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Where are the hard knapsack problems
 Computers and Operations Research
, 2005
"... The knapsack problem is believed to be one of the “easier ” N Phard problems. Not only can it be solved in pseudopolynomial time, but also decades of algorithmic improvements have made it possible to solve nearly all standard instances from the literature. The purpose of this paper is to give an o ..."
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Cited by 28 (2 self)
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The knapsack problem is believed to be one of the “easier ” N Phard problems. Not only can it be solved in pseudopolynomial time, but also decades of algorithmic improvements have made it possible to solve nearly all standard instances from the literature. The purpose of this paper is to give an overview of all recent exact solution approaches, and to show that the knapsack problem still is hard to solve for these algorithms for a variety of new test problems. These problems are constructed either by using standard benchmark instances with larger coefficients, or by introducing new classes of instances for which most upper bounds perform badly. The first group of problems challenge the dynamic programming algorithms while the other group of problems are focused towards branchandbound algorithms. Numerous computational experiments with all recent stateofart codes are used to show that the (KP) is still difficult to solve for a wide number of problems. One could say that the previous benchmark tests were limited to a few highly structured instances, which do not show the full characteristics of knapsack problems. 1
Saturation and Stability in the Theory of Computation over the Reals
, 1997
"... This paper was motivated by the following two questions which arise in the theory of complexity for computation over ordered rings in the now famous computational model introduced by Blum, Shub and Smale: (i) is the answer to the question P =? NP the same in every realclosed field ? (ii) if P 6= ..."
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Cited by 15 (10 self)
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This paper was motivated by the following two questions which arise in the theory of complexity for computation over ordered rings in the now famous computational model introduced by Blum, Shub and Smale: (i) is the answer to the question P =? NP the same in every realclosed field ? (ii) if P 6= NP for R, does there exist a problem of R which is NP but not NPcomplete ? Some unclassical complexity classes arise naturally in the study of these questions. They are still open, but we could obtain unconditional results of independent interest. Michaux introduced =const complexity classes in an effort to attack question (i). We show that AR =const = AR , answering a question of his. Here A is the class of real problems which are algorithmic in bounded time. We also prove the stronger result: PARR =const = PARR , where PAR stands for parallel polynomial time. In our terminology, we say that R is Asaturated and PARsaturated. We also prove, at the nonuniform level, the above results for...
Naurois. The complexity of semilinear problems in succinct representation
 Computational Complexity
"... Abstract. We prove completeness results for twentythree problems in semilinear geometry. These results involve semilinear sets given by additive circuits as input data. If arbitrary real constants are allowed in the circuit, the completeness results are for the BlumShubSmale additive model of com ..."
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Cited by 6 (3 self)
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Abstract. We prove completeness results for twentythree problems in semilinear geometry. These results involve semilinear sets given by additive circuits as input data. If arbitrary real constants are allowed in the circuit, the completeness results are for the BlumShubSmale additive model of computation. If, in contrast, the circuit is constantfree, then the completeness results are for the Turing model of computation. One such result, the P NP[log]completeness of deciding Zariski irreducibility, exhibits for the first time a problem with a geometric nature complete in this class.
Topological Complexity of the Range Searching
"... We prove an existence of a topological decision tree which solves the range searching problem for a system of real polynomials, in other words, the tree finds all feasible signs vectors of these polynomials, with the (topological) complexity logarithmic in the number of signs vectors. This answer ..."
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Cited by 5 (0 self)
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We prove an existence of a topological decision tree which solves the range searching problem for a system of real polynomials, in other words, the tree finds all feasible signs vectors of these polynomials, with the (topological) complexity logarithmic in the number of signs vectors. This answers the problem posed in [FK98]. 1 Range Searching Problem Let polynomials f 1 ; : : : ; fm 2 R[X 1 ; : : : ; X n ]. Our purpose is to solve the range searching problem [FK98] by means of topological decision trees (TDT) [S87]. Namely, TDT allows tests of the form "P (x) ? 0?" for arbitrary polynomials P 2 R[X 1 ; : : : ; X n ] (thus, we ignore the cost of the computations). We say that a TDT solves the range searching problem for the polynomials f 1 ; : : : ; fm if any two input points x; y 2 R n with different signs vectors (sgn(f 1 ); : : : ; sgn(f m ))(x) 6= (sgn(f 1 ); : : : ; sgn(f m ))(y) arrive to different leaves of the TDT. As usual, sgn could attain three values. By the topologi...
Circuits versus Trees in Algebraic Complexity
 In Proc. STACS 2000
, 2000
"... . This survey is devoted to some aspects of the \P = NP ?" problem over the real numbers and more general algebraic structures. We argue that given a structure M , it is important to nd out whether NPM problems can be solved by polynomial depth computation trees, and if so whether these trees ca ..."
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Cited by 5 (4 self)
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. This survey is devoted to some aspects of the \P = NP ?" problem over the real numbers and more general algebraic structures. We argue that given a structure M , it is important to nd out whether NPM problems can be solved by polynomial depth computation trees, and if so whether these trees can be eciently simulated by circuits. Point location, a problem of computational geometry, comes into play in the study of these questions for several structures of interest. 1 Introduction In algebraic complexity one measures the complexity of an algorithm by the number of basic operations performed during a computation. The basic operations are usually arithmetic operations and comparisons, but sometimes transcendental functions are also allowed [2123, 26]. Even when the set of basic operations has been xed, the complexity of a problem depends on the particular model of computation considered. The two main categories of interest for this paper are circuits and trees. In section 2 and...
Lower Bounds Are not Easier over the Reals: Inside PH
, 1999
"... We prove that all NP problems over the reals with addition and order can be solved in polynomial time with the help of a boolean NP oracle. As a consequence, the "P = NP?" question over the reals with addition and order is equivalent to the classical question. For the reals with addition and equalit ..."
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Cited by 2 (0 self)
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We prove that all NP problems over the reals with addition and order can be solved in polynomial time with the help of a boolean NP oracle. As a consequence, the "P = NP?" question over the reals with addition and order is equivalent to the classical question. For the reals with addition and equality only, the situation is quite different since P is known to be different from NP. Nevertheless, we prove similar transfer theorems for the polynomial hierarchy.
Counting complexity classes over the reals I: The additive case
 In Proc. 14th ISAAC 2003, number 2906 in LNCS
, 2003
"... Abstract. We define a counting class #Padd in the BlumShubSmalesetting of additive computations over the reals. Structural properties of this class are studied, including a characterization in terms of the classical counting class #P introduced by Valiant. We also establish transfer theorems for b ..."
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Cited by 1 (1 self)
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Abstract. We define a counting class #Padd in the BlumShubSmalesetting of additive computations over the reals. Structural properties of this class are studied, including a characterization in terms of the classical counting class #P introduced by Valiant. We also establish transfer theorems for both directions between the real additive and the discrete setting. Then we characterize in terms of completeness results the complexity of computing basic topological invariants of semilinear sets given by additive circuits. It turns out that the computation of the Euler characteristic is FP #Padd addcomplete, while for fixed k, the computation of the kth Betti number is FPARaddcomplete. Thus the latter is more difficult under standard complexity theoretic assumptions. We use all the above to prove some analogous completeness results in the classical setting. 1
COUNTING COMPLEXITY CLASSES FOR NUMERIC COMPUTATIONS I: SEMILINEAR SETS ∗
"... Abstract. We define a counting class #Padd in the BlumShubSmalesetting of additive computations over the reals. Structural properties of this class are studied, including a characterization in terms of the classical counting class #P introduced by Valiant. We also establish transfer theorems for ..."
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Abstract. We define a counting class #Padd in the BlumShubSmalesetting of additive computations over the reals. Structural properties of this class are studied, including a characterization in terms of the classical counting class #P introduced by Valiant. We also establish transfer theorems for both directions between the real additive and the discrete setting. Then we characterize in terms of completeness results the complexity of computing basic topological invariants of semilinear sets given by additive circuits. It turns out that the computation of the Euler characteristic is FP #Padd addcomplete, while for fixed k, the computation of the kth Betti number is FPARaddcomplete. Thus the latter is more difficult under standard complexity theoretic assumptions. We use all the above to prove some analogous completeness results in the classical setting.