More Like this

1. A Unified Approach for Resilience and Causal Responsibility with Integer Linear Programming (ILP) and LP Relaxations

   https://doi.org/10.1145/3626715  

   Makhija, Neha ; Gatterbauer, Wolfgang ( December 2023 , Proceedings of the ACM on Management of Data)

   What is a minimal set of tuples to delete from a database in order to eliminate all query answers? This problem is called the resilience of a query and is one of the key algorithmic problems underlying various forms of reverse data management, such as view maintenance, deletion propagation and causal responsibility. A long-open question is determining the conjunctive queries (CQs) for which resilience can be solved in PTIME.

   We shed new light on this problem by proposing a unified Integer Linear Programming (ILP) formulation. It is unified in that it can solve both previously studied restrictions (e.g., self-join-free CQs under set semantics that allow a PTIME solution) and new cases (all CQs under set or bag semantics). It is also unified in that all queries and all database instances are treated with the same approach, yet the algorithm is guaranteed to terminate in PTIME for all known PTIME cases. In particular, we prove that for all known easy cases, the optimal solution to our ILP is identical to a simpler Linear Programming (LP) relaxation, which implies that standard ILP solvers return the optimal solution to the original ILP in PTIME.

   Our approach allows us to explore new variants and obtain new complexity results. 1) It works under bag semantics, for which we give the first dichotomy results in the problem space. 2) We extend our approach to the related problem of causal responsibility and give a more fine-grained analysis of its complexity. 3) We recover easy instances for generally hard queries, including instances with read-once provenance and instances that become easy because of Functional Dependencies in the data. 4) We solve an open conjecture about a unified hardness criterion from PODS 2020 and prove the hardness of several queries of previously unknown complexity. 5) Experiments confirm that our findings accurately predict the asymptotic running times, and that our universal ILP is at times even quicker than a previously proposed dedicated flow algorithm.

    

   more » « less
2. From Shapley Value to Model Counting and Back

   https://doi.org/10.1145/3651142  

   Kara, Ahmet ; Olteanu, Dan ; Suciu, Dan ( May 2024 , Proceedings of the ACM on Management of Data)

   In this paper we investigate the problem of quantifying the contribution of each variable to the satisfying assignments of a Boolean function based on the Shapley value. Our main result is a polynomial-time equivalence between computing Shapley values and model counting for any class of Boolean functions that are closed under substitutions of variables with disjunctions of fresh variables. This result settles an open problem raised in prior work, which sought to connect the Shapley value computation to probabilistic query evaluation.

   We show two applications of our result. First, the Shapley values can be computed in polynomial time over deterministic and decomposable circuits, since they are closed under OR-substitutions. Second, there is a polynomial-time equivalence between computing the Shapley value for the tuples contributing to the answer of a Boolean conjunctive query and counting the models in the lineage of the query. This equivalence allows us to immediately recover the dichotomy for Shapley value computation in case of self-join-free Boolean conjunctive queries; in particular, the hardness for non-hierarchical queries can now be shown using a simple reduction from the \#P-hard problem of model counting for lineage in positive bipartite disjunctive normal form.

    

   more » « less
3. Lifting with simple gadgets and applications to circuit and proof complexity

   Meir, O ; Nördstrom, J ; Pitassi, T ; Robere, R ; de Rezende, Susanna F ; Vinyals, M. ( February 2020 , Electronic colloquium on computational complexity)

   We significantly strengthen and generalize the theorem lifting Nullstellensatz degree to monotone span program size by Pitassi and Robere (2018) so that it works for any gadget with high enough rank, in particular, for useful gadgets such as equality and greater-than. We apply our generalized theorem to solve two open problems: • We present the first result that demonstrates a separation in proof power for cutting planes with unbounded versus polynomially bounded coefficients. Specifically, we exhibit CNF formulas that can be refuted in quadratic length and constant line space in cutting planes with unbounded coefficients, but for which there are no refutations in subexponential length and subpolynomial line space if coefficients are restricted to be of polynomial magnitude. • We give the first explicit separation between monotone Boolean formulas and monotone real formulas. Specifically, we give an explicit family of functions that can be computed with monotone real formulas of nearly linear size but require monotone Boolean formulas of exponential size. Previously only a non-explicit separation was known. An important technical ingredient, which may be of independent interest, is that we show that the Nullstellensatz degree of refuting the pebbling formula over a DAG G over any field coincides exactly with the reversible pebbling price of G. In particular, this implies that the standard decision tree complexity and the parity decision tree complexity of the corresponding falsified clause search problem are equal. 

   more » « less
4. Constraint Optimization over Semirings

   https://doi.org/10.1609/aaai.v37i4.25522  

   Pavan, A. ; Meel, Kuldeep S. ; Vinodchandran, N. V. ; Bhattacharyya, Arnab ( June 2023 , Proceedings of the AAAI Conference on Artificial Intelligence)

   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. 

   more » « less
5. Constraint Optimization over Semirings

   A. Pavan ; Kuldeep S Meel ; N. V. Vinodchandranl Arnab Bhattacharyya ( February 2023 , Proceedings of the AAAI Conference on Artificial Intelligence)

   Williams Brian ; Chen Yiling ; Neville Jennifer (Ed.)

   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 $\varphi$ over $n$ variable and a semiring $(K,+,\cdot,0,1)$, find the maximum value over all possible interpretations of $\varphi$ 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 \optrustval\ and \optrust. We first show that for general propositional formulas in negation normal form, \optrustval\ and {\optrust} are in ${\mathrm{FP}}^{\mathrm{NP}}$. We then investigate {\optrust} when the input formula $\varphi$ is represented in the conjunctive normal form. For CNF formulae, we first derive an upper bound on the value of {\optrust} 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 $\optrust$ value is at most $1/4^{m-r}$. Building on this we establish that {\optrust} for CNF formulae is hard for the complexity class ${\mathrm{FP}}^{\mathrm{NP}[\log]}$. We also design polynomial-time approximation algorithms and establish an inapproximability for {\optrustval}. We establish similar complexity results for these optimization problems over other semirings including tropical, fuzzy, and access control semirings. 

   more » « less