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On the Relative Complexity of Resolution Refinements and Cutting Planes Proof Systems
, 2000
"... An exponential lower bound for the size of treelike Cutting Planes refutations of a certain family of CNF formulas with polynomial size resolution refutations is proved. This implies an exponential separation between the treelike versions and the daglike versions of resolution and Cutting Planes. ..."
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Cited by 38 (6 self)
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An exponential lower bound for the size of treelike Cutting Planes refutations of a certain family of CNF formulas with polynomial size resolution refutations is proved. This implies an exponential separation between the treelike versions and the daglike versions of resolution and Cutting Planes. In both cases only superpolynomial separations were known [29, 18, 8]. In order to prove these separations, the lower bounds on the depth of monotone circuits of Raz and McKenzie in [25] are extended to monotone real circuits. An exponential separation is also proved between treelike resolution and several refinements of resolution: negative resolution and regular resolution. Actually this last separation also provides a separation between treelike resolution and ordered resolution, thus the corresponding superpolynomial separation of [29] is extended. Finally, an exponential separation between ordered resolution and unrestricted resolution (also negative resolution) is proved. Only a superpolynomial separation between ordered and unrestricted resolution was previously known [13].
Exponential Separations between Restricted Resolution and Cutting Planes Proof Systems
, 1998
"... We prove an exponential lower bound for treelike Cutting Planes refutations of a set of clauses which has polynomial size resolution refutations. This implies an exponential separation between treelike and daglike proofs for both CuttingPlanes and resolution; in both cases only superpolynomial se ..."
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Cited by 24 (5 self)
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We prove an exponential lower bound for treelike Cutting Planes refutations of a set of clauses which has polynomial size resolution refutations. This implies an exponential separation between treelike and daglike proofs for both CuttingPlanes and resolution; in both cases only superpolynomial separations were known before [30, 20, 10]. In order to prove this, we extend the lower bounds on the depth of monotone circuits of Raz and McKenzie [26] to monotone real circuits. In the case of resolution, we further improve this result by giving an exponential separation of treelike resolution from (daglike) regular resolution proofs. In fact, the refutation provided to give the upper bound respects the stronger restriction of being a DavisPutnam resolution proof. This extends the corresponding superpolynomial separation of [30]. Finally, we prove an exponential separation between DavisPutnam resolution and unrestricted resolution proofs; only a superpolynomial separation was previously...
JAMRESISTANT COMMUNICATION WITHOUT SHARED SECRETS THROUGH THE USE OF CONCURRENT CODES
, 2007
"... We consider the problem of establishing jamresistant, wireless, omnidirectional communication channels when there is no initial shared secret. No existing system achieves this. We propose a general algorithm for this problem, the BBC algorithm, and give several instantiations of it. We develop an ..."
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Cited by 19 (9 self)
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We consider the problem of establishing jamresistant, wireless, omnidirectional communication channels when there is no initial shared secret. No existing system achieves this. We propose a general algorithm for this problem, the BBC algorithm, and give several instantiations of it. We develop and analyze this algorithm within the framework of a new type of code, concurrent codes, which are those superimposed codes that allow efficient decoding. Finally, we propose the Universal Concurrent Code algorithm, and prove that it covers all possible concurrent codes, and give connections between its theory and that of monotone Boolean functions.
Jam resistant communications without shared secrets
 in Proceedings of the 3 rd International Conference on Information Warfare and Security
, 2008
"... Distribution A, Approved for public release, distribution unlimited Abstract. We consider the problem of establishing jamresistant, wireless, omnidirectional communication channels when there is no initial shared secret. No existing system achieves this. We propose a general algorithm for this prob ..."
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Cited by 11 (1 self)
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Distribution A, Approved for public release, distribution unlimited Abstract. We consider the problem of establishing jamresistant, wireless, omnidirectional communication channels when there is no initial shared secret. No existing system achieves this. We propose a general algorithm for this problem, the BBC algorithm, and give several instantiations of it. We develop and analyze this algorithm within the framework of a new type of code, concurrent codes, which are those superimposed codes that allow efficient decoding. Finally, we propose the Universal Concurrent Code algorithm, and prove that it covers all possible concurrent codes, and give connections between its theory and that of monotone Boolean functions.
Lower Bounds for Tropical Circuits and Dynamic Programs
"... Tropical circuits are circuits with Min and Plus, or Max and Plus operations as gates. Their importance stems from their intimate relation to dynamic programming algorithms. The power of tropical circuits lies somewhere between that of monotone boolean circuits and monotone arithmetic circuits. In ..."
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Cited by 3 (1 self)
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Tropical circuits are circuits with Min and Plus, or Max and Plus operations as gates. Their importance stems from their intimate relation to dynamic programming algorithms. The power of tropical circuits lies somewhere between that of monotone boolean circuits and monotone arithmetic circuits. In this paper we present some lower bounds arguments for tropical circuits, and hence, for dynamic programs.
Lower Bounds for Monotone Counting Circuits
"... A {+,×}circuit counts a given multivariate polynomial f, if its values on 01 inputs are the same as those of f; on other inputs the circuit may output arbitrary values. Such a circuit counts the number of monomials of f evaluated to 1 by a given 01 input vector (with multiplicities given by their ..."
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A {+,×}circuit counts a given multivariate polynomial f, if its values on 01 inputs are the same as those of f; on other inputs the circuit may output arbitrary values. Such a circuit counts the number of monomials of f evaluated to 1 by a given 01 input vector (with multiplicities given by their coefficients). A circuit decides f if it has the same 01 roots as f. We first show that some multilinear polynomials can be exponentially easier to count than to compute them, and can be exponentially easier to decide than to count them. Then we give general lower bounds on the size of counting circuits.
Cutting Planes Cannot Approximate Some Integer ProgramsI
"... For every positive integer l, we consider a zeroone linear program describing the following optimization problem: maximize the number of nodes in a clique of an nvertex graph whose chromatic number does not exceed l. Although l is a trivial solution for this problem, we show that any cuttingplane ..."
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For every positive integer l, we consider a zeroone linear program describing the following optimization problem: maximize the number of nodes in a clique of an nvertex graph whose chromatic number does not exceed l. Although l is a trivial solution for this problem, we show that any cuttingplane proof certifying that no such graph can have a clique on more than r l vertices must generate an exponential in min{l, n/r l}1/4 number of inequalities. We allow Gomory–Chvátal cuts and even the more powerful split cuts. This extends the results of Pudlák [J. Symb. Log. 62:3 (1997) 981–998] and Dash [Math. of Operations Research 30:3 (2005) 678–700; Oper. Res. Lett. 38:2 (2010), 109–114] who proved exponential lower bounds for the case when l = n2/3 and r = 1. Key words: Cutting planes, monotone circuits, interpolation, clique problem 1.