A Second Step Towards Complexity-Theoretic Analogs of Rice's Theorem

Rice's Theorem states that every nontrivial language property of the recursively enumerable sets is undecidable. Borchert and Stephan initiated the search for complexity-theoretic analogs of Rice's Theorem. In particular, they proved that every nontrivial counting property of circuits is UP-hard, and that a number of closely related problems are SPP-hard. The present paper studies whether their UP-hardness result itself can be improved to SPP-hardness. We show that their UP-hardness result cannot be strengthened to SPP-hardness unless unlikely complexity class containments hold. Nonetheless, we prove that every P-constructibly bi-infinite counting property of circuits is SPP-hard. We also raise their general lower bound from unambiguous nondeterminism to constant-ambiguity nondeterminism.Comment: 14 pages. To appear in Theoretical Computer Scienc

Exponential Time Complexity of the Permanent and the Tutte Polynomial

We show conditional lower bounds for well-studied #P-hard problems: (a) The number of satisfying assignments of a 2-CNF formula with n variables cannot be counted in time exp(o(n)), and the same is true for computing the number of all independent sets in an n-vertex graph. (b) The permanent of an n x n matrix with entries 0 and 1 cannot be computed in time exp(o(n)). (c) The Tutte polynomial of an n-vertex multigraph cannot be computed in time exp(o(n)) at most evaluation points (x,y) in the case of multigraphs, and it cannot be computed in time exp(o(n/polylog n)) in the case of simple graphs. Our lower bounds are relative to (variants of) the Exponential Time Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF formulas cannot be decided in time exp(o(n)). We relax this hypothesis by introducing its counting version #ETH, namely that the satisfying assignments cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds, we transfer the sparsification lemma for d-CNF formulas to the counting setting

Upper and lower bounds for dynamic data structures on strings

We consider a range of simply stated dynamic data structure problems on strings. An update changes one symbol in the input and a query asks us to compute some function of the pattern of length $m$ and a substring of a longer text. We give both conditional and unconditional lower bounds for variants of exact matching with wildcards, inner product, and Hamming distance computation via a sequence of reductions. As an example, we show that there does not exist an $O(m^{1/2-\varepsilon})$ time algorithm for a large range of these problems unless the online Boolean matrix-vector multiplication conjecture is false. We also provide nearly matching upper bounds for most of the problems we consider.Comment: Accepted at STACS'1

On Generalizations of Network Design Problems with Degree Bounds

Iterative rounding and relaxation have arguably become the method of choice in dealing with unconstrained and constrained network design problems. In this paper we extend the scope of the iterative relaxation method in two directions: (1) by handling more complex degree constraints in the minimum spanning tree problem (namely, laminar crossing spanning tree), and (2) by incorporating `degree bounds' in other combinatorial optimization problems such as matroid intersection and lattice polyhedra. We give new or improved approximation algorithms, hardness results, and integrality gaps for these problems.Comment: v2, 24 pages, 4 figure

The Geometry of Differential Privacy: the Sparse and Approximate Cases

In this work, we study trade-offs between accuracy and privacy in the context of linear queries over histograms. This is a rich class of queries that includes contingency tables and range queries, and has been a focus of a long line of work. For a set of $d$ linear queries over a database $x \in \R^N$, we seek to find the differentially private mechanism that has the minimum mean squared error. For pure differential privacy, an $O(\log^2 d)$ approximation to the optimal mechanism is known. Our first contribution is to give an $O(\log^2 d)$ approximation guarantee for the case of (\eps,\delta)-differential privacy. Our mechanism is simple, efficient and adds correlated Gaussian noise to the answers. We prove its approximation guarantee relative to the hereditary discrepancy lower bound of Muthukrishnan and Nikolov, using tools from convex geometry. We next consider this question in the case when the number of queries exceeds the number of individuals in the database, i.e. when $d > n \triangleq \|x\|_1$. It is known that better mechanisms exist in this setting. Our second main contribution is to give an (\eps,\delta)-differentially private mechanism which is optimal up to a \polylog(d,N) factor for any given query set $A$ and any given upper bound $n$ on $\|x\|_1$. This approximation is achieved by coupling the Gaussian noise addition approach with a linear regression step. We give an analogous result for the \eps-differential privacy setting. We also improve on the mean squared error upper bound for answering counting queries on a database of size $n$ by Blum, Ligett, and Roth, and match the lower bound implied by the work of Dinur and Nissim up to logarithmic factors. The connection between hereditary discrepancy and the privacy mechanism enables us to derive the first polylogarithmic approximation to the hereditary discrepancy of a matrix $A$

Model Checking Lower Bounds for Simple Graphs

A well-known result by Frick and Grohe shows that deciding FO logic on trees involves a parameter dependence that is a tower of exponentials. Though this lower bound is tight for Courcelle's theorem, it has been evaded by a series of recent meta-theorems for other graph classes. Here we provide some additional non-elementary lower bound results, which are in some senses stronger. Our goal is to explain common traits in these recent meta-theorems and identify barriers to further progress. More specifically, first, we show that on the class of threshold graphs, and therefore also on any union and complement-closed class, there is no model-checking algorithm with elementary parameter dependence even for FO logic. Second, we show that there is no model-checking algorithm with elementary parameter dependence for MSO logic even restricted to paths (or equivalently to unary strings), unless E=NE. As a corollary, we resolve an open problem on the complexity of MSO model-checking on graphs of bounded max-leaf number. Finally, we look at MSO on the class of colored trees of depth d. We show that, assuming the ETH, for every fixed d>=1 at least d+1 levels of exponentiation are necessary for this problem, thus showing that the (d+1)-fold exponential algorithm recently given by Gajarsk\`{y} and Hlin\u{e}n\`{y} is essentially optimal