It is well known that for many NP-complete problems, such as K-Sat, etc., typical cases are easy to solve; so that computationally hard cases must be rare (assuming P = NP). This paper shows that NP-complete problems can be summarized by at least one "order parameter", and that the hard problems occur at a critical value of such a parameter. This critical value separates two regions of characteristically different properties. For example, for K-colorability, the critical value separates overconstrained from underconstrained random graphs, and it marks the value at which the probability of a solution changes abruptly from near 0 to near 1. It is the high density of wellseparated almost solutions (local minima) at this boundary that cause search algorithms to "thrash". This boundary is a type of phase transition and we show that it is preserved under mappings between problems. We show that for some P problems either there is no phase transition or it occurs for bounded N (and so bounds the cost). These results suggest a way of deciding if a problem is in P or NP and why they are different. 1

. We outline a technique for studying phase transition behaviour in computational problems using number partitioning as a case study. We first build an "annealed" theory that assumes independence between parts of the number partition problem. Using this theory, we identify a parameter which represents the "constrainedness" of a problem. We determine experimentally the critical value of this parameter at which a rapid transition between soluble and insoluble problems occurs. Finite-size scaling methods developed in statistical mechanics describe the behaviour around the critical value. We identify phase transition behaviour in both the decision and optimization versions of number partitioning, in the size of the optimal partition, and in the quality of heuristic solutions. This case study demonstrates how annealed theories and finite-size scaling allows us to compare algorithms and heuristics in a precise and quantitative manner. 1 Introduction Phase transition behaviour has recently r...

We illustrate the use of phase transition behavior in the study of heuristics. Using an “annealed ” theory, we define a parameter that measures the “constrainedness ” of an ensemble of number partitioning problems. We identify a phase transition at a critical value of constrainedness. We then show that constrainedness can be used to analyze and compare algorithms and heuristics for number partitioning in a precise and quantitative manner. For example, we demonstrate that on uniform random problems both the Karmarkar–Karp and greedy heuristics minimize the constrainedness, but that the decisions made by the Karmarkar–Karp heuristic are superior at reducing constrainedness. This supports the better performance observed experimentally for the Karmarkar–Karp heuristic. Our results refute a conjecture of Fu that phase transition behavior does not occur in number partitioning. Additionally, they demonstrate that phase transition behavior is useful for more than just simple benchmarking. It can, for instance, be used to analyze heuristics, and to compare the quality of heuristic solutions. Key words: heuristics, number partitioning, phase transitions. 1.

We report numerical simulations on the number of local minima in the landscape of the Graph Bipartitioning Problem and provide an explanation in terms of the correlation length of its landscape.

We consider the problem of partitioning n integers chosen randomly between 1 and 2 m into two subsets such that the discrepancy, the absolute value of the difference of their sums, is minimized. A partition is called perfect if the optimum discrepancy is 0 when the sum of all n integers in the original set is even, or 1 when the sum is odd. Parameterizing the random problem in terms of κ = m/n, we prove that the problem has a sharp threshold at κ = 1, in the sense that for κ < 1, there are many perfect partitions with probability tending to 1 as n → ∞, while for κ> 1, there are no perfect partitions with probability tending to 1. Moreover, we show that the derivative of the so-called entropy is discontinuous at κ = 1. We also determine the scaling window about the transition point: κn = 1 − (2n) −1 log 2 n + λn/n, by showing that the probability of a perfect partition tends to 0, 1, or some explicitly computable p(λ) ∈ (0, 1), depending on whether λn tends to −∞, ∞, or λ ∈ (−∞, ∞), respectively. For λn → − ∞ fast enough, we show that the number of perfect partitions is Gaussian in the limit. For λn → ∞, we prove that with high probability the optimum partition is unique, and that the optimum discrepancy is Θ(λn). Within the window, i.e., if |λn | is bounded, we prove that the optimum discrepancy is bounded. Both for λn → ∞ and within the window, the limiting distribution of the (scaled) discrepancy is found.

We identify a natural parameter for random number partitioning, and show that there is a rapid transition between soluble and insoluble problems at a critical value of this parameter. Hard problems are associated with this transition. As is seen in other computational problems, finite-size scaling methods developed in statistical mechanics describe the behaviour around the critical value. Such phase transition phenomena appear to be ubiquitous. Indeed, we have yet to find a NP-complete problem which lacks a similar phase transition. The identification of a phase transition in number partitioning is of particular interest since it had been suggested that one would not occur. Department of Computer Science, University of Strathclyde, Glasgow G1 1XH, Scotland. ipg@cs.strath.ac.uk y Mechanized Reasoning Group, IRST, Loc. Pante di Povo, 38100 Trento & DIST, University of Genoa, 16143 Genoa, Italy. toby@irst.it RR-95-185 University of Strathclyde 1 Introduction Phase transitions have ...

We consider the problem of partitioning n integers into two subsets of given cardinalities such that the discrepancy, the absolute value of the difference of their sums, is minimized. The integers are i.i.d. random variables chosen uniformly from the set {1,..., M}. We study how the typical behavior of the optimal partition depends on n, M and the bias s, the difference between the cardinalities of the two subsets in the partition. In particular, we rigorously establish this typical behavior as a function of the two parameters κ:= n −1 log 2 M and b: = |s|/n by proving the existence of three distinct “phases ” in the κbplane, characterized by the value of the discrepancy and the number of optimal solutions: a “perfect phase ” with exponentially many optimal solutions with discrepancy 0 or 1; a “hard phase ” with minimal discrepancy of order Me −Θ(n) ; and a “sorted phase ” with an unique optimal partition of order Mn, obtained by putting the (s + n)/2 smallest integers in one subset. Our phase diagram covers all but a relatively small region in the κb-plane. We also show that the three phases can be alternatively characterized by the number of basis solutions of the associated linear programming problem, and by the fraction of these basis solutions whose ±1-valued components form optimal integer partitions of the subproblem with the corresponding weights. We show in particular that this fraction is one in the sorted phase, and exponentially small in both the perfect and hard phases, and strictly exponentially smaller in the hard phase than in the perfect phase. Open problems are discussed, and numerical experiments are presented.

The number partitioning problem is a classic problem of combinatorial optimization in which a set of n numbers is partitioned into two subsets such that the sum of the numbers in one subset is as close as possible to the sum of the numbers in the other set. When the n numbers are i.i.d. variables drawn from some distribution, the partitioning problem turns out to be equivalent to a mean-field antiferromagnetic Ising spin glass. In the spin glass representation, it is natural to define energies – corresponding to the costs of the partitions, and overlaps – corresponding to the correlations between partitions. Although the energy levels of this model are a priori highly correlated, a surprising recent conjecture asserts that the energy spectrum of number partitioning is locally that of a random energy model (REM): the spacings between nearby energy levels are uncorrelated. In other words, the properly scaled energies converge to a Poisson process. The conjecture also asserts that the corresponding spin configurations are uncorrelated, indicating vanishing overlaps in the spin glass representation. In this paper, we prove these two claims, collectively known as the local REM conjecture.

It is well known that for many NP-complete problems, such as K-Sat, etc., typical cases are easy to solve; so that computationally hard cases must be rare (assuming P 6= NP ). This paper shows that NP-complete problems can be summarized by at least one "order parameter ", and that the hard problems occur at a critical value of such a parameter. This critical value separates two regions of characteristically different properties. For example, for K-colorability, the critical value separates overconstrained from underconstrained random graphs, and it marks the value at which the probability of a solution changes abruptly from near 0 to near 1. It is the high density of wellseparated almost solutions (local minima) at this boundary that cause search algorithms to "thrash". This boundary is a type of phase transition and we show that it is preserved under mappings between problems. We show that for some P problems either there is no phase transition or it occurs for bounded N (and so bound...

Phase transitions in combinatorial problems have recently been shown [MZK 99] to be useful in locating \hard" instances of combinatorial problems. The connection between computational complexity and the existence of phase transitions has been addressed in Statistical Mechanics [MZK 99] and Arti cial Intelligence [CKT91], but not studied rigorously.