A New Look at the Easy-Hard-Easy Pattern of Combinatorial Search Difficulty

The easy-hard-easy pattern in the difficulty of combinatorial search problems as constraints are added has been explained as due to a competition between the decrease in number of solutions and increased pruning. We test the generality of this explanation by examining one of its predictions: if the number of solutions is held fixed by the choice of problems, then increased pruning should lead to a monotonic decrease in search cost. Instead, we find the easy-hard-easy pattern in median search cost even when the number of solutions is held constant, for some search methods. This generalizes previous observations of this pattern and shows that the existing theory does not explain the full range of the peak in search cost. In these cases the pattern appears to be due to changes in the size of the minimal unsolvable subproblems, rather than changing numbers of solutions.Comment: See http://www.jair.org/ for any accompanying file

A Good Age

The difference between a dream and reality, in many cases, is a simple but difficult thing, hard work. In the following pages, Bert Kruger Smith chronicles the efforts of one very determined group of people in converting their dream into reality. Her story is an important one for at least two reasons. First is its message that amazing things are possible if you just believe hard enough, work hard enough, and endure. Its second lesson, however, may be the more important. I believe that the Austin Groups for the Elderly history reminds us that it is still possible to find creative solutions to difficult problems. It underscores the ability that people have to face a crisis and convert it into a vision. And the crisis that the founders of AGE faced is not a unique one, nor is the solution one applicable only to agencies serving the elderly. Indeed, if anything, all those who are attempting to deliver services in this time of such great need and so little plenty should study carefully the efforts of these dedicated people. As I read the AGE story, I thought of the old song "High Hopes." It is very easy nowadays to dwell on the negative, to mourn the unchangeable "way things are," and forget that "high in the sky, apple pie hopes" still can make differences. Austin is a better place in which to live, and, even better, in which to grow old because of Austin Groups for the Elderly.Hogg Foundation for Mental Healt

Phase Transitions and Backbones of Constraint Minimization Problems

Many real-world problems involve constraints that cannot be all satisfied. The goal toward an overconstrained problem is to find solutions minimizing the total number of constraints violated. We call such a problem constraint minimization problem (CMP). We study the behavior of the phase transitions and backbones of CMP. We first investigate the relationship between the phase transitions of Boolean satisfiability, or precisely 3-SAT (a well-studied NP-complete decision problem), and the phase transitions of MAX 3-SAT (an NP-hard optimization problem). To bridge the gap between the easy-hard-easy phase transitions of 3-SAT, in which solutions of bounded quality, e.g., solutions with at most a constant number of constraints violated, are sufficient. We show that phase transitions are persistent in bounded 3-SAT and are similar to that of 3-SAT. We then study backbones of MAX 3-SAT, which are critically constrained variables that have fixed values in all optimal solutions. Our experimental results show that backbones of MAX 3-SAT emerge abruptly and experience sharp transitions from nonexistence when underconstrained to almost complete when overconstrained. More interestingly, the phase transitions of MAX 3-SAT backbones surprisingly concur with the phase transitions of satisfiability of 3-SAT. Specifically, the backbone of MAX 3-SAT with size 0.5 approximately collocates with the 0.5 satisfiablity of 3-SAT, and hte backbone and satisfiability seems to follow a linear correlation near this 0.5-0.5 collocation

Phase transition and landscape statistics of the number partitioning problem

The phase transition in the number partitioning problem (NPP), i.e., the transition from a region in the space of control parameters in which almost all instances have many solutions to a region in which almost all instances have no solution, is investigated by examining the energy landscape of this classic optimization problem. This is achieved by coding the information about the minimum energy paths connecting pairs of minima into a tree structure, termed a barrier tree, the leaves and internal nodes of which represent, respectively, the minima and the lowest energy saddles connecting those minima. Here we apply several measures of shape (balance and symmetry) as well as of branch lengths (barrier heights) to the barrier trees that result from the landscape of the NPP, aiming at identifying traces of the easy/hard transition. We find that it is not possible to tell the easy regime from the hard one by visual inspection of the trees or by measuring the barrier heights. Only the {\it difficulty} measure, given by the maximum value of the ratio between the barrier height and the energy surplus of local minima, succeeded in detecting traces of the phase transition in the tree. In adddition, we show that the barrier trees associated with the NPP are very similar to random trees, contrasting dramatically with trees associated with the $p$ spin-glass and random energy models. We also examine critically a recent conjecture on the equivalence between the NPP and a truncated random energy model

New Problems for Subsidized Speech

The constitutionality of conditional offers from the government is a transsubstantive issue with broad and growing practical implications, but it has always been a particular problem for free speech. Recent developments suggest at least three new approaches to the problem, but no easy solutions to it. The first approach would permit conditions that define the limits of the government spending program, while forbidding conditions that leverage funding so as to regulate speech outside the contours of the program. This is an appealing distinction, but runs into some of the same challenges as public forum analysis. The second approach would treat conditional offers to purchase speech like other proposed economic transactions, invalidating them when they are coercive. This principle helps explain many recent cases, including the healthcare decision. And yet the Court’s willingness to find coercion in cases involving conditional offers from the government is hard to square with its approach to campaign finance law and its apparent faith in markets more generally. The third and final approach would treat limits on conditional offers not as individual rights, but as structural limitations on the scope of government. This approach, too, points in the direction of possible solutions—and also further problems—for analyzing the constitutionality of subsidized speech

Diversification and Intensification in Hybrid Metaheuristics for Constraint Satisfaction Problems

Metaheuristics are used to find feasible solutions to hard Combinatorial Optimization Problems (COPs). Constraint Satisfaction Problems (CSPs) may be formulated as COPs, where the objective is to reduce the number of violated constraints to zero. The popular puzzle Sudoku is an NP-complete problem that has been used to study the effectiveness of metaheuristics in solving CSPs. Applying the Simulated Annealing (SA) metaheuristic to Sudoku has been shown to be a successful method to solve CSPs. However, the ‘easy-hard-easy’ phase-transition behavior frequently attributed to a certain class of CSPs makes finding a solution extremely difficult in the hard phase because of the vast search space, the small number of solutions and a fitness landscape marked by many plateaus and local minima. Two key mechanisms that metaheuristics employ for searching are diversification and intensification. Diversification is the method of identifying diverse promising regions of the search space and is achieved through the process of heating/reheating. Intensification is the method of finding a solution in one of these promising regions and is achieved through the process of cooling. The hard phase area of the search terrain makes traversal without becoming trapped very challenging. Running the best available method - a Constraint Propagation/Depth-First Search algorithm - against 30,000 benchmark problem-instances, 20,240 remain unsolved after ten runs at one minute per run which we classify as very hard. This dissertation studies the delicate balance between diversification and intensification in the search process and offers a hybrid SA algorithm to solve very hard instances. The algorithm presents (a) a heating/reheating strategy that incorporates the lowest solution cost for diversification; (b) a more complex two-stage cooling schedule for faster intensification; (c) Constraint Programming (CP) hybridization to reduce the search space and to escape a local minimum; (d) a three-way swap, secondary neighborhood operator for a low expense method of diversification. These techniques are tested individually and in hybrid combinations for a total of 11 strategies, and the effectiveness of each is evaluated by percentage solved and average best run-time to solution. In the final analysis, all strategies are an improvement on current methods, but the most remarkable results come from the application of the “Quick Reset” technique between cooling stages

A Framework for Data-Driven Explainability in Mathematical Optimization

Advancements in mathematical programming have made it possible to efficiently tackle large-scale real-world problems that were deemed intractable just a few decades ago. However, provably optimal solutions may not be accepted due to the perception of optimization software as a black box. Although well understood by scientists, this lacks easy accessibility for practitioners. Hence, we advocate for introducing the explainability of a solution as another evaluation criterion, next to its objective value, which enables us to find trade-off solutions between these two criteria. Explainability is attained by comparing against (not necessarily optimal) solutions that were implemented in similar situations in the past. Thus, solutions are preferred that exhibit similar features. Although we prove that already in simple cases the explainable model is NP-hard, we characterize relevant polynomially solvable cases such as the explainable shortest-path problem. Our numerical experiments on both artificial as well as real-world road networks show the resulting Pareto front. It turns out that the cost of enforcing explainability can be very small