Phase Transitions and Backbones of the Asymmetric Traveling Salesman Problem

In recent years, there has been much interest in phase transitions of combinatorial problems. Phase transitions have been successfully used to analyze combinatorial optimization problems, characterize their typical-case features and locate the hardest problem instances. In this paper, we study phase transitions of the asymmetric Traveling Salesman Problem (ATSP), an NP-hard combinatorial optimization problem that has many real-world applications. Using random instances of up to 1,500 cities in which intercity distances are uniformly distributed, we empirically show that many properties of the problem, including the optimal tour cost and backbone size, experience sharp transitions as the precision of intercity distances increases across a critical value. Our experimental results on the costs of the ATSP tours and assignment problem agree with the theoretical result that the asymptotic cost of assignment problem is pi ^2 /6 the number of cities goes to infinity. In addition, we show that the average computational cost of the well-known branch-and-bound subtour elimination algorithm for the problem also exhibits a thrashing behavior, transitioning from easy to difficult as the distance precision increases. These results answer positively an open question regarding the existence of phase transitions in the ATSP, and provide guidance on how difficult ATSP problem instances should be generated

Intermediate Phases, structural variance and network demixing in chalcogenides: the unusual case of group V sulfides

We review Intermediate Phases (IPs) in chalcogenide glasses and provide a structural interpretation of these phases. In binary group IV selenides, IPs reside in the 2.40 < r < 2.54 range, and in binary group V selenides they shift to a lower r, in the 2.29< r < 2.40 range. Here r represents the mean coordination number of glasses. In ternary alloys containing equal proportions of group IV and V selenides, IPs are wider and encompass ranges of respective binary glasses. These data suggest that the local structural variance contributing to IP widths largely derives from four isostatic local structures of varying connectivity r; two include group V based quasi-tetrahedral (r = 2.29) and pyramidal (r = 2.40) units, and the other two are group IV based corner-sharing (r = 2.40) and edge-sharing (r = 2.67) tetrahedral units. Remarkably, binary group V (P, As) sulfides exhibit IPs that are shifted to even a lower r than their selenide counterparts; a result that we trace to excess Sn chains either partially (As-S) or completely (P-S) demixing from network backbone, in contrast to excess Sen chains forming part of the backbone in corresponding selenide glasses. In ternary chalcogenides of Ge with the group V elements (As, P), IPs of the sulfides are similar to their selenide counterparts, suggesting that presence of Ge serves to reign in the excess Sn chain fragments back in the backbone as in their selenide counterparts

Limit Crossing for Decision Problems

Limit crossing is a methodology in which modified versions of a problem are solved and compared, yielding useful information about the original problem. Pruning rules that are used to exclude portions of search trees are excellent examples of the limit-crossing technique. In our previous work, we examined limit crossing for optimization problems. In this paper, we extend this methodology to decision problems. We demonstrate the use of limit crossing in our design of a tool for identifying K-SAT backbones. This tool is guaranteed to identify all of the backbone variables by solving at most n+1 formulae, where n is the total number of variables. While previous 3-SAT backbone research was limited to 28 variables, we have computed backbones for 200 variables. In addition to being useful for identifying backbones, this code can be used directly to solve a special class of QBF problem

A Preference-Based Approach to Backbone Computation with Application to Argumentation

The backbone of a constraint satisfaction problem consists of those variables that take the same value in all solutions. Algorithms for determining the backbone of propositional formulas, i.e., Boolean satisfiability (SAT) instances, find various real-world applications. From the knowledge representation and reasoning (KRR) perspective, one interesting connection is that of backbones and the so-called ideal semantics in abstract argumentation. In this paper, we propose a new backbone algorithm which makes use of a "SAT with preferences" solver, i.e., a SAT solver which is guaranteed to output a most preferred satisfying assignment w.r.t. a given preference over literals of the SAT instance at hand. We also show empirically that the proposed approach is specifically effective in computing the ideal semantics of argumentation frameworks, noticeably outperforming an other state-of-the-art backbone solver as well as the winning approach of the recent ICCMA 2017 argumentation solver competition in the ideal semantics track.Peer reviewe

CadiBack: Extracting Backbones with CaDiCaL

The backbone of a satisfiable formula is the set of literals that are true in all its satisfying assignments. Backbone computation can improve a wide range of SAT-based applications, such as verification, fault localization and product configuration. In this tool paper, we introduce a new backbone extraction tool called CadiBack. It takes advantage of unique features available in our state-of-the-art SAT solver CaDiCaL including transparent inprocessing and single clause assumptions, which have not been evaluated in this context before. In addition, CaDiCaL is enhanced with an improved algorithm to support model rotation by utilizing watched literal data structures. In our comprehensive experiments with a large number of benchmarks, CadiBack solves 60% more instances than the state-of-the-art backbone extraction tool MiniBones. Our tool is thoroughly tested with fuzzing, internal correctness checking and cross-checking on a large benchmark set. It is publicly available as open source, well documented and easy to extend