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 ...

We dene the `frozen development' of coloring random graphs. We identify two nodes in a graph as frozen if they are the same color in all legal colorings. This is analogous to studies of the development of a backbone or spine in SAT (the Satisability problem). We rst describe in detail the algorithmic techniques used to study frozen development. We present strong empirical evidence that freezing in 3-coloring is sudden. A single edge typically causes the size of the graph to collapse in size by 28%. We also use the frozen development to calculate unbiased estimates of probability of colorability in random graphs, even where this probability is as low as 10 300 . We investigate the links between frozen development and the solution cost of graph coloring. In SAT, a discontinuity in the order parameter has been correlated with the hardness of SAT instances, and our data for coloring is suggestive of an asymptotic discontinuity. The uncolorability threshold is known to give rise to har...

In this paper we propose a new type of random CSP model, called Model RB, which is a revision to the standard Model B. It is proved that phase transitions from a region where almost all problems are satis able to a region where almost all problems are unsatis able do exist for Model RB as the number of variables approaches in nity. Moreover, the critical values at which the phase transitions occur are also known exactly. By relating the hardness of Model RB to Model B, it is shown that there exist a lot of hard instances in Model RB.

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We show that problems at the uncolorability phase transition are well out of reach of intelligent algorithms. Since there are not small and easily checkable subgraphs which can be used to confirm uncolorability quickly, we cannot hope to build more intelligent algorithms to avoid hard problems at the phase transition. Also, our results suggest that a conjectured double phase transition in graph coloring occurs only in small graphs. Similar results are likely in other NP-complete problems where instances from phase transitions are hard for all known algorithms, and will help to explain the phenomenon. Furthermore, our results help to elucidate the distinction between polynomial and non-polynomial search behavior.

This paper analyzes the resolution complexity of two random CSP models, i.e. Model RB/RD for which we can establish the existence of phase transitions and identify the threshold points exactly. By encoding CSPs into CNF formulas, this paper proves that almost all instances of Model RB/RD have no tree-like resolution proofs of less than exponential size. Thus, we not only introduce new families of CNF formulas hard for resolution, which is a central task of Proof-Complexity theory, but also propose models with both many hard instances and exact phase transitions. Moreover, it is shown both theoretically and experimentally that an application of RB/RD might be in the generation of hard satisfiable instances, which is not only of practical importance but also related to some open problems in cryptography such as generating one-way functions. Finally, conclusions are presented, as well as a detailed comparison of RB/RD with some well-studied models such as the Hamiltonian cycle problem and random 3-SAT.

In this paper, we analyze the decision version of the NK landscape model from the perspective of threshold phenomena and phase transitions under two random distributions, the uniform probability model and the fixed ratio model. For the uniform probability model, we prove that the phase transition is easy in the sense that there is a polynomial algorithm that can solve a random instance of the problem with the probability asymptotic to 1 as the problem size tends to infinity. For the fixed ratio model, we establish several upper bounds for the solubility threshold, and prove that random instances with parameters above these upper bounds can be solved polynomially. This, together with our empirical study for random instances generated below and in the phase transition region, suggests that the phase transition of the fixed ratio model is also easy.

In recent years, there has been much interest in phase transitions of combinatorial problems.

Universities aim for good “Space Management ” so as to use the teaching space efficiently. Part of this task is to assign rooms and time-slots to teaching activities with limited numbers and capacities of lecture theaters, seminar rooms, etc. It is also common that some teaching activities require splitting into multiple events. For example, lectures can be too large to fit in one room or good teaching practice requires that seminars/tutorials are taught in small groups. Then, space management involves decisions on splitting as well as the assignments to rooms and time-slots. These decisions must be made whilst satisfying the pedagogic requirements of the institution and constraints on space resources. The efficiency of such management can be measured by the “utilisation”: the percentage of available seat-hours actually used. In many institutions, the observed utilisation is unacceptably low, and this provides our underlying motivation: to study the factors that affect teaching space utilisation, with the goal of improving it. We give a brief introduction to our work in this area, and then introduce a specific model for splitting. We present experimental results that show threshold phenomena and associated easy-hard-easy patterns of computational difficulty. We discuss why such behaviour is of importance for space management. Contact Author. 1 1

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