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Free, publiclyaccessible full text available October 1, 2023

A graph profile records all possible densities of a fixed finite set of graphs. Profiles can be extremely complicated; for instance the full profile of any triple of connected graphs is not known, and little is known about hypergraph profiles. We introduce the tropicalization of graph and hypergraph profiles. Tropicalization is a wellstudied operation in algebraic geometry, which replaces a variety (the set of real or complex solutions to a finite set of algebraic equations) with its “combinatorial shadow”. We prove that the tropicalization of a graph profile is a closed convex cone, which still captures interesting combinatorial information. We explicitly compute these tropicalizations for arbitrary sets of complete and star hypergraphs. We show they are rational polyhedral cones even though the corresponding profiles are not even known to be semialgebraic in some of these cases. We then use tropicalization to prove strong restrictions on the power of the sums of squares method, equivalently CauchySchwarz calculus, to test (which is weaker than certification) the validity of graph density inequalities. In particular, we show that sums of squares cannot test simple binomial graph density inequalities, or even their approximations. Small concrete examples of such inequalities are presented, and include the famousmore »

We study optimal design problems in which the goal is to choose a set of linear measurements to obtain the most accurate estimate of an unknown vector. We study the [Formula: see text]optimal design variant where the objective is to minimize the average variance of the error in the maximum likelihood estimate of the vector being measured. We introduce the proportional volume sampling algorithm to obtain nearly optimal bounds in the asymptotic regime when the number [Formula: see text] of measurements made is significantly larger than the dimension [Formula: see text] and obtain the first approximation algorithms whose approximation factor does not degrade with the number of possible measurements when [Formula: see text] is small. The algorithm also gives approximation guarantees for other optimal design objectives such as [Formula: see text]optimality and the generalized ratio objective, matching or improving the previously bestknown results. We further show that bounds similar to ours cannot be obtained for [Formula: see text]optimal design and that [Formula: see text]optimal design is NPhard to approximate within a fixed constant when [Formula: see text].

Scrubbing sensitive data before releasing memory is a widely accepted but often ignored programming practice for developing secure software. Consequently, confidential data such as cryptographic keys, passwords, and personal data, can remain in memory indefinitely, thereby increasing the risk of exposure to hackers who can retrieve the data using memory dumps or exploit vulnerabilities such as Heartbleed and Etherleak. We propose an approach for detecting a specific memory safety bug called Improper Clearing of Heap Memory Before Release, also known as Common Weakness Enumeration 244, in C programs. The CWE244 bug in a program allows the leakage of confidential information when a variable is not wiped before heap memory is freed. Our approach combines taint analysis and model checking to detect this weakness. We have three main phases: (1) perform a coarse flowinsensitive interprocedural static analysis on the program to construct a set of pointer variables that could point to sensitive data; (2) instrument the program with required dynamic variable tracking, and assertion logic for memory wiping before deallocation; and (3) invoke a model checker, the CBounded Model Checker (CBMC) in our case, to detect assertion violation in the instrumented program. We develop a tool, \toolname, implementing our instrumentation basedmore »

Constrained submodular function maximization has been used in subset selection problems such as selection of most informative sensor locations. Although these models have been quite popular, the solutions obtained via this approach are unstable to perturbations in data defining the submodular functions. Robust submodular maximization has been proposed as a richer model that aims to overcome this discrepancy as well as increase the modeling scope of submodular optimization. In this work, we consider robust submodular maximization with structured combinatorial constraints and give efficient algorithms with provable guarantees. Our approach is applicable to constraints defined by single or multiple matroids and knapsack as well as distributionally robust criteria. We consider both the offline setting where the data defining the problem are known in advance and the online setting where the input data are revealed over time. For the offline setting, we give a general (nearly) optimal bicriteria approximation algorithm that relies on new extensions of classical algorithms for submodular maximization. For the online version of the problem, we give an algorithm that returns a bicriteria solution with sublinear regret. Summary of Contribution: Constrained submodular maximization is one of the core areas in combinatorial optimization with a wide variety of applications inmore »

Experimental design is a classical statistics problem, and its aim is to estimate an unknown vector from linear measurements where a Gaussian noise is introduced in each measurement. For the combinatorial experimental design problem, the goal is to pick a subset of experiments so as to make the most accurate estimate of the unknown parameters. In this paper, we will study one of the most robust measures of error estimation—the Doptimality criterion, which corresponds to minimizing the volume of the confidence ellipsoid for the estimation error. The problem gives rise to two natural variants depending on whether repetitions of experiments are allowed or not. We first propose an approximation algorithm with a 1/eapproximation for the Doptimal design problem with and without repetitions, giving the first constantfactor approximation for the problem. We then analyze another sampling approximation algorithm and prove that it is asymptotically optimal. Finally, for Doptimal design with repetitions, we study a different algorithm proposed by the literature and show that it can improve this asymptotic approximation ratio. All the sampling algorithms studied in this paper are shown to admit polynomialtime deterministic implementations.

Cloud computing has motivated renewed interest in resource allocation problems with new consumption models. A common goal is to share a resource, such as CPU or I/O bandwidth, among distinct users with different demand patterns as well as different quality of service requirements. To ensure these service requirements, cloud offerings often come with a service level agreement (SLA) between the provider and the users. A SLA specifies the amount of a resource a user is entitled to utilize. In many cloud settings, providers would like to operate resources at high utilization while simultaneously respecting individual SLAs. There is typically a tradeoff between these two objectives; for example, utilization can be increased by shifting away resources from idle users to “scavenger” workload, but with the risk of the former then becoming active again. We study this fundamental tradeoff by formulating a resource allocation model that captures basic properties of cloud computing systems, including SLAs, highly limited feedback about the state of the system, and variable and unpredictable input sequences. Our main result is a simple and practical algorithm that achieves nearoptimal performance on the above two objectives. First, we guarantee nearly optimal utilization of the resource even if compared with themore »