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Inductive TimeSpace Lower Bounds for SAT and Related Problems
 Computational Complexity
, 2005
"... Abstract. We improve upon indirect diagonalization arguments for lower bounds on explicit problems within the polynomial hierarchy. Our contributions are summarized as follows. 1. We present a technique that uniformly improves upon most known nonlinear time lower bounds for nondeterminism andalterna ..."
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Cited by 13 (4 self)
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Abstract. We improve upon indirect diagonalization arguments for lower bounds on explicit problems within the polynomial hierarchy. Our contributions are summarized as follows. 1. We present a technique that uniformly improves upon most known nonlinear time lower bounds for nondeterminism andalternating computation, on both subpolynomial (n o(1) ) space RAMs and sequential onetape machines with random access to the input. We obtain improved lower bounds for Boolean satisfiability (SAT), as well as all NPcomplete problems that have efficient reductions from SAT, and ΣkSAT, for constant k ≥ 2. For example, SAT cannot be solved by random access machines using n √ 3 time and subpolynomial space. 2. We show how indirect diagonalization leads to timespace lower bounds for computation with bounded nondeterminism. For both the random access and multitape Turing machine models, we prove that for all k ≥ 1, there is a constant ck> 1 such that linear time with n 1/k nondeterministic bits is not contained in deterministic n ck time with subpolynomial space. This is used to prove that satisfiability of Boolean circuits with n inputs and n k size cannot be solved by deterministic multitape Turing machines running in n k·ck time and subpolynomial space.
Algorithms and Resource Requirements for Fundamental Problems
, 2007
"... no. DGE0234630. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any other entity. ..."
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Cited by 10 (6 self)
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no. DGE0234630. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any other entity.
AlternationTrading Proofs, Linear Programming, and Lower Bounds
"... A fertile area of recent research has demonstrated concrete polynomial time lower bounds for solving natural hard problems on restricted computational models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path, MOD6SAT, MajorityofMajoritySAT, and Tautologies, to name a few. The ..."
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Cited by 3 (2 self)
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A fertile area of recent research has demonstrated concrete polynomial time lower bounds for solving natural hard problems on restricted computational models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path, MOD6SAT, MajorityofMajoritySAT, and Tautologies, to name a few. The proofs of these lower bounds follow a certain proofbycontradiction strategy, which we call “resourcetrading” or “alternationtrading.” An important open problem is to determine how powerful such proofs can possibly be. We propose a methodology for studying these proofs that makes them amenable to both formal analysis and automated theorem proving. Formalizing the framework, we prove that the search for better lower bounds can often be turned into a problem of solving a large series of linear programming instances. We implement a smallscale theorem prover and report surprising results, which allow us to extract new humanreadable time lower bounds for several problems. We also use the framework to prove concrete limitations on the current techniques.
Relationships among Time and Space Complexity Classes
, 2001
"... Multitape Turing machines are the canonical mathematical model for studying the time and space requirements of problems. Most computational problems that are ecient in practice have ecient algorithms on multitape Turing machines, and vice versa. ..."
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Cited by 1 (0 self)
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Multitape Turing machines are the canonical mathematical model for studying the time and space requirements of problems. Most computational problems that are ecient in practice have ecient algorithms on multitape Turing machines, and vice versa.
Better TimeSpace Lower Bounds for SAT and Related Problems
"... We make several improvements on time lower bounds for concrete problems in NP and PH. 1. We present an elementary technique based on “indirect diagonalization ” that uniformly improves upon the known nonlinear time lower bounds for nondeterminism and alternating computation, on both sublinear (n o(1 ..."
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We make several improvements on time lower bounds for concrete problems in NP and PH. 1. We present an elementary technique based on “indirect diagonalization ” that uniformly improves upon the known nonlinear time lower bounds for nondeterminism and alternating computation, on both sublinear (n o(1) ) space RAMs and sequential worktape machines with random access to the input. We obtain better lower bounds for SAT, as well as all NPcomplete problems that have efficient reductions from SAT, and ΣkSAT, for constant k ≥ 2. For example, SAT cannot be solved by random access machines using n √ 3 time and n o(1) space. The technique is a natural inductive approach, for which previous work is essentially its base case. 2. We show how indirect diagonalization can also yield timespace lower bounds for computation with bounded nondeterminism. One corollary is that for all k, there exists a constant ck> 1 such that satisfiability of Boolean circuits with n inputs and n k gates cannot be solved in deterministic time n k·ck and n o(1) space. 1
Relationships among Time and Space Complexity Classes
, 2001
"... Abstract Multitape Turing machines are the canonical mathematical model for studying the time and space requirements of problems. Most computational problems that are efficient in practice have efficient algorithms on multitape Turing machines, and vice versa. We survey the literature on relationshi ..."
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Abstract Multitape Turing machines are the canonical mathematical model for studying the time and space requirements of problems. Most computational problems that are efficient in practice have efficient algorithms on multitape Turing machines, and vice versa. We survey the literature on relationships among time and space classes defined using multitape Turing machines. We discuss the result of Paul, Hopcroft and Valiant that deterministic space T is more powerful than deterministic time T and the result of Paul, Pippenger, Szemeredi and Trotter that nondeterministic linear time is more powerful than deterministic linear time. We also discuss techniques to simulate Turing machines by Turing machines with different sets of resources, and techniques to diagonalize against complexity classes, with specific reference to the recent research by Fortnow et al. on timespace tradeoffs for the Satisfiability problem. Finally, we mention a few results on models other than Turing machines and resources other than time and space, and list the major open problems in this area. Contents 1 Introduction 1 2 The Valiant Paradigm 5