Ahwcz--The knapsack problem is aa Np-complete combinatorial problem that is strongly believed to be computationally difficult to solve in general. Specific instances of this problem tbat appear very difficult to solve unless one pawses “trapdoor information ” used in the design of the problem are demonstrated. Because only the designer can easily solve problems, others can send bim ioformation hidden in the solution to the problems without fear that au eavesdropper will be able to extract the information. This approach differs from usual cryptograpkic systems in that a secret key is not needed. Conversely, only the designer can generate signature8 for messages, but anyone can easily check their authenticity. G I.

We propose a definition of `constrainedness' that unifies two of the most common but informal uses of the term. These are that branching heuristics in search algorithms often try to make the most "constrained" choice, and that hard search problems tend to be "critically constrained". Our definition of constrainedness generalizes a number of parameters used to study phase transition behaviour in a wide variety of problem domains. As well as predicting the location of phase transitions in solubility, constrainedness provides insight into why problems at phase transitions tend to be hard to solve. Such problems are on a constrainedness "knife-edge", and we must search deep into the problem before they look more or less soluble. Heuristics that try to get off this knife-edge as quickly as possible by, for example, minimizing the constrainedness are often very effective. We show that heuristics from a wide variety of problem domains can be seen as minimizing the constrainedness (or proxies ...

Abstract. We discuss fast exponential time solutions for NP-complete problems. We survey known results and approaches, we provide pointers to the literature, and we discuss several open problems in this area. The list of discussed NP-complete problems includes the travelling salesman problem, scheduling under precedence constraints, satisfiability, knapsack, graph coloring, independent sets in graphs, bandwidth of a graph, and many more. 1

Although several large sized 0-1 Knapsack Problems (KP) may be easily solved, it is often the case that most of the computational eort is used for preprocessing, i.e. sorting and reduction. In order to avoid this problem it has been proposed to solve the so-called core of the problem: A Knapsack Problem de ned on a small subset of the variables. But the exact core cannot be identi ed without solving KP, so till now approximated core sizes had to be used.

We present the first average-case analysis proving a polynomial upper bound on the expected running time of an exact algorithm for the 0/1 knapsack problem. In particular, we prove for various input distributions, that the number of Pareto-optimal knapsack fillings is polynomially bounded in the number of available items. An algorithm by Nemhauser and Ullmann can enumerate these solutions very efficiently so that a polynomial upper bound on the number of Pareto-optimal solutions implies an algorithm with expected polynomial running time. The random input model underlying our analysis is quite general and not restricted to a particular input distribution. We assume adversarial weights and randomly drawn profits (or vice versa). Our analysis covers general probability distributions with finite mean and, in its most general form, can even handle different probability distributions for the profits of different items. This feature enables us to study the effects of correlations between profits and weights. Our analysis confirms and explains practical studies showing that so-called strongly correlated instances are harder to solve than weakly correlated ones.

While the 1980s were focused on the solution of large sized "easy" knapsack problems, this decade has brought several new algorithms, which are able to solve "hard" large sized instances. We will give an overview of the recent techniques for solving hard knapsack problems, with special emphasis on the addition of cardinality constraints, dynamic programming, and rudimentary divisibility. Computational results, comparing all recent algorithms, are presented. 1 Introduction We consider the classical 0-1 Knapsack Problem (KP) where a subset of n given items has to be packed in a knapsack of capacity c. Each item has a profit p j and a weight w j and the problem is to select a subset of the items whose total weight does not exceed c and whose total profit is a maximum. We assume, without loss of generality, that all input data are positive integers. Introducing the binary decision variables x j , with x j = 1 if item j is selected, and x j = 0 otherwise, we get the ILP-model: maximize z =...

We show that the treewidth and the minimum fill-in of an n-vertex graph can be computed in time O(1.8899 n). Our results are based on combinatorial proofs that an n-vertex graph has O(1.7087 n) minimal separators and O(1.8135 n) potential maximal cliques. We also show that for the class of AT-free graphs the running time of our algorithms can be reduced to O(1.4142 n).

Given a set of numbers, the two-way partitioning problem is to divide them into two subsets, so that the sum of the numbers in each subset are as nearly equal as possible. The problem is NP-complete, and is contained in many scheduling applications. Based on a polynomial-time heuristic due to Karmarkar and Karp, we present a new algorithm, called Complete Karmarkar Karp (CKK), that op-timally solves the general number-partitioning problem, and signif-icantly outperforms the best previously-known algorithm for large problem instances. By restricting the numbers to twelve signi cant digits, CKK can optimally solve two-way partitioning problems of ar-bitrary size in practice. For numbers with greater precision, CKK rst returns the Karmarkar-Karp solution, then continues to nd better so-lutions as time allows. Over seven orders of magnitude improvement in solution quality is obtained in less than an hour of running time. CKK is directly applicable to the subset sum problem, by reducing it to number partitioning. Rather than building a single solution one element at a time, or modifying a complete solution, CKK constructs subsolutions, and combines them together in all possible ways. This approach may be e ective for other NP-hard problems as well. 1 1

The Multiple Knapsack Problem is the problem of assigning a subset of n items to m distinct knapsacks, such that the total profit sum of the selected items is maximized, without exceeding the capacity of each of the knapsacks. The problem has several applications in naval as well as financial management.

Abstract. We present a novel method for exactly solving (in fact, counting solutions to) general constraint satisfaction optimization with at most two variables per constraint (e.g. MAX-2-CSP and MIN-2-CSP), which gives the first exponential improvement over the trivial algorithm. More precisely, the runtime bound is a constant factor improvement in the base of the exponent: the algorithm can count the number of optima in MAX-2-SAT and MAX-CUT instances in O(m 3 2 ωn/3) time, where ω < 2.376 is the matrix product exponent over a ring. When constraints have arbitrary weights, there is a (1 + ɛ)-approximation with roughly the same runtime, modulo polynomial factors. Our construction shows that improvement in the runtime exponent of either k-clique solution (even when k = 3) or matrix multiplication over GF(2) would improve the runtime exponent for solving 2-CSP optimization. Our approach also yields connections between the complexity of some (polynomial time) high dimensional search problems and some NP-hard problems. For example, if there are sufficiently faster algorithms for computing the diameter of n points in ℓ1, then there is an (2 − ɛ) n algorithm for MAX-LIN. These results may be construed as either lower bounds on the high-dimensional problems, or hope that better algorithms exist for the corresponding hard problems. 1