Exact Combinatorial Algorithm Research Articles

Optimal algorithms for inverse obnoxious center location problems under the weighted Chebyshev and Hamming cost norms on networks

This article deals with the inverse obnoxious center location problem on general networks in which the edge lengths are modified at the minimum overall cost with respect to given modification bounds such that a predetermined facility point becomes an obnoxious center location under the modified edge lengths on the perturbed network. Exact combinatorial algorithms with linear time complexities are developed for obtaining the optimal solutions of the problem under the weighted Chebyshev, bottleneck-type Hamming and sum-type Hamming cost norms.

An exact lexicographic approach for the maximally risk-disjoint/minimal cost path pair problem in telecommunication networks

The paper addresses the lexicographically maximal risk-disjoint/minimal cost path pair problem that aims at finding a pair of paths between two given nodes, which is the shortest (in terms of cost) among those that have the fewest risks in common. This problem is of particular importance in telecommunication network design, namely concerning resilient routing models where both a primary and a backup path have to be calculated to minimize the risk of failure of a connection between origin and terminal nodes, in case of failure along the primary path and where bandwidth routing costs should also be minimized. An exact combinatorial algorithm is proposed for solving this problem which combines a path ranking method and a path labelling algorithm. Also an integer linear programming (ILP) formulation is shown for comparison purposes. After a theoretical justification of the algorithm foundations, this is described and tested, together with the ILP procedure, for a set of reference networks in telecommunications, considering randomly generated risks, associated with Shared Risk Link Groups (SRLGs) and arc costs. Both methods were capable of solving the problem instances in relatively short times and, in general, the proposed algorithm was clearly faster than the ILP formulation excepting for the networks with the greatest dimension and connectivity.

Generalized Langevin Equations for Systems with Local Interactions

We present a new method to approximate the Mori-Zwanzig (MZ) memory integral in generalized Langevin equations (GLEs) describing the evolution of smooth observables in high-dimensional nonlinear systems with local interactions. Building upon the Faber operator series we recently developed for the orthogonal dynamics propagator, and an exact combinatorial algorithm that allows us to compute memory kernels from first principles, we demonstrate that the proposed method is effective in computing auto-correlation functions, intermediate scattering functions and other important statistical properties of the observable. We also develop a new stochastic process representation of the MZ fluctuation term for systems in statistical equilibrium. Numerical applications are presented for the Fermi-Pasta-Ulam model, and for random wave propagation in homogeneous media.

Exact algorithms for finding well-connected 2-clubs in sparse real-world graphs: Theory and experiments

Finding large “cliquish” subgraphs is a central topic in graph mining and community detection. A popular clique relaxation are 2-clubs: instead of asking for subgraphs of diameter one (these are cliques), one asks for subgraphs of diameter at most two (these are 2-clubs). A drawback of the 2-club model is that it produces star-like hub-and-spoke structures as maximum-cardinality solutions. Hence, we study 2-clubs with the additional constraint to be well-connected. More specifically, we investigate the algorithmic complexity for three variants of well-connected 2-clubs, all established in the literature: robust, hereditary, and “connected” 2-clubs. Finding these more cohesive 2-clubs is NP-hard; nevertheless, we develop an exact combinatorial algorithm, extensively using efficient data reduction rules. Besides several theoretical insights we provide a number of empirical results based on an engineered implementation of our exact algorithm. In particular, the algorithm significantly outperforms existing algorithms on almost all (sparse) real-world graphs we considered.

Optimal Algorithms for Integer Inverse Undesirable p-Median Location Problems on Weighted Extended Star Networks

This paper is concerned with the problem of modifying the edge lengths of a weighted extended star network with n vertices by integer amounts at the minimum total cost subject to be given modification bounds so that a set of p prespecified vertices becomes an undesirable p-median location on the perturbed network. We call this problem as the integer inverse undesirable p-median location model. Exact combinatorial algorithms with $$ {\mathcal {O}}\left( p^2 n \log n\right) $$ and $${{\mathcal {O}}}\left( p^{2}(n \log n+ n \log \eta _{\max })\right) $$ running times are proposed for solving the problem under the weighted rectilinear and weighted Chebyshev norms, respectively. Furthermore, it is shown that the problem under the weighted sum-type Hamming distance with uniform modification bounds can be solved in $$ {\mathcal {O}}(p^2n \log n)$$ time.

On Finding the Maximum Feasible Subsystem of a System of Linear Inequalities

Some methods for finding the maximum feasible subsystems of systems of linear inequalities are considered. The problem of finding the most accurate algorithm in a parametric family of linear classification algorithms is one of the most important problems in machine learning. In order to solve this discrete optimization problem, an exact (combinatorial) algorithm, its approximations (relaxation and greedy combinatorial descent algorithms), and the approximation algorithm are given. The latter consists in replacing the original discrete optimization problem with a nonlinear programming problem by changing from linear inequalities to their sigmoid functions. The initial results of their comparison are presented.

An efficient algorithm for the single facility location problem with polyhedral norms and disk-shaped demand regions

The single facility location problem with demand regions seeks for a facility location minimizing the sum of the distances from n demand regions to the facility. The demand regions represent sales markets where the transportation costs are negligible. In this paper, we assume that all demand regions are disks of the same radius, and the distances are measured by a rectilinear norm, e.g. ell _1 or ell _infty . We develop an exact combinatorial algorithm running in time O(nlog ^c n) for some c dependent only on the space dimension. The algorithm is generalizable to the other polyhedral norms.

Linear time optimal approaches for reverse obnoxious center location problems on networks

This paper is concerned with a reverse obnoxious (undesirable) center location problem on networks in which the aim is to modify the edge lengths within an associated budget such that a predetermined facility location on the underlying network becomes as far as possible from the existing customer points under the new edge lengths. Exact combinatorial algorithms with linear time complexities are developed for the problem under the weighted rectilinear norm and the weighted Hamming distance. Furthermore, it is shown that the problem with integer decision variables can also be solved in linear time.

Combinatorial approach to exactly solving discrete and hybrid berth allocation problem

This paper presents an exact combinatorial algorithm for solving the Discrete Berth Allocation Problem (DBAP) and the Hybrid Berth Allocation Problem (HBAP) with fixed handling times of vessels based on the original algorithm for solving combinatorial problems called Sedimentation Algorithm. We address the issues of DBAP and HBAP according to the Rashidi and Tsang model. To the best of our knowledge, the proposed algorithm is the first exact combinatorial algorithm for solving the general DBAP and HBAP based on Rashidi and Tsang model. Computational results prove the superiority of the proposed algorithms compared with the exact solvers based on the Mixed Integer Programming (MIP) models. Efficient C implementation enabled us to solve instances with up to 65 vessels. This resolves most of the real life problems, even in large ports.

An exact combinatorial algorithm for minimum graph bisection

We present a novel exact algorithm for the minimum graph bisection problem, whose goal is to partition a graph into two equally-sized cells while minimizing the number of edges between them. Our algorithm is based on the branch-and-bound framework and, unlike most previous approaches, it is fully combinatorial. We introduce novel lower bounds based on packing trees, as well as a new decomposition technique that contracts entire regions of the graph while preserving optimality guarantees. Our algorithm works particularly well on graphs with relatively small minimum bisections, solving to optimality several large real-world instances (with up to millions of vertices) for the first time.

An exact algorithm for the maximum probabilistic clique problem

The maximum clique problem is a classical problem in combinatorial optimization that has a broad range of applications in graph-based data mining, social and biological network analysis and a variety of other fields. This article investigates the problem when the edges fail independently with known probabilities. This leads to the maximum probabilistic clique problem, which is to find a subset of vertices of maximum cardinality that forms a clique with probability at least $$\theta \in [0,1]$$ ? ? [ 0 , 1 ] , which is a user-specified probability threshold. We show that the probabilistic clique property is hereditary and extend a well-known exact combinatorial algorithm for the maximum clique problem to a sampling-free exact algorithm for the maximum probabilistic clique problem. The performance of the algorithm is benchmarked on a test-bed of DIMACS clique instances and on a randomly generated test-bed.

Information-theoretic approach to ground-state phase transitions for two- and three-dimensional frustrated spin systems

The information-theoretic observables entropy (a measure of disorder), excess entropy (a measure of complexity), and multi-information are used to analyze ground-state spin configurations for disordered and frustrated model systems in two and three dimensions. For both model systems, ground-state spin configurations can be obtained in polynomial time via exact combinatorial optimization algorithms, which allowed us to study large systems with high numerical accuracy. Both model systems exhibit a continuous transition from an ordered to a disordered ground state as a model parameter is varied. By using the above information-theoretic observables it is possible to detect changes in the spatial structure of the ground states as the critical point is approached. It is further possible to quantify the scaling behavior of the information-theoretic observables in the vicinity of the critical point. For both model systems considered, the estimates of critical properties for the ground-state phase transitions are in good agreement with existing results reported in the literature.

Exact combinatorial algorithms and experiments for finding maximum k-plexes

We propose new practical algorithms to find maximum-cardinality k-plexes in graphs. A k-plex denotes a vertex subset in a graph inducing a subgraph where every vertex has edges to all but at most k vertices in the k-plex. Cliques are 1-plexes. In analogy to the special case of finding maximum-cardinality cliques, finding maximum-cardinality k-plexes is NP-hard. Complementing previous work, we develop exact combinatorial algorithms, which are strongly based on methods from parameterized algorithmics. The experiments with our freely available implementation indicate the competitiveness of our approach, for many real-world graphs outperforming the previously used methods.

Topological optimization of the large-scale data transmission networks

Consideration was given to the design of the topological structure of the doubly connected large-scale networks by the cost criterion and constraints on the diameter. An exact combinatorial algorithm to solve the problem and a parallel algorithm to carry out calculations on the multiprocessor data-flow computer system were proposed. Parallelization was shown to provide a substantial gain in the time of computations in the optimal topological structure of the large-scale networks.

A primal-dual algorithm for computing Fisher equilibrium in the absence of gross substitutability property

We provide the first strongly polynomial time exact combinatorial algorithm to compute Fisher equilibrium for the case when utility functions do not satisfy the Gross substitutability property. The motivation for this comes from the work of Kelly, Maulloo, and Tan [F.P. Kelly, A.K. Maulloo, D.K.H. Tan, Rate control for communication networks: Shadow prices, proportional fairness and stability, Journal of Operational Research (1998)] and Kelly and Vazirani [F.P. Kelly, Vijay V. Vazirani, Rate control as a market equilibrium (2003) (in preparation)] on rate control in communication networks. We consider a tree like network in which root is the source and all the leaf nodes are the sinks. Each sink has got a fixed amount of money which it can use to buy the capacities of the edges in the network. The edges of the network sell their capacities at certain prices. The objective of each edge is to fix a price that can fetch the maximum money for it, and the objective of each sink is to buy capacities on edges in such a way that it can facilitate the sink to pull maximum flow from the source. In this problem, the edges and the sinks play precisely the role of sellers and buyers, respectively, in Fisher’s market model. The utility of a buyer (or sink) takes the form of a Leontief function which is known for not satisfying Gross substitutability property. We develop an O ( m 3 ) exact combinatorial algorithm for computing equilibrium prices of the edges. The time taken by our algorithm is independent of the values of sink money and edge capacities. A corollary of our algorithm is that equilibrium prices and flows are rational numbers. Although there are algorithms to solve this problem they are all based on convex programming techniques. To the best of our knowledge, ours is the first strongly polynomial time exact combinatorial algorithm for computing equilibrium prices of Fisher’s model under the case when buyers’ utility functions do not satisfy gross substitutability property.

Application of exact combinatorial optimization algorithms to the physics of disordered systems

We discuss various applications of exact combinatorial optimization methods in the statistical physics of disordered systems. The physical systems we consider are elastic manifolds and elastic media with and without periodic potential. The algorithms we use find the exact ground states of these systems in polynomial time and allow for a thorough finite size scaling analysis that gives accurate results for the universal properties, i.e. in particular critical exponents, of these systems.