalization performance of machine learning algorithms trained with conventional learning paradigms such as empirical risk minimization. Acknowledging this spurious effect, we develop a new learning principle inspired by causal insights that provably generalizes to test data sampled from a larger set of distributions different from the training distribution. In the last chapter we consider the influence of unobserved confounders for causal discovery. We show that with some assumptions on the type and influence on the nature of unobserved confounding one may develop provably consistent causal discovery algorithms, formulated as a solution to a continuous optimization program

Efficient Interconnection Schemes for VLSI and Parallel Computation

This thesis is primarily concerned with two problems of interconnecting components in VLSI technologies. In the first case, the goal is to construct efficient interconnection networks for general-purpose parallel computers. The second problem is a more specialized problem in the design of VLSI chips, namely multilayer channel routing. In addition, a final part of this thesis provides lower bounds on the area required for VLSI implementations of finite-state machines. This thesis shows that networks based on Leiserson\u27s fat-tree architecture are nearly as good as any network built in a comparable amount of physical space. It shows that these universal networks can efficiently simulate competing networks by means of an appropriate correspondence between network components and efficient algorithms for routing messages on the universal network. In particular, a universal network of area A can simulate competing networks with O(lg^3A) slowdown (in bit-times), using a very simple randomized routing algorithm and simple network components. Alternatively, a packet routing scheme of Leighton, Maggs, and Rao can be used in conjunction with more sophisticated switching components to achieve O(lg^2 A) slowdown. Several other important aspects of universality are also discussed. It is shown that universal networks can be constructed in area linear in the number of processors, so that there is no need to restrict the density of processors in competing networks. Also results are presented for comparisons between networks of different size or with processors of different sizes (as determined by the amount of attached memory). Of particular interest is the fact that a universal network built from sufficiently small processors can simulate (with the slowdown already quoted) any competing network of comparable size regardless of the size of processors in the competing network. In addition, many of the results given do not require the usual assumption of unit wire delay. Finally, though most of the discussion is in the two-dimensional world, the results are shown to apply in three dimensions by way of a simple demonstration of general results on graph layout in three dimensions. The second main problem considered in this thesis is channel routing when many layers of interconnect are available, a scenario that is becoming more and more meaningful as chip fabrication technologies advance. This thesis describes a system MulCh for multilayer channel routing which extends the Chameleon system developed at U. C. Berkeley. Like Chameleon, MulCh divides a multilayer problem into essentially independent subproblems of at most three layers, but unlike Chameleon, MulCh considers the possibility of using partitions comprised of a single layer instead of only partitions of two or three layers. Experimental results show that MulCh often performs better than Chameleon in terms of channel width, total net length, and number of vias. In addition to a description of MulCh as implemented, this thesis provides improved algorithms for subtasks performed by MulCh, thereby indicating potential improvements in the speed and performance of multilayer channel routing. In particular, a linear time algorithm is given for determining the minimum width required for a single-layer channel routing problem, and an algorithm is given for maintaining the density of a collection of nets in logarithmic time per net insertion. The last part of this thesis shows that straightforward techniques for implementing finite-state machines are optimal in the worst case. Specifically, for any s and k, there is a deterministic finite-state machine with s states and k symbols such that any layout algorithm requires (ks lg s) area to lay out its realization. For nondeterministic machines, there is an analogous lower bound of (ks^2) area

SoK: A Stratified Approach to Blockchain Decentralization

Decentralization has been touted as the principal security advantage which propelled blockchain systems at the forefront of developments in the financial technology space. Its exact semantics nevertheless remain highly contested and ambiguous, with proponents and critics disagreeing widely on the level of decentralization offered. To address this, we put forth a systematization of the current landscape with respect to decentralization and we derive a methodology that can help direct future research towards defining and measuring decentralization. Our approach dissects blockchain systems into multiple layers, or strata, each possibly encapsulating multiple categories, and enables a unified method for measuring decentralization in each one. Our layers are (1) hardware, (2) software, (3) network, (4) consensus, (5) economics ("tokenomics"), (6) API, (7) governance, and (8) geography. Armed with this stratification, we examine for each layer which pertinent properties of distributed ledgers (safety, liveness, privacy, stability) can be at risk due to centralization and in what way. Our work highlights the challenges in measuring and achieving decentralization, points to the degree of (de)centralization of various existing systems, where such assessment can be made from presently available public information, and suggests potential metrics and directions where future research is needed. We also introduce the "Minimum Decentralization Test", as a way to assess the decentralization state of a blockchain system and, as an exemplary case, we showcase how it can be applied to Bitcoin

Practical Graphs for Optimal Side-Channel Resistant Memory-Hard Functions

A memory-hard function (MHF) $f_n$ with parameter $n$ can be computed in sequential time and space $n$. Simultaneously, a high amortized parallel area-time complexity (aAT) is incurred per evaluation. In practice, MHFs are used to limit the rate at which an adversary (using a custom computational device) can evaluate a security sensitive function that still occasionally needs to be evaluated by honest users (using an off-the-shelf general purpose device). The most prevalent examples of such sensitive functions are Key Derivation Functions (KDFs) and password hashing algorithms where rate limits help mitigate off-line dictionary attacks. As the honest users\u27 inputs to these functions are often (low-entropy) passwords special attention is given to a class of side-channel resistant MHFs called iMHFs. Essentially all iMHFs can be viewed as some mode of operation (making $n$ calls to some round function) given by a directed acyclic graph (DAG) with very low indegree. Recently, a combinatorial property of a DAG has been identified (called ``depth-robustness\u27\u27) which results in good provable security for an iMHF based on that DAG. Depth-robust DAGs have also proven useful in other cryptographic applications. Unfortunately, up till now, all known very depth-robust DAGs are impractically complicated and little is known about their exact (i.e. non-asymptotic) depth-robustness both in theory and in practice. In this work we build and analyze (both formally and empirically) several exceedingly simple and efficient to navigate practical DAGs for use in iMHFs and other applications. For each DAG we: - Prove that their depth-robustness is asymptotically maximal. - Prove bounds of at least $3$ orders of magnitude better on their exact depth-robustness compared to known bounds for other practical iMHF. - Implement and empirically evaluate their depth-robustness and aAT against a variety of state-of-the art (and several new) depth-reduction and low aAT attacks. We find that, against all attacks, the new DAGs perform significantly better in practice than Argon2i, the most widely deployed iMHF in practice. Along the way we also improve the best known empirical attacks on the aAT of Argon2i by implementing and testing several heuristic versions of a (hitherto purely theoretical) depth-reduction attack. Finally, we demonstrate practicality of our constructions by modifying the Argon2i code base to use one of the new high aAT DAGs. Experimental benchmarks on a standard off-the-shelf CPU show that the new modifications do not adversely affect the impressive throughput of Argon2i (despite seemingly enjoying significantly higher aAT)