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Lower Bounds on the Complexity of MSO1 ModelChecking
"... One of the most important algorithmic metatheorems is a famous result by Courcelle which states that any graph problem definable in monadic secondorder logic with edgeset quantifications (MSO2) is decidable in linear time on any class of graphs of bounded treewidth. In the parlance of parameteri ..."
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One of the most important algorithmic metatheorems is a famous result by Courcelle which states that any graph problem definable in monadic secondorder logic with edgeset quantifications (MSO2) is decidable in linear time on any class of graphs of bounded treewidth. In the parlance of parameterized complexity, this means that MSO2 is FPTtractable wrt. the treewidth as parameter. Recently, Kreutzer and Tazari [13] proved a corresponding complexity lowerbound— that MSO2 modelchecking is not even in XP wrt. the formula size as parameter for graph classes that are subgraphclosed and whose treewidth is polylogarithmically unbounded. Of course, this is not an unconditional result but holds modulo a certain complexitytheoretic assumption, namely, the Exponential Time Hypothesis (ETH). In this paper we present a closely related result. We show that even MSO1 modelchecking with a fixed set of vertex labels, but without edgeset quantifications, is not in XP wrt. the formula size as parameter for graph classes which are subgraphclosed and whose treewidth is polylogarithmically unbounded unless the nonuniform ETH fails. In comparison to Kreutzer and Tazari, (1) we use a stronger prerequisite, namely nonuniform instead of uniform ETH, to avoid
On TC0 Lower Bounds for the Permanent
 In Proc. of COCOON, 420–432
, 2012
"... Abstract In this paper we consider the problem of proving lower bounds for the permanent. An ongoing line of research has shown superpolynomial lower bounds for slightlynonuniform smalldepth threshold and arithmetic circuits [All99, KP09, JS11, JS12]. We prove a new parameterized lower bound th ..."
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Abstract In this paper we consider the problem of proving lower bounds for the permanent. An ongoing line of research has shown superpolynomial lower bounds for slightlynonuniform smalldepth threshold and arithmetic circuits [All99, KP09, JS11, JS12]. We prove a new parameterized lower bound that includes each of the previous results as subcases. Our main result implies that the permanent does not have Boolean threshold circuits of the following kinds.
Tighter Connections between Derandomization and Circuit Lower Bounds *BY 18th Int'l Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX'15) / 19th Int'l Workshop on Randomization and Computation (RANDOM'15)
"... Abstract We tighten the connections between circuit lower bounds and derandomization for each of the following three types of derandomization: general derandomization of promiseBPP (connected to Boolean circuits), derandomization of Polynomial Identity Testing (PIT) over fixed finite fields (conne ..."
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Abstract We tighten the connections between circuit lower bounds and derandomization for each of the following three types of derandomization: general derandomization of promiseBPP (connected to Boolean circuits), derandomization of Polynomial Identity Testing (PIT) over fixed finite fields (connected to arithmetic circuit lower bounds over the same field), and derandomization of PIT over the integers (connected to arithmetic circuit lower bounds over the integers). We show how to make these connections uniform equivalences, although at the expense of using somewhat less common versions of complexity classes and for a less studied notion of inclusion. Our main results are as follows: 1. We give the first proof that a nontrivial (nondeterministic subexponentialtime) algorithm for PIT over a fixed finite field yields arithmetic circuit lower bounds. 2. We get a similar result for the case of PIT over the integers, strengthening a result of Jansen and Santhanam ACM Subject Classification F.2.3 Tradeoffs between Complexity Measures
Lecture 4
"... Circuits are directed, acyclic graphs where nodes are called gates and edges are called wires. Input gates are gates with indegree zero, and we will take the output gate of a circuit to be a gate with outdegree zero. (For circuits having multiple outputs this is not necessarily the case; however, ..."
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Circuits are directed, acyclic graphs where nodes are called gates and edges are called wires. Input gates are gates with indegree zero, and we will take the output gate of a circuit to be a gate with outdegree zero. (For circuits having multiple outputs this is not necessarily the case; however, we will only be concerned with circuits having a singlebit output.) Input gates are labeled with bits of the input (in a onetoone fashion); each noninput gate is labeled with a value from a given (finite) basis, where a basis may contain functions and/or families of functions. One common basis is B0 = {¬, ∨, ∧}, the standard bounded fanin basis. Another example is B1 = {¬, ( ∨ n)n ∈ , ( ∧ n)n ∈ }, the standard unbounded fanin basis. An important point is that gates may have unbounded fanout (even over B0), unless explicitly stated otherwise. A circuit C with n input gates defines a function fC: {0, 1} n → {0, 1} in the natural way: for a given input x = x1 · · · xn we inductively define the value at any gate, and the output of the circuit on that input is simply the value at the output gate. Two important complexity measures for circuits (which somewhat parallel time and space for Turing machines) are size and depth, where the size of a circuit is the number of noninput gates it has and the depth of a circuit is the length of the longest path (from an input gate to the output gate) in the underlying directed graph
Notes on Complexity Theory Last updated: September, 2011 Lecture 4
"... In this lecture and the next one, we discuss two types of results that are related by the technique used in their proofs. Both kinds of results are also fundamental in their own right. The common proof technique is called diagonalization. It is somewhat difficult to formally define the term, but rou ..."
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In this lecture and the next one, we discuss two types of results that are related by the technique used in their proofs. Both kinds of results are also fundamental in their own right. The common proof technique is called diagonalization. It is somewhat difficult to formally define the term, but roughly the idea is that we want to show the existence of some language L (with certain properties) that cannot be decided by any Turing machine within some set S = {M1,...}. 1 We do so by starting with some L0 (with the property we want) and then, for i = 1,... changing Li−1 to Li such that none of M1,...,Mi decide Li. Of course part of the difficulty is to make sure that Li has the property we want also. Actually, one can also prove the existence of an undecidable language using this technique (though not quite as explicitly as stated above). Consider an enumeration x1,... of all binary strings, and an enumeration M1,... of all Turing machines. 2 Define L as follows: xi 6 ∈ L iff Mi(xi) = 1. (A picture really helps here. For those who have seen it before, this is exactly analogous to the proof that there is no bijection from the integers to the reals. In fact, that