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Constraint satisfaction problems (CSPs) and data stream models are two powerful abstractions to capture a wide variety of problems arising in different domains of computer science. Developments in the two communities have mostly occurred independently and with little interaction between them. In this work, we seek to investigate whether bridging the seeming communication gap between the two communities may pave the way to richer fundamental insights. To this end, we focus on two foundational problems: model counting for CSP’s and computation of zeroth frequency moments (F₀) for data streams.

Our investigations lead us to observe a striking similarity in the core techniques employed in the algorithmic frameworks that have evolved separately for model counting andF₀computation. We design a recipe for translating algorithms developed forF₀estimation to model counting, resulting in new algorithms for model counting. We also provide a recipe for transforming sampling algorithm over streams to constraint sampling algorithms. We then observe that algorithms in the context of distributed streaming can be transformed into distributed algorithms for model counting. We next turn our attention to viewing streaming from the lens of counting and show that framingF₀estimation as a special case of #DNF counting allows us to obtain a general recipe for a rich class of streaming problems, which had been subjected to case-specific analysis in prior works. In particular, our view yields an algorithm for multidimensional range efficientF₀estimation with a simpler analysis.

 

Constraint satisfaction problems (CSPs) and data stream models are two powerful abstractions to capture a wide variety of problems arising in different domains of computer science. Developments in the two communities have mostly occurred independently and with little interaction between them. In this work, we seek to investigate whether bridging the seeming communication gap between the two communities may pave the way to richer fundamental insights. To this end, we focus on two foundational problems: model counting for CSPs and the computation of the number of distinct elements in a data stream, also known as the zeroth frequency moment (F₀) of a data stream.

Our investigations lead us to observe striking similarity in the core techniques employed in the algorithmic frameworks that have evolved separately for model counting and distinct elements computation. We design a recipe for the translation of algorithms developed for distinct elements estimation to that of model counting, resulting in new algorithms for model counting. We then observe that algorithms in the context of distributed streaming can be transformed into distributed algorithms for model counting. We next turn our attention to viewing streaming from the lens of counting and show that framing distinct elements estimation as a special case of #DNF counting allows us to obtain a general recipe for a rich class of streaming problems, which had been subjected to case-specific analysis in prior works.

 

We consider the problem of maximizing submodular functions under submodular constraints by formulating the problem in two ways: \SCSKC and \DiffC. Given two submodular functions $f$ and $g$ where $f$ is monotone, the objective of \SCSKC problem is to find a set $S$ of size at most $k$ that maximizes $f(S)$ under the constraint that $g(S)\leq \theta$, for a given value of $\theta$. The problem of \DiffC focuses on finding a set $S$ of size at most $k$ such that $h(S) = f(S)-g(S)$ is maximized. It is known that these problems are highly inapproximable and do not admit any constant factor multiplicative approximation algorithms unless NP is easy. Known approximation algorithms involve data-dependent approximation factors that are not efficiently computable. We initiate a study of the design of approximation algorithms where the approximation factors are efficiently computable. For the problem of \SCSKC, we prove that the greedy algorithm produces a solution whose value is at least $(1-1/e)f(\OPT) - A$, where $A$ is the data-dependent additive error. For the \DiffC problem, we design an algorithm that uses the \SCSKC greedy algorithm as a subroutine. This algorithm produces a solution whose value is at least $(1-1/e)h(\OPT)-B$, where $B$ is also a data-dependent additive error. A salient feature of our approach is that the additive error terms can be computed efficiently, thus enabling us to ascertain the quality of the solutions produced. 

Total variation distance (TV distance) is a fundamental notion of distance between probability distributions. In this work, we introduce and study the problem of computing the TV distance of two product distributions over the domain {0,1}^n. In particular, we establish the following results.1. The problem of exactly computing the TV distance of two product distributions is #P-complete. This is in stark contrast with other distance measures such as KL, Chi-square, and Hellinger which tensorize over the marginals leading to efficient algorithms.2. There is a fully polynomial-time deterministic approximation scheme (FPTAS) for computing the TV distance of two product distributions P and Q where Q is the uniform distribution. This result is extended to the case where Q has a constant number of distinct marginals. In contrast, we show that when P and Q are Bayes net distributions the relative approximation of their TV distance is NP-hard.

 

We investigate the problems of maximizing k-submodular functions over total size constraints and over individual size constraints. k-submodularity is a generalization of submodularity beyond just picking items of a ground set, instead associating one of k types to chosen items. For sensor selection problems, for instance, this enables modeling of which type of sensor to put at a location, not simply whether to put a sensor or not. We propose and analyze threshold-greedy algorithms for both types of constraints. We prove that our proposed algorithms achieve the best known approximation ratios for both constraint types, up to a user-chosen parameter that balances computational complexity and the approximation ratio, while only using a number of function evaluations that depends linearly (up to poly-logarithmic terms) on the number of elements n, the number of types k, and the inverse of the user chosen parameter. Other algorithms that achieve the best-known deterministic approximation ratios require a number of function evaluations that depend linearly on the budget B, while our methods do not. We empirically demonstrate our algorithms' performance in applications of sensor placement with k types and influence maximization with k topics. 

Interpretations of logical formulas over semirings (other than the Boolean semiring) have applications in various areas of computer science including logic, AI, databases, and security. Such interpretations provide richer information beyond the truth or falsity of a statement. Examples of such semirings include Viterbi semiring, min-max or access control semiring, tropical semiring, and fuzzy semiring. The present work investigates the complexity of constraint optimization problems over semirings. The generic optimization problem we study is the following: Given a propositional formula phi over n variable and a semiring (K,+, . ,0,1), find the maximum value over all possible interpretations of phi over K. This can be seen as a generalization of the well-known satisfiability problem (a propositional formula is satisfiable if and only if the maximum value over all interpretations/assignments over the Boolean semiring is 1). A related problem is to find an interpretation that achieves the maximum value. In this work, we first focus on these optimization problems over the Viterbi semiring, which we call optConfVal and optConf. We first show that for general propositional formulas in negation normal form, optConfVal and optConf are in FP^NP. We then investigate optConf when the input formula phi is represented in the conjunctive normal form. For CNF formulae, we first derive an upper bound on the value of optConf as a function of the number of maximum satisfiable clauses. In particular, we show that if r is the maximum number of satisfiable clauses in a CNF formula with m clauses, then its optConf value is at most 1/4^(m-r). Building on this we establish that optConf for CNF formulae is hard for the complexity class FP^NP[log]. We also design polynomial-time approximation algorithms and establish an inapproximability for optConfVal. We establish similar complexity results for these optimization problems over other semirings including tropical, fuzzy, and access control semirings. 

A probabilistic algorithm A is pseudodeterministic if, on every input, there exists a canonical value that is output with high probability. If the algorithm outputs one of k canonical values with high probability, then it is called a k-pseudodeterministic algorithm. In the study of pseudodeterminism, the Acceptance Probability Estimation Problem (APEP), which is to additively approximate the acceptance probability of a Boolean circuit, is emerging as a central computational problem. This problem admits a 2-pseudodeterministic algorithm. Recently, it was shown that a pseudodeterministic algorithm for this problem would imply that any multi-valued function that admits a k-pseudodeterministic algorithm for a constant k (including approximation algorithms) also admits a pseudodeterministic algorithm (Dixon, Pavan, Vinodchandran; ITCS 2021). The contribution of the present work is two-fold. First, as our main conceptual contribution, we establish that the existence of a pseudodeterministic algorithm for APEP is fundamentally related to the gap between probabilistic promise classes and the corresponding standard complexity classes. In particular, we show the following equivalence: APEP has a pseudodeterministic approximation algorithm if and only if every promise problem in PromiseBPP has a solution in BPP. A conceptual interpretation of this equivalence is that the algorithmic gap between 2-pseudodeterminism and pseudodeterminism is equivalent to the gap between PromiseBPP and BPP. Based on this connection, we show that designing pseudodeterministic algorithms for APEP leads to the solution of some open problems in complexity theory, including new Boolean circuit lower bounds. This equivalence also explains how multi-pseudodeterminism is connected to problems in SearchBPP. In particular, we show that if APEP has a pseudodeterministic algorithm, then every problem that admits a k(n)-pseudodeterministic algorithm (for any polynomial k) is in SearchBPP and admits a pseudodeterministic algorithm. Motivated by this connection, we also explore its connection to probabilistic search problems and establish that APEP is complete for certain notions of search problems in the context of pseudodeterminism. Our second contribution is establishing query complexity lower bounds for multi-pseudodeterministic computations. We prove that for every k ≥ 1, there exists a problem whose (k+1)-pseudodeterministic query complexity, in the uniform query model, is O(1) but has a k-pseudodeterministic query complexity of Ω(n), even in the more general nonadaptive query model. A key contribution of this part of the work is the utilization of Sperner’s lemma in establishing query complexity lower bounds. 

Constraint satisfaction problems (CSPs) and data stream models are two powerful abstractions to capture a wide variety of problems arising in different domains of computer science. Developments in the two communities have mostly occurred independently and with little interaction between them. In this work, we seek to investigate whether bridging the seeming communication gap between the two communities may pave the way to richer fundamental insights. To this end, we focus on two foundational problems: model counting for CSPs and computation of zeroth frequency moments (F0) for data streams. 

Constraint satisfaction problems (CSP's) and data stream models are two powerful abstractions to capture a wide variety of problems arising in different domains of computer science. Developments in the two communities have mostly occurred independently and with little interaction between them. In this work, we seek to investigate whether bridging the seeming communication gap between the two communities may pave the way to richer fundamental insights. To this end, we focus on two foundational problems: model counting for CSP's and computation of zeroth frequency moments $(F_0)$ for data streams. Our investigations lead us to observe striking similarity in the core techniques employed in the algorithmic frameworks that have evolved separately for model counting and $F_0$ computation. We design a recipe for translation of algorithms developed for $F_0$ estimation to that of model counting, resulting in new algorithms for model counting. We then observe that algorithms in the context of distributed streaming can be transformed to distributed algorithms for model counting. We next turn our attention to viewing streaming from the lens of counting and show that framing $F_0$ estimation as a special case of DNF counting allows us to obtain a general recipe for a rich class of streaming problems, which had been subjected to case-specific analysis in prior works. In particular, our view yields a state-of-the art algorithm for multidimensional range efficient $F_0$ estimation with a simpler analysis.