Breaking Instance-Independent Symmetries In Exact Graph Coloring

Code optimization and high level synthesis can be posed as constraint satisfaction and optimization problems, such as graph coloring used in register allocation. Graph coloring is also used to model more traditional CSPs relevant to AI, such as planning, time-tabling and scheduling. Provably optimal solutions may be desirable for commercial and defense applications. Additionally, for applications such as register allocation and code optimization, naturally-occurring instances of graph coloring are often small and can be solved optimally. A recent wave of improvements in algorithms for Boolean satisfiability (SAT) and 0-1 Integer Linear Programming (ILP) suggests generic problem-reduction methods, rather than problem-specific heuristics, because (1) heuristics may be upset by new constraints, (2) heuristics tend to ignore structure, and (3) many relevant problems are provably inapproximable. Problem reductions often lead to highly symmetric SAT instances, and symmetries are known to slow down SAT solvers. In this work, we compare several avenues for symmetry breaking, in particular when certain kinds of symmetry are present in all generated instances. Our focus on reducing CSPs to SAT allows us to leverage recent dramatic improvement in SAT solvers and automatically benefit from future progress. We can use a variety of black-box SAT solvers without modifying their source code because our symmetry-breaking techniques are static, i.e., we detect symmetries and add symmetry breaking predicates (SBPs) during pre-processing. An important result of our work is that among the types of instance-independent SBPs we studied and their combinations, the simplest and least complete constructions are the most effective. Our experiments also clearly indicate that instance-independent symmetries should mostly be processed together with instance-specific symmetries rather than at the specification level, contrary to what has been suggested in the literature

Strengthening Canonical Pattern Databases with Structural Symmetries

Symmetry-based state space pruning techniques have proved to greatly improve heuristic search based classical planners. Similarly, abstraction heuristics in general and pattern databases in particular are key ingredients of such planners. However, only little work has dealt with how the abstraction heuristics behave under symmetries. In this work, we investigate the symmetry properties of the popular canonical pattern databases heuristic. Exploiting structural symmetries, we strengthen the canonical pattern databases by adding symmetric pattern databases, making the resulting heuristic invariant under structural symmetry, thus making it especially attractive for symmetry-based pruning search methods. Further, we prove that this heuristic is at least as informative as using symmetric lookups over the original heuristic. An experimental evaluation confirms these theoretical results

Solution methods for the tray optimization problem

In order to perform medical surgeries, hospitals keep large inventories of surgical in- struments. These instruments need to be sterilized before each surgery. Typically the instruments are kept in trays. Multiple trays may be required for a single surgery, while a single tray may contain instruments that are required for multiple surgical procedures. The tray optimization problem (TOP) consists of three main decisions: (i) the assignment of instruments to trays, (ii) the assignment of trays to surgeries, and (iii) the number of trays to keep in inventory. The TOP decisions have to be made such that total operating costs are minimized and such that for every surgery sufficient instruments are available. This paper presents and evaluates several exact and heuristic solution methods for the TOP. We compare solution methods on computation time and solution quality. Moreover, we conduct simulations to evaluate the performance of the solutions in the long run. The novel methods that are provided are the first methods that are capable of solving instances of realistic size. The most promising method consists of a highly scalable advanced greedy algorithm. Our results indicate that the outcomes of this method are, on average, very close to the outcomes of the other methods investigated, while it may be easily applied by (large) hospitals. The findings are robust with respect to fluctuations in long term OR schedules

Stochastic programming for City Logistics: new models and methods

The need for mobility that emerged in the last decades led to an impressive increase in the number of vehicles as well as to a saturation of transportation infrastructures. Consequently, traffic congestion, accidents, transportation delays, and polluting emissions are some of the most recurrent concerns transportation and city managers have to deal with. However, just building new infrastructures might be not sustainable because of their cost, the land usage, which usually lacks in metropolitan regions, and their negative impact on the environment. Therefore, a different way of improving the performance of transportation systems while enhancing travel safety has to be found in order to make people and good transportation operations more efficient and support their key role in the economic development of either a city or a whole country. The concept of City Logistics (CL) is being developed to answer to this need. Indeed, CL focus on reducing the number of vehicles operating in the city, controlling their dimension and characteristics. CL solutions do not only improve the transportation system but the whole logistics system within an urban area, trying to integrate interests of the several. This global view challenges researchers to develop planning models, methods and decision support tools for the optimization of the structures and the activities of the transportation system. In particular, this leads researchers to the definition of strategic and tactical problems belonging to well-known problem classes, including network design problem, vehicle routing problem (VRP), traveling salesman problem (TSP), bin packing problem (BPP), which typically act as sub-problems of the overall CL system optimization. When long planning horizons are involved, these problems become stochastic and, thus, must explicitly take into account the different sources of uncertainty that can affect the transportation system. Due to these reasons and the large-scale of CL systems, the optimization problems arising in the urban context are very challenging. Their solution requires investigations in mathematical and combinatorial optimization methods as well as the implementation of efficient exact and heuristic algorithms. However, contributions answering these challenges are still limited number. This work contributes in filling this gap in the literature in terms of both modeling framework for new planning problems in CL context and developing new and effective heuristic solving methods for the two-stage formulation of these problems. Three stochastic problems are proposed in the context of CL: the stochastic variable cost and size bin packing problem (SVCSBPP), the multi-handler knapsack problem under uncertainty (MHKPu) and the multi-path traveling salesman problem with stochastic travel times (mpTSPs). The SVCSBPP arises in supply-chain management, in which companies outsource the logistics activities to a third-party logistic firm (3PL). The procurement of sufficient capacity, expressed in terms of vehicles, containers or space in a warehouse for varying periods of time to satisfy the demand plays a crucial role. The SVCSBPP focuses on the relation between a company and its logistics capacity provider and the tactical-planning problem of determining the quantity of capacity units to secure for the next period of activity. The SVCSBPP is the first attempt to introduce a stochastic variant of the variable cost and size bin packing problem (VCSBPP) considering not only the uncertainty on the demand to deliver, but also on the renting cost of the different bins and their availability. A large number of real-life situations can be satisfactorily modeled as a MHKPu, in particular in the last mile delivery. Last mile delivery may involve different sequences of consolidation operations, each handled by different workers with different skill levels and reliability. The improper management of consolidation operations can cause delay in the operations reducing the overall profit of the deliveries. Thus, given a set of potential logistics handlers and a set of items to deliver, characterized by volume and random profit, the MHKPu consists in finding a subset of items which maximizes the expected total profit. The profit is given by the sum of a deterministic profit and a stochastic profit oscillation, with unknown probability distribution, due to the random handling costs of the handlers.The mpTSPs arises mainly in City Logistics applications. Cities offer several services, such as garbage collection, periodic delivery of goods in urban grocery distribution and bike sharing services. These services require the planning of fixed and periodic tours that will be used from one to several weeks. However, the enlarged time horizon as well as strong dynamic changes in travel times due to traffic congestion and other nuisances typical of the urban transportation induce the presence of multiple paths with stochastic travel times. Given a graph characterized by a set of nodes connected by arcs, mpTSPs considers that, for every pair of nodes, multiple paths between the two nodes are present. Each path is characterized by a random travel time. Similarly to the standard TSP, the aim of the problem is to define the Hamiltonian cycle minimizing the expected total cost. These planning problems have been formulated as two-stage integer stochastic programs with recourse. Discretization methods are usually applied to approximate the probability distribution of the random parameters. The resulting approximated program becomes a deterministic linear program with integer decision variables of generally very large dimensions, beyond the reach of exact methods. Therefore, heuristics are required. For the MHKPu, we apply the extreme value theory and derive a deterministic approximation, while for the SVCSBPP and the mpTSPs we introduce effective and accurate heuristics based on the progressive hedging (PH) ideas. The PH mitigates the computational difficulty associated with large problem instances by decomposing the stochastic program by scenario. When effective heuristic techniques exist for solving individual scenario, that is the case of the SVCSBPP and the mpTSPs, the PH further reduces the computational effort of solving scenario subproblems by means of a commercial solver. In particular, we propose a series of specific strategies to accelerate the search and efficiently address the symmetry of solutions, including an aggregated consensual solution, heuristic penalty adjustments, and a bundle fixing technique. Yet, although solution methods become more powerful, combinatorial problems in the CL context are very large and difficult to solve. Thus, in order to significantly enhance the computational efficiency, these heuristics implement parallel schemes. With the aim to make a complete analysis of the problems proposed, we perform extensive numerical experiments on a large set of instances of various dimensions, including realistic setting derived by real applications in the urban area, and combinations of different levels of variability and correlations in the stochastic parameters. The campaign includes the assessment of the efficiency of the meta-heuristic, the evaluation of the interest to explicitly consider uncertainty, an analysis of the impact of problem characteristics, the structure of solutions, as well as an evaluation of the robustness of the solutions when used as decision tool. The numerical analysis indicates that the stochastic programs have significant effects in terms of both the economic impact (e.g. cost reduction) and the operations management (e.g. prediction of the capacity needed by the firm). The proposed methodologies outperform the use of commercial solvers, also when small-size instances are considered. In fact, they find good solutions in manageable computing time. This makes these heuristics a strategic tool that can be incorporated in larger decision support systems for CL

Star-topology decoupled state-space search in AI planning and model checking

State-space search is a widely employed concept in many areas of computer science. The well-known state explosion problem, however, imposes a severe limitation to the effective implementation of search in state spaces that are exponential in the size of a compact system description, which captures the state-transition semantics. Decoupled state-space search, decoupled search for short, is a novel approach to tackle the state explosion. It decomposes the system such that the dependencies between components take the form of a star topology with a center and several leaf components. Decoupled search exploits that the leaves in that topology are conditionally independent. Such independence naturally arises in many kinds of factored model representations, where the overall state space results from the product of several system components. In this work, we introduce decoupled search in the context of artificial intelligence planning and formal verification using model checking. Building on common formalisms, we develop the concept of the decoupled state space and prove its correctness with respect to capturing reachability of the underlying model exactly. This allows us to connect decoupled search to any search algorithm, and, important for planning, adapt any heuristic function to the decoupled state representation. Such heuristics then guide the search towards states that satisfy a desired goal condition. In model checking, we address the problems of verifying safety properties, which express system states that must never occur, and liveness properties, that must hold in any infinite system execution. Many approaches have been proposed in the past to tackle the state explosion problem. Most prominently partial-order reduction, symmetry breaking, Petri-net unfolding, and symbolic state representations. Like decoupled search, all of these are capable of exponentially reducing the search effort, either by pruning part of the state space (the former two), or by representing large state sets compactly (the latter two). For all these techniques, we prove that decoupled search can be exponentially more efficient, confirming that it is indeed a novel concept that exploits model properties in a unique way. Given such orthogonality, we combine decoupled search with several complementary methods. Empirically, we show that decoupled search favourably compares to state-of-the-art planners in common algorithmic planning problems using standard benchmarks. In model checking, decoupled search outperforms well-established tools, both in the context of the verification of safety and liveness properties.Die Zustandsraumsuche ist ein weit verbreitetes Konzept in vielen Bereichen der Informatik, deren effektive Anwendung jedoch durch das Problem der Zustandsexplosion deutlich erschwert wird. Die Zustandsexplosion ist dadurch charakterisiert dass kompakte Systemmodelle exponentiell große Zustandsräume beschreiben. Entkoppelte Zustandsraumsuche (entkoppelte Suche) beschreibt einen neuartigen Ansatz der Zustandsexplosion entgegenzuwirken indem die Struktur des Modells, insbesondere die bedingte Unabhängigkeit von Systemkomponenten in einer Sterntopologie, ausgenutzt wird. Diese Unabhängigkeit ergibt sich bei vielen faktorisierten Modellen deren Zustandsraum sich aus dem Produkt mehrerer Komponenten zusammensetzt. In dieser Arbeit wird die entkoppelte Suche in der Planung, als Teil der Künstlichen Intelligenz, und der Verifikation mittels Modellprüfung eingeführt. In etablierten Formalismen wird das Konzept des entkoppelten Zustandsraums entwickelt und dessen Korrektheit bezüglich der exakten Erfassung der Erreichbarkeit von Modellzuständen bewiesen. Dies ermöglicht die Kombination der entkoppelten Suche mit beliebigen Suchalgorithmen. Wichtig für die Planung ist zudem die Nutzung von Heuristiken, die die Suche zu Zuständen führen, die eine gewünschte Zielbedingung erfüllen, mit der entkoppelten Zustandsdarstellung. Im Teil zur Modellprüfung wird die Verifikation von Sicherheits- sowie Lebendigkeitseigenschaften betrachtet, die unerwünschte Zustände, bzw. Eigenschaften, die bei unendlicher Systemausführung gelten müssen, beschreiben. Es existieren diverse Ansätze um die Zustandsexplosion anzugehen. Am bekanntesten sind die Reduktion partieller Ordnung, Symmetriereduktion, Entfaltung von Petri-Netzen und symbolische Suche. Diese können, wie die entkoppelte Suche, den Suchaufwand exponentiell reduzieren. Dies geschieht durch Beschneidung eines Teils des Zustandsraums, oder durch die kompakte Darstellung großer Zustandsmengen. Für diese Verfahren wird bewiesen, dass die entkoppelte Suche exponentiell effizienter sein kann. Dies belegt dass es sich um ein neuartiges Konzept handelt, das sich auf eigene Art der Modelleigenschaften bedient. Auf Basis dieser Beobachtung werden, mit Ausnahme der Entfaltung, Kombinationen mit entkoppelter Suche entwickelt. Empirisch kann die entkoppelte Suche im Vergleich zu modernen Planern zu deutlichen Vorteilen führen. In der Modellprüfung werden, sowohl bei der Überprüfung von Sicherheit-, als auch Lebendigkeitseigenschaften, etablierte Programme übertroffen.Deutsche Forschungsgesellschaft; Star-Topology Decoupled State Space Searc

A novel multi-objective Interactive Coral Reefs Optimization algorithm for the Unequal Area Facility Layout Problem

The Unequal Area Facility Layout Problem (UA-FLP) has been widely analyzed in the literature using several heuristics and meta-heuristics to optimize some qualitative criteria, taking into account different restrictions and constraints. Nevertheless, the subjective opinion of the designer (Decision Maker, DM) has never been considered along with the quantitative criteria and restrictions. This work proposes a novel approach for the UA-FLP based on an Interactive Coral Reefs Optimization (ICRO) algorithm, which combines the simultaneous consideration of both quantitative and qualitative (DM opinion) features. The algorithm implementation is explained in detail, including the way of jointly considering quantitative and qualitative aspects in the fitness function of the problem. The experimental part of the paper illustrates the effect of including qualitative aspects in UA-FLP problems, considering three different hard UA-FLP instances. Empirical results show that the proposed approach is able to incorporate the DM preferences in the obtained layouts, without affecting much to the quantitative part of the solutions