Combinatorial Search Problems Research Articles

Enhancing set constraint solvers with lexicographic bounds

Since their beginning in constraint programming, set solvers have been applied to a wide range of combinatorial search problems, such as bin-packing, set partitioning, circuit and combinatorial design. In this paper we present and evaluate a new means towards improving the practical reasoning power of Finite Set (FS) constraint solvers to better address such set problems with a particular attention to the challenging symmetrical set problems often cast as Combinatorial Design Problems (CDPs). While CDPs find a natural formulation in the language of sets, in constraint programming literature, alternative models are often used due to a lack of efficiency of traditional FS solvers. We first identify the main structural components of CDPs that render their modeling suitable to set languages but their solving a technical challenge. Our new prototype solver extends the traditional subset variable domain with lexicographic bounds that better approximate a set domain by satisfying the cardinality restrictions applied to the variable, and allow for active symmetry breaking using ordering constraints. Our contribution includes the formal and practical framework of the new solver implemented on top of a traditional set solver, and an empirical evaluation on benchmark CDPs.

Large Scale Transductive SVMs

We show how the concave-convex procedure can be applied to transductive SVMs, which traditionally require solving a combinatorial search problem. This provides for the first time a highly scalable ...

Combinatorial optimization in DNA mapping — a computational thread of the Simplified Partial Digest Problem

In the paper, the problem of the genome mapping of DNA molecules, is presented. In particular, the new approach — the Simplified Partial Digest Problem (SPDP), is analyzed. This approach, although easy in laboratory implementation and robust with respect to measurement errors, when formulated in terms of a combinatorial search problem, is proved to be strongly NP-hard for the general error-free case. For a subproblem of the SPDP, a simple O(n logn )-time algorithm is given, where n is a number of restriction sites.

Optimal Boundary Triangulations of an Interpolating Ruled Surface

Abstract We investigate how to define a triangulated ruled surface interpolating two polygonal directrices that will meet a variety of optimization objectives which originate from many CAD/CAM and geometric modeling applications. This optimal triangulation problem is formulated as a combinatorial search problem whose search space however has the size tightly factorial to the numbers of points on the two directrices. To tackle this bound, we introduce a novel computational tool called multilayer directed graph and establish an equivalence between the optimal triangulation and the single-source shortest path problem on the graph. Well known graph search algorithms such as the Dijkstra’s are then employed to solve the single-source shortest path problem, which effectively solves the optimal triangulation problem in O(mn) time, where n and m are the numbers of vertices on the two directrices respectively. Numerous experimental examples are provided to demonstrate the usefulness of the proposed optimal triangulation problem in a variety of engineering applications.

Quantum search by measurement

We propose a quantum algorithm for solving combinatorial search problems that uses only a sequence of measurements. The algorithm is similar to quantum computation by adiabatic evolution, in that the goal is to remain in the ground state of a time-varying Hamiltonian. Indeed, we show that the running times of the two algorithms are closely related. We also show how to achieve the quadratic speedup for Grover's unstructured search problem with only two measurements. Finally, we discuss some similarities and differences between the adiabatic and measurement algorithms.

On a combinatorial search problem

Abstract In this paper, we consider the problem of searching for undirected Hamiltonian circuits in the complete graph on n vertices with the use of unconditional edge tests. We prove that the minimal test contains exactly n(n - 3)/2 - [n/3] + 1 edges. We propose an explicit characterisation of all minimal difference sets of edges. This research was supported by the Russian Foundation for Basic Research, grants 02-01-00985 and 00-15-96103, by the Program ‘Universities of Russia,’ and the Federal Program ‘Integration.’

Finding cliques by quantum adiabatic evolution

Quantum adiabatic evolution provides a general technique for the solution of combinatorial search problems on quantum computers. We present the results of a numerical study of a particular application of quantum adiabatic evolution, the problem of finding the largest clique in a random graph. An n-vertex random graph has each edge included with probability 1/2, and a clique is a completely connected subgraph. There is no known classical algorithm that finds the largest clique in a random graph with high probability and runs in a time polynomial in n. For the small graphs we are able to investigate ($n \le 18$), the quantum algorithm appears to require only a quadratic run time.

Robustness of adiabatic quantum computation

We study the fault tolerance of quantum computation by adiabatic evolution, a quantum algorithm for solving various combinatorial search problems. We describe an inherent robustness of adiabatic computation against two kinds of errors, unitary control errors and decoherence, and we study this robustness using numerical simulations of the algorithm.

A combinatorial search problem on matroids

Abstract In this paper we consider the worst-case adaptive complexity of the search problem (S, F ) , where F is also the set of independent sets of a matroid over S. We give a formula for the number of questions needed and an algorithm to find the optimal search algorithm for any matroid. This algorithm uses only O(|S|3) steps (i.e. questions to the independence oracle). This is also the length of Edmonds’ partitioning algorithm for matroids, which does not seem to be avoidable.

Modelling protein side-chain conformations using constraint logic programming

Side-chain placement is an important sub-task in protein modelling. Selecting conformations for side-chains is a difficult problem because of the large search space to be explored. This problem can be addressed using constraint logic programming (CLP), which is an artificial intelligence technique developed to solve large combinatorial search problems. The side-chain placement problem can be expressed as a CLP program in which rotamer conformations are used as values for finite domain variables, and bad steric contacts involving rotamers are represented as constraints. This paper introduces the concept of null rotamers, and shows how these can be used in implementing a novel iterative approach. We present results that compare the accuracy of models constructed using different rotamer libraries and different domain variable enumeration heuristics. The results obtained using this CLP-based approach compare favourably with those obtained by other methods.

Modular lazy search for Constraint Satisfaction Problems

We describe a unified, lazy, declarative framework for solving constraint satisfaction problems, an important subclass of combinatorial search problems. These problems are both practically significant and computationally hard. Finding solutions involves combining good general-purpose search algorithms with problem-specific heuristics. Conventional imperative algorithms are usually implemented and presented monolithically, which makes them hard to understand and reuse, even though new algorithms often are combinations of simpler ones. Lazy functional languages, such as Haskell, encourage modular structuring of search algorithms by separating the generation and testing of potential solutions into distinct functions communicating through an explicit, lazy intermediate data structure. But only relatively simple search algorithms have been treated this way in the past. Our framework uses a generic generation and pruning algorithm parameterized by a labeling function that annotates search trees with conflict sets. We show that many advanced imperative search algorithms, including conflict-directed backjumping, backmarking, minimal forward checking, and fail-first dynamic variable ordering, can be obtained by suitable instantiation of the labeling function. More importantly, arbitrary combinations of these algorithms can be built by simply composing their labeling functions. Our modular algorithms are as efficient as the monolithic imperative algorithms in the sense that they make the same number of consistency checks, and most of our algorithms are within a constant factor of their imperative counterparts in runtime and space usage. We believe our framework is especially well-suited for experimenting to find good combinations of algorithms for specific problems.

Algorithm portfolios

Stochastic algorithms are among the best methods for solving computationally hard search and reasoning problems. The run time of such procedures can vary significantly from instance to instance and, when using different random seeds, on the same instance. One can take advantage of such differences by combining several algorithms into a portfolio, and running them in parallel or interleaving them on a single processor. We provide an evaluation of the portfolio approach on distributions of hard combinatorial search problems. We show under what conditions the portfolio approach can have a dramatic computational advantage over the best traditional methods. In particular, we will see how, in a portfolio setting, it can be advantageous to use a more “risk-seeking” strategy with a high variance in run time, such as a randomized depth-first search approach in mixed integer programming versus the more traditional best-bound approach. We hope these insights will stimulate the development of novel randomized combinatorial search methods.

Scheduling a Major College Basketball Conference—Revisited

Nemhauser and Trick presented the problem of finding a timetable for the 1997/1998 Atlantic Coast Conference (ACC) in basketball. Their solution, found with a combination of integer programming and exhaustive enumeration, was accepted by the ACC. Finite-domain constraint programming is another programming technique that can be used for solving combinatorial search problems such as sports tournament scheduling. This paper presents a solution of round-robin tournament planning based on finite-domain constraint programming. The approach yields a dramatic performance improvement, which makes an integrated interactive software solution feasible.

Enhancing the performance of constraint programming through the introduction of linear programming

Constraint Programming (CP) has been successful in a number of combinatorial search and discrete optimisation problems. Yet other more traditional approaches, such as Integer Programming (IP), can still give a better performance on the same problem types. Central to IP's success is its reliance on a fast Linear Programming (LP) solver providing solutions during the search to the corresponding relaxed problems. These solutions are used to guide the search within IP as well as a means of detecting infeasibility and integrality. This paper shows that there is scope also to include LP within the CP framework, in order to similarly guide the CP search. The problems examined here are one for which CP on its own had proved markedly inferior to IP. Hence a hybrid solver based on the CP search and using an LP solver is configured and run on these problems. The outcome shows that using the LP solver within the CP search is a valuable addition to the available search strategies. An improved performance over the CP-only strategies is obtained and, further, comparable results are obtained to those from IP. Overall, CP+LP can be considered as a more robust approach than either CP or IP on their own on a variety of combinatorial search problems.

Complexity-theoretic models of phase transitions in search problems

In recent years, numerous studies have observed that many hard combinatorial decision problems exhibit behaviour described as a ‘ phase-transition’. This is the phenomenon whereby typical instances of a problem display a dramatic shift in certain characteristics as some parameter of the instances is varied. Such characteristics include the likelihood of an instance having a solution and the time taken by a search algorithm. The apparent pervasiveness of phase-transitions in hard combinatorial search problems has led to contrasting claims being advanced concerning to what extent all NP-complete problems exhibit phase-transitions. The established importance of exploiting phase-transition effects in the design of search heuristics provides a strong motivation for assessing how valid such claims may be. In this paper we argue that questions concerning the generality of phase-transition phenomena are, at present, ill-defined. In order to address this difficulty, we propose and examine rigorous complexity-theoretic models of the statement ‘the decision problem D has a phase-transition’. Within these models it is proved that for certain ‘natural’ definitions contrasting results about phase-transition behaviour can be proved.

Building tractable disjunctive constraints

Many combinatorial search problems can be expressed as 'constraint satisfaction problems'. This class of problems is known to be NP-hard in general, but a number of restricted constraint classes have been identified which ensure tractability. This paper presents the first general results on combining tractable constraint classes to obtain larger, more general, tractable classes. We give examples to show that many known examples of tractable constraint classes, from a wide variety of different contexts, can be constructed from simpler tractable classes using a general method. We also construct several new tractable classes that have not previously been identified.

Constraint propagation techniques for the disjunctive scheduling problem

Constraint propagation is an elementary method for reducing the search space of combinatorial search and optimization problems which has become more and more important in the last decades. The basic idea of constraint propagation is to detect and remove inconsistent variable assignments that cannot participate in any feasible solution through the repeated analysis and evaluation of the variables, domains and constraints describing a specific problem instance. The contribution of this paper is twofold. The first contribution is a description of efficient constraint propagation methods also known as consistency tests for the disjunctive scheduling problem (DSP) which is a generalization of the classical job shop scheduling problem (JSP). Applying an elementary constraint based approach involving a limited number of search variables, we will derive consistency tests that ensure 3 -b-consistency. We will further present and analyze both new and classical consistency tests which to some extent are generalizations of the aforementioned consistency tests involving a higher number of variables, but still can be implemented efficiently with a polynomial time complexity. Further, the concepts of energetic reasoning and shaving are analyzed and discussed. The other contribution is a classification of the consistency tests derived according to the domain reduction achieved. The particular strength of using consistency tests is based on their repeated application, so that the knowledge derived is propagated, i.e., reused for acquiring additional knowledge. The deduction of this knowledge can be described as the computation of a fixed point. Since this fixed point depends upon the order of the application of the tests, we first derive a necessary condition for its uniqueness. We then develop a concept of dominance which enables the comparison of different consistency tests as well as a simple method for proving dominance. An extensive comparison of all consistency tests is given. Quite surprisingly, we will find out that some apparently stronger consistency tests are subsumed by apparently weaker ones. At the same time an open question regarding the effectiveness of energetic reasoning is answered.

LOCAL++: A C++ framework for local search algorithms

Local search is an emerging paradigm for combinatorial search which has recently been shown to be very effective for a large number of combinatorial problems. It is based on the idea of navigating the search space by iteratively stepping from one solution to one of its neighbors, which are obtained by applying a simple local change to it. In this paper we present LOCAL++, an object-oriented framework to be used as a general tool for the development and implementation of local search algorithms in C++. The framework comprises a hierarchy of abstract template classes, one for each local search technique taken into account (i.e. hill-climbing, simulated annealing and tabu search). Each class specifies and implements the invariant part of the algorithm built according to the technique, and is supposed to be specialized by a concrete class once a given search problem is considered, so as to implement the problem-dependent part of the algorithm. LOCAL++ comprises also a set of abstract classes for creating new techniques by combining different search techniques and different neighborhood relations. The architecture of LOCAL++ provides a principled modularization for the solution of combinatorial search problems, and helps the designer deriving a neat conceptual scheme of the application, thus facilitating the development and debugging phases. LOCAL++ proved to be flexible enough for the implementation of the algorithms solving various scheduling problems. Copyright © 2000 John Wiley & Sons, Ltd.

LOCAL++: A C++ framework for local search algorithms

Local search is an emerging paradigm for combinatorial search which has recently been shown to be very effective for a large number of combinatorial problems. It is based on the idea of navigating the search space by iteratively stepping from one solution to one of its neighbors, which are obtained by applying a simple local change to it. In this paper we present LOCAL++, an object-oriented framework to be used as a general tool for the development and implementation of local search algorithms in C++. The framework comprises a hierarchy of abstract template classes, one for each local search technique taken into account (i.e. hill-climbing, simulated annealing and tabu search). Each class specifies and implements the invariant part of the algorithm built according to the technique, and is supposed to be specialized by a concrete class once a given search problem is considered, so as to implement the problem-dependent part of the algorithm. LOCAL++ comprises also a set of abstract classes for creating new techniques by combining different search techniques and different neighborhood relations. The architecture of LOCAL++ provides a principled modularization for the solution of combinatorial search problems, and helps the designer deriving a neat conceptual scheme of the application, thus facilitating the development and debugging phases. LOCAL++ proved to be flexible enough for the implementation of the algorithms solving various scheduling problems. Copyright © 2000 John Wiley & Sons, Ltd.

Minimization of Makespan in Generalized Assignment Problem

A quantitative combinatorial search problem consisting of m sources supplying in bulk to n(> m) destinations is considered in this paper. Each destination receives its full quota of a homogeneous product from a single source but a source can supply to many destinations subject to its capacity restrictions. If tij, which without loss of generality can be assumed to be a positive integer, is the time of transportation for shipping the goods from the source i to the destination j, then the total time used by the ith source is sum of the transportation time to the destinations that are served by this source. The time of transportation, called makespan, for a feasible solution of this generalized assignment problem (also called bulk transportation problem) is the maximum total time used by a source. The objective is to find a feasible solution that minimizes the makespan. A lexicographic search algorithm is developed which avoids scanning through all the possible feasible solutions though the search is exhaustive enough to obtain an optimal feasible solution at an early stage of the search. A starting upper bound on the value of objective function computed heauristically is used as a supplementary aid in reducing the search. Some results are established which, though of computational nature, help to reduce both the memory requirements and computational time.