arXiv subjects

Abraham P. Punnen

Publications and source records attributed to Abraham P. Punnen.

24 records · Page 2Linked to original sources

arXiv2013

Integrating tabu search and VLSN search to develop enhanced algorithms: A case study using bipartite boolean quadratic programs

The bipartite boolean quadratic programming problem (BBQP) is a generalization of the well studied boolean quadratic programming problem. The model has a variety of real life applications; however, empirical studies of the model are not available in the literature, except in a few isolated instances. In this paper, we develop efficient heuristic algorithms based on tabu search, very large scale neighborhood (VLSN) search, and a hybrid algorithm that integrates the two. The computational study establishes that effective integration of simple tabu search with VLSN search results in superior outcomes, and suggests the value of such an integration in other settings. Complexity analysis and implementation details are provided along with conclusions drawn from experimental analysis. In addition, we obtain solutions better than the best previously known for almost all medium and large size benchmark instances.

cs.AI↗

arXiv2013

Average value of solutions for the bipartite boolean quadratic programs and rounding algorithms

We consider domination analysis of approximation algorithms for the bipartite boolean quadratic programming problem (BBQP) with m+n variables. A closed form formula is developed to compute the average objective function value A of all solutions in O(mn) time. However, computing the median objective function value of the solutions is shown to be NP-hard. Also, we show that any solution with objective function value no worse than A dominates at least 2^{m+n-2} solutions and this bound is the best possible. Further, we show that such a solution can be identified in O(mn) time and hence the dominance ratio of this algorithm is at least 1/4. We then show that for any fixed rational number a > 1, no polynomial time approximation algorithm exists for BBQP with dominance ratio larger than 1-2^{(m+n)(1-a)/a}, unless P=NP. We then analyze some powerful local search algorithms and show that they can get trapped at a local maximum with objective function value less than A. One of our approximation algorithms has an interesting rounding property which provides a data dependent lower bound on the optimal objective function value. A new integer programming formulation of BBQP is also given and computational results with our rounding algorithms are reported.

math.OC↗

arXiv2012

A reduced integer programming model for the ferry scheduling problem

We present an integer programming model for the ferry scheduling problem, improving existing models in various ways. In particular, our model has reduced size in terms of the number of variables and constraints compared to existing models by a factor of approximately O(n), where n being the number of ports. The model also handles efficiently load/unload time constraints, crew scheduling and passenger transfers. Experiments using real world data produced high quality solutions in 12 hours using CPLEX 12.4 with a performance guarantee of within 15% of optimality, on average. This establishes that using a general purpose integer programming solver is a viable alternative in solving the ferry scheduling problem of moderate size.

cs.DM↗

arXiv2007

Bottleneck flows in networks

The bottleneck network flow problem (BNFP) is a generalization of several well-studied bottleneck problems such as the bottleneck transportation problem (BTP), bottleneck assignment problem (BAP), bottleneck path problem (BPP), and so on. In this paper we provide a review of important results on this topic and its various special cases. We observe that the BNFP can be solved as a sequence of $O(\log n)$ maximum flow problems. However, special augmenting path based algorithms for the maximum flow problem can be modified to obtain algorithms for the BNFP with the property that these variations and the corresponding maximum flow algorithms have identical worst case time complexity. On unit capacity network we show that BNFP can be solved in $O(\min \{{m(n\log n)}^{2/3}, m^{3/2}\sqrt{\log n}\})$. This improves the best available algorithm by a factor of $\sqrt{\log n}$. On unit capacity simple graphs, we show that BNFP can be solved in $O(m \sqrt {n \log n})$ time. As a consequence we have an $O(m \sqrt {n \log n})$ algorithm for the BTP with unit arc capacities.

cs.DS↗

arXiv2007

On the approximability of the vertex cover and related problems

In this paper we show that the problem of identifying an edge $(i,j)$ in a graph $G$ such that there exists an optimal vertex cover $S$ of $G$ containing exactly one of the nodes $i$ and $j$ is NP-hard. Such an edge is called a weak edge. We then develop a polynomial time approximation algorithm for the vertex cover problem with performance guarantee $2-\frac{1}{1+σ}$, where $σ$ is an upper bound on a measure related to a weak edge of a graph. Further, we discuss a new relaxation of the vertex cover problem which is used in our approximation algorithm to obtain smaller values of $σ$. We also obtain linear programming representations of the vertex cover problem for special graphs. Our results provide new insights into the approximability of the vertex cover problem - a long standing open problem.

cs.DS↗

arXiv2007

A polynomial time $\frac 3 2$ -approximation algorithm for the vertex cover problem on a class of graphs

We develop a polynomial time 3/2-approximation algorithm to solve the vertex cover problem on a class of graphs satisfying a property called ``active edge hypothesis''. The algorithm also guarantees an optimal solution on specially structured graphs. Further, we give an extended algorithm which guarantees a vertex cover $S_1$ on an arbitrary graph such that $|S_1|\leq {3/2} |S^*|+ξ$ where $S^*$ is an optimal vertex cover and $ξ$ is an error bound identified by the algorithm. We obtained $ξ= 0$ for all the test problems we have considered which include specially constructed instances that were expected to be hard. So far we could not construct a graph that gives $ξ\not= 0$.

cs.DS↗