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64
A Separator Theorem for Planar Graphs
, 1977
"... Let G be any nvertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than 2& & vertices. We exhibit an algorithm which ..."
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Cited by 465 (1 self)
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Let G be any nvertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than 2& & vertices. We exhibit an algorithm which finds such a partition A, B, C in O(n) time.
SEARCHING AND PEBBLING
, 1986
"... We relate the search number of an undirected graph G with the minimum and maximum of the progressive pebble demands of the directed acyclic graphs obtained by orienting (7. Towards this end, we introduce nodesearching, a slight variant of searching, in which an edge is cleared by placing searchers ..."
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Cited by 99 (1 self)
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We relate the search number of an undirected graph G with the minimum and maximum of the progressive pebble demands of the directed acyclic graphs obtained by orienting (7. Towards this end, we introduce nodesearching, a slight variant of searching, in which an edge is cleared by placing searchers on both of its endpoints. We also show that the minimum number of searchers necessary to nodesearch a graph equals its vertex separator plus one. Key words. Searching a graph, pebble games, layout parameters of a graph, vertex separator of a graph.
Provably efficient scheduling for languages with finegrained parallelism
 IN PROC. SYMPOSIUM ON PARALLEL ALGORITHMS AND ARCHITECTURES
, 1995
"... Many highlevel parallel programming languages allow for finegrained parallelism. As in the popular worktime framework for parallel algorithm design, programs written in such languages can express the full parallelism in the program without specifying the mapping of program tasks to processors. A ..."
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Cited by 94 (27 self)
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Many highlevel parallel programming languages allow for finegrained parallelism. As in the popular worktime framework for parallel algorithm design, programs written in such languages can express the full parallelism in the program without specifying the mapping of program tasks to processors. A common concern in executing such programs is to schedule tasks to processors dynamically so as to minimize not only the execution time, but also the amount of space (memory) needed. Without careful scheduling, the parallel execution on p processors can use a factor of p or larger more space than a sequential implementation of the same program. This paper first identifies a class of parallel schedules that are provably efficient in both time and space. For any
A subexponentialtime quantum algorithm for the dihedral hidden subgroup problem
, 2003
"... Abstract. We present a quantum algorithm for the dihedral hidden subgroup problem (DHSP) with time and query complexity 2O(√log N). In this problem an oracle computes a function f on the dihedral group DN which is invariant under a hidden reflection in DN. By contrast, the classical query complexity ..."
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Cited by 76 (0 self)
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Abstract. We present a quantum algorithm for the dihedral hidden subgroup problem (DHSP) with time and query complexity 2O(√log N). In this problem an oracle computes a function f on the dihedral group DN which is invariant under a hidden reflection in DN. By contrast, the classical query complexity of DHSP is O ( √ N). The algorithm also applies to the hidden shift problem for an arbitrary finitely generated abelian group. The algorithm begins as usual with a quantum character transform, which in the case of DN is essentially the abelian quantum Fourier transform. This yields the name of a group representation of DN, which is not by itself useful, and a state in the representation, which is a valuable but indecipherable qubit. The algorithm proceeds by repeatedly pairing two unfavorable qubits to make a new qubit in a more favorable representation of DN. Once the algorithm obtains certain target representations, direct measurements reveal the hidden subgroup.
Algebrization: A new barrier in complexity theory
 MIT Theory of Computing Colloquium
, 2007
"... Any proof of P � = NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linearsize circuits) that overcome both barriers simultaneously. So the question arises of whether there is a ..."
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Cited by 51 (3 self)
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Any proof of P � = NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linearsize circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in complexity theory. In this paper we present such a barrier, which we call algebraic relativization or algebrization. The idea is that, when we relativize some complexity class inclusion, we should give the simulating machine access not only to an oracle A, but also to a lowdegree extension of A over a finite field or ring. We systematically go through basic results and open problems in complexity theory to delineate the power of the new algebrization barrier. First, we show that all known nonrelativizing results based on arithmetization—both inclusions such as IP = PSPACE and MIP = NEXP, and separations such as MAEXP � ⊂ P/poly —do indeed algebrize. Second, we show that almost all of the major open problems—including P versus NP, P versus RP, and NEXP versus P/poly—will require nonalgebrizing techniques. In some cases algebrization seems to explain exactly why progress stopped where it did: for example, why we have superlinear circuit lower bounds for PromiseMA but not for NP. Our second set of results follows from lower bounds in a new model of algebraic query complexity, which we introduce in this paper and which is interesting in its own right. Some of our lower bounds use direct combinatorial and algebraic arguments, while others stem from a surprising connection between our model and communication complexity. Using this connection, we are also able to give an MAprotocol for the Inner Product function with O ( √ n log n) communication (essentially matching a lower bound of Klauck), as well as a communication complexity conjecture whose truth would imply NL � = NP. 1
Size Space tradeoffs for Resolution
, 2002
"... We investigate tradeoffs of various important complexity measures such as size, space and width. We show examples of CNF formulas that have optimal proofs with respect to any one of these parameters, but optimizing one parameter must cost an increase in the other. These results, the first of their ..."
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Cited by 26 (2 self)
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We investigate tradeoffs of various important complexity measures such as size, space and width. We show examples of CNF formulas that have optimal proofs with respect to any one of these parameters, but optimizing one parameter must cost an increase in the other. These results, the first of their kind, have implications on the efficiency (or rather, inefficiency) of some commonly used SAT solving heuristics. Our proof
On the Complexity of SAT
, 1999
"... We show that nondeterministic time NT IME(n) is not contained in deterministic time n # 2# and polylogarithmic space, for any # > 0. This implies that (infinitely often) satisfiability cannot be solved in time O(n # 2# ) and polylogarithmic space. A similar result is presented for uniform cir ..."
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Cited by 25 (1 self)
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We show that nondeterministic time NT IME(n) is not contained in deterministic time n # 2# and polylogarithmic space, for any # > 0. This implies that (infinitely often) satisfiability cannot be solved in time O(n # 2# ) and polylogarithmic space. A similar result is presented for uniform circuits.
Short proofs may be spacious: An optimal separation of space and length in resolution
, 2008
"... A number of works have looked at the relationship between length and space of resolution proofs. A notorious question has been whether the existence of a short proof implies the existence of a proof that can be verified using limited space. In this paper we resolve the question by answering it negat ..."
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Cited by 22 (10 self)
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A number of works have looked at the relationship between length and space of resolution proofs. A notorious question has been whether the existence of a short proof implies the existence of a proof that can be verified using limited space. In this paper we resolve the question by answering it negatively in the strongest possible way. We show that there are families of 6CNF formulas of size n, for arbitrarily large n, that have resolution proofs of length O(n) but for which any proof requires space Ω(n / log n). This is the strongest asymptotic separation possible since any proof of length O(n) can always be transformed into a proof in space O(n / log n). Our result follows by reducing the space complexity of so called pebbling formulas over a directed acyclic graph to the blackwhite pebbling price of the graph. The proof is somewhat simpler than previous results (in particular, those reported in [Nordström 2006, Nordström and H˚astad 2008]) as it uses a slightly different flavor of pebbling formulas which allows for a rather straightforward reduction of proof space to standard blackwhite pebbling price.
A Guide for New Referees in Theoretical Computer Science
, 1994
"... Your success as a scientist will in part be measured by the quality of your research publications in highquality journals and conference proceedings. Of the three classical rhetorical techniques, it is logos, rather than pathos or ethos, which is most commonly associated with scientific publication ..."
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Cited by 21 (1 self)
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Your success as a scientist will in part be measured by the quality of your research publications in highquality journals and conference proceedings. Of the three classical rhetorical techniques, it is logos, rather than pathos or ethos, which is most commonly associated with scientific publications. In the mathematical sciences the paradigm for publication is to describe the mathematical proofs of propositions in sufficient detail to allow duplication by interested readers. Quality control is achieved by a system of peer review commonly referred to as refereeing. This guide is an attempt to distill the experience of the theoretical computer science community on the subject of refereeing into a convenient form which can be easily distributed to students and other inexperienced referees. Although aimed primarily at theoretical computer scientists, it contains advice which maybe relevant to other mathematical sciences. It may also be of some use to new authors who are unfamiliar with the peer review process. However, it must be understood that this is not a guide on how to write papers. Authors who are interested in improving their writing skills can consult the "Further Reading" section. The main part of this guide is divided into nine sections. The first section describes the
Narrow proofs may be spacious: Separating space and width in resolution (Extended Abstract)
 REVISION 02, ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY (ECCC
, 2005
"... The width of a resolution proof is the maximal number of literals in any clause of the proof. The space of a proof is the maximal number of clauses kept in memory simultaneously if the proof is only allowed to infer new clauses from clauses currently in memory. Both of these measures have previously ..."
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Cited by 20 (7 self)
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The width of a resolution proof is the maximal number of literals in any clause of the proof. The space of a proof is the maximal number of clauses kept in memory simultaneously if the proof is only allowed to infer new clauses from clauses currently in memory. Both of these measures have previously been studied and related to the resolution refutation size of unsatisfiable CNF formulas. Also, the refutation space of a formula has been proven to be at least as large as the refutation width, but it has been open whether space can be separated from width or the two measures coincide asymptotically. We prove that there is a family of kCNF formulas for which the refutation width in resolution is constant but the refutation space is nonconstant, thus solving a problem mentioned in several previous papers.