Aleksandar Shurbevski

Approximating the Bipartite TSP and Its Biased Generalization

We examine a generalization of the symmetric bipartite traveling salesman problem (TSP) with quadrangle inequality, by extending the cost function of a Hamiltonian tour to include a bias factor β ≥ 1. The bias factor is known and given as a part of the input. We propose a novel heuristic procedure for building Hamiltonian cycles in bipartite graphs, and show that it is an approximation algorithm for the generalized problem with an approximation ratio of \(1+\frac{1+\lambda}{\beta+\lambda}\), where λ is a real parameter dependent on the problem instance. This expression is bounded above by a constant 2, for any positive real λ and β ≥ 1, which improves a previously reported approximation ratio of 16/7. As a part of a composite heuristic, the proposed procedure can contribute to an approximation ratio of \(1+\frac{2}{\zeta+\beta(2-\zeta)}\), where ζ is an approximation ratio for the metric TSP.

Optimization Techniques for Robot Path Planning

We present a method for robot path planning in the robot’s configuration space, in the presence of fixed obstacles. Our method employs both combinatorial and gradient-based optimization techniques, but most distinguishably, it employs a Multi-sphere Scheme purposefully developed for two and three-dimensional packing problems. This is a singular feature which not only enables us to use a particularly high-grade implementation of a packing-problem solver, but can also be utilized as a model to reduce computational effort with other path-planning or obstacle avoidance methods.

Enumerating Chemical Mono-Block 3-Augmented Trees with Two Junctions

Enumerating chemical graphs with given constraints on their topological structure is a fundamental problem in chemoinformatics and bioinformatics, with a wide range of applications, including structure determination and designing novel chemical compounds. We specify the constraint on the structure by the frequencies of all paths up to a fixed length in the graph. Given upper and lower bounds, we consider the problem of enumerating chemical graphs whose path frequency satisfies one withing the given bounds. A k-augmented tree is a connected multigraph where the number of pairs of adjacent vertices minus that of vertices is k-1, and a mono-block k-augmented tree is a k-augmented tree that contains one biconnected component. The biconnected component of a mono-block 3-augmented tree with two junctions contains two vertices joined by four internally disjoint paths. We design an algorithm for enumerating all mono-block 3-augmented trees with two junctions that satisfy given upper and lower bounds on path frequencies. Experimental results indicate that our algorithm performs favorably when compared to a state-of-the-art production program, and we obtain large numbers of structures with up to 35 atoms excluding hydrogen, which has not been possible with existing methods.

Routing of Carrier-vehicle Systems with Dedicated Last-stretch Delivery Vehicle and Fixed Carrier Route

We examine a routing problem that arises when an unmanned aerial vehicle (UAV), or drone, is used in the last-stretch of parcel delivery to end customers. In the scenario that we study, a delivery truck is dispatched carrying a shipment of parcels to be delivered to customers. While the truck is following a predetermined route, a drone is charged with making the last-stretch delivery of a parcel from the truck to a customer’s doorstep. Given a set of customers to be served and a set of rendezvous points where the drone can meet with the truck to pick up a parcel, we ask what the quickest way is of delivering all parcels to the end customers. We model this problem as a problem of finding a special type of a path in a graph of a special structure, and show that the graph problem is NP-hard even when all edge weights are restricted to be 1 or 2. Furthermore, we identify a special instance type that can be solved optimally in polynomial time. Finally, we propose a polynomial-time approximation algorithm for the graph problem in metric graphs, and show that its approximation ratio is bounded above by 2 in restricted metric graphs.

An Improved-time Polynomial-space Exact Algorithm for TSP in Degree-5 Graphs

The Traveling Salesman Problem (TSP) is one of the most well-known NP-hard optimization problems. Following a recent trend of research which focuses on developing algorithms for special types of TSP instances, namely graphs of limited degree, in an attempt to reduce a part of the time and space complexity, we present a polynomial-space branching algorithm for the TSP in an n-vertex graph with degree at most 5, and show that it has a running time of O∗(2.3500n), which improves the previous best known time bound of O∗(2.4723n) given by the authors (the 12th International Symposium on Operations Research and Its Application (ISORA 2015), pp.45–58, 2015). While the base of the exponent in the running time bound of our algorithm is greater than 2, it still outperforms Gurevich and Shelah’s O∗(4nnlog n) polynomial-space exact algorithm for the TSP in general graphs (SIAM Journal of Computation, Vol.16, No.3, pp.486–502, 1987). In the analysis of the running time, we use the measure-and-conquer method, and we develop a set of branching rules which foster the analysis of the running time.

Parameterization of Strategy-Proof Mechanisms in the Obnoxious Facility Game

In the obnoxious facility game, a location for an undesirable facility is to be determined based on the voting of selfish agents. The design of group strategy proof mechanisms has been extensively studied, and it is known that there exists a gap between the social benefit (i.e., the sum of individual benefits) by a facility location determined by any group strategy proof mechanism and the maximum social benefit over all choices of facility locations; their ratio, called the benefit ratio can be 3 in the line metric space. In this paper, we investigate a trade-off between the benefit ratio and a possible relaxation of group strategy proofness, taking 2-candidate mechanisms for the obnoxious facility game in the line metric as an example. Given a real \(\lambda \ge 1\) as a parameter, we introduce a new strategy proofness, called “\(\lambda \)-group strategy-proofness,” so that each coalition of agents has no incentive to lie unless every agent in the group can increase her benefit by strictly more than \(\lambda \) times by doing so, where the 1-group strategy-proofness is the previously known group strategy-proofness. We next introduce “masking zone mechanisms,” a new notion on structure of mechanisms, and prove that every \(\lambda \)-group strategy-proof (\(\lambda \)-GSP) mechanism is a masking zone mechanism. We then show that, for any \(\lambda \ge 1\), there exists a \(\lambda \)-GSP mechanism whose benefit ratio is at most \(1+\frac{2}{\lambda }\), which converges to 1 as \(\lambda \) becomes infinitely large. Finally we prove that the bound is nearly tight: given \(n \ge 1\) selfish agents, the benefit ratio of \(\lambda \)-GSP mechanisms cannot be better than \(1+\frac{2}{\lambda }\) when n is even, and \(1 + \frac{2n-2}{\lambda n + 1}\) when n is odd.

A Polynomial-space Exact Algorithm for TSP in Degree-6 Graphs

This paper presents the first polynomial-space exact algorithm specialized for the TSP in graphs with degree at most 6. We develop a set of branching rules to aid the analysis of the branching algorithm. Using the measure-and-conquer method, we show that when applied to an n-vertex graph with degree at most 6, the algorithm has a running time of \(O^*(3.0335^n)\), which is still advantageous over other known polynomial-space algorithms for the TSP in general graphs.