arXiv subjects

Abraham P. Punnen

Publications and source records attributed to Abraham P. Punnen.

At least 19 recordsLinked to original sources

arXiv2024

Revisiting some classical linearizations of the quadratic binary optimization problem

In this paper, we present several new linearizations of a quadratic binary optimization problem (QBOP), primarily using the method of aggregations. Although aggregations were studied in the past in the context of solving system of Diophantine equations in non-negative variables, none of the approaches developed produced practical models, particularly due to the large size of associate multipliers. Exploiting the special structure of QBOP we show that selective aggregation of constraints provide valid linearizations with interesting properties. For our aggregations, multipliers can be any non-zero real numbers. Moreover, choosing the multipliers appropriately, we demonstrate that the resulting LP relaxations have value identical to the corresponding non-aggregated models. We also provide a review of existing explicit linearizations of QBOP and presents the first systematic study of such models. Theoretical and experimental comparisons of new and existing models are also provided.

cs.DM↗

arXiv2022

Multi-strip observation scheduling problem for ac-tive-imaging agile earth observation satellites

Active-imaging agile earth observation satellite (AI-AEOS) is a new generation agile earth observation satellite (AEOS). With renewed capabilities in observation and active im-aging, AI-AEOS improves upon the observation capabilities of AEOS and provide additional ways to observe ground targets. This however makes the observation scheduling problem for these agile earth observation satellite more complex, especially when considering multi-strip ground targets. In this paper, we investigate the multi-strip observation scheduling problem for an active-image agile earth observation satellite (MOSP). A bi-objective optimization model is presented for MOSP along with an adaptive bi-objective memetic algorithm which integrates the combined power of an adaptive large neighborhood search algorithm (ALNS) and a nondominated sorting genetic algorithm II (NSGA-II). Results of extensive computa-tional experiments are presented which disclose that ALNS and NSGA-II when worked in unison produced superior outcomes. Our model is more versatile than existing models and provide enhanced capabilities in applied problem solving.

cs.AI↗

arXiv2022

An adaptive bi-objective optimization algorithm for the satellite image data downlink scheduling problem considering request split

The satellite image data downlink scheduling problem (SIDSP) is well studied in literature for traditional satellites. With recent developments in satellite technology, SIDSP for modern satellites became more complicated, adding new dimensions of complexities and additional opportunities for the effective use of the satellite. In this paper, we introduce the dynamic two-phase satellite image data downlink scheduling problem (D-SIDSP) which combines two interlinked operations of image data segmentation and image data downlink, in a dynamic way, and thereby offering additional modelling flexibility and renewed capabilities. D-SIDSP is formulated as a bi-objective problem of optimizing the image data transmission rate and the service-balance degree. Harnessing the power of an adaptive large neighborhood search algorithm (ALNS) with a nondominated sorting genetic algorithm II (NSGA-II), an adaptive bi-objective memetic algorithm, ALNS+NSGA-II, is developed to solve D-SIDSP. Results of extensive computational experiments carried out using benchmark instances are also presented. Our experimental results disclose that the algorithm ALNS+NSGA-II is a viable alternative to solve D-SIDSP more efficiently and demonstrates superior outcomes based on various performance metrics. The paper also offers new benchmark instances for D-SIDSP that can be used in future research works on the topic.

cs.CV↗

arXiv2018

A characterization of Linearizable instances of the Quadratic Traveling Salesman Problem

We consider the linearization problem associated with the quadratic traveling salesman problem (QTSP). Necessary and sufficient conditions are given for a cost matrix $Q$ of QTSP to be linearizable. It is shown that these conditions can be verified in $O(n^5)$ time. Some simpler sufficient conditions for linearization are also given along with related open problems.

cs.DM↗

arXiv2018

Representations of quadratic combinatorial optimization problems: A case study using the quadratic set covering problem

The objective function of a quadratic combinatorial optimization problem (QCOP) can be represented by two data points, a quadratic cost matrix Q and a linear cost vector c. Different, but equivalent, representations of the pair (Q, c) for the same QCOP are well known in literature. Research papers often state that without loss of generality we assume Q is symmetric, or upper-triangular or positive semidefinite, etc. These representations however have inherently different properties. Popular general purpose 0-1 QCOP solvers such as GUROBI and CPLEX do not suggest a preferred representation of Q and c. Our experimental analysis discloses that GUROBI prefers the upper triangular representation of the matrix Q while CPLEX prefers the symmetric representation in a statistically significant manner. Equivalent representations, although preserve optimality, they could alter the corresponding lower bound values obtained by various lower bounding schemes. For the natural lower bound of a QCOP, symmetric representation produced tighter bounds, in general. Effect of equivalent representations when CPLEX and GUROBI run in a heuristic mode are also explored. Further, we review various equivalent representations of a QCOP from the literature that have theoretical basis to be viewed as strong and provide new theoretical insights for generating such equivalent representations making use of constant value property and diagonalization (linearization) of QCOP instances.

math.OC↗

arXiv2017

Bilinear Assignment Problem: Large Neighborhoods and Experimental Analysis of Algorithms

The bilinear assignment problem (BAP) is a generalization of the well-known quadratic assignment problem (QAP). In this paper, we study the problem from the computational analysis point of view. Several classes of neigborhood structures are introduced for the problem along with some theoretical analysis. These neighborhoods are then explored within a local search and a variable neighborhood search frameworks with multistart to generate robust heuristic algorithms. Results of systematic experimental analysis have been presented which divulge the effectiveness of our algorithms. In addition, we present several very fast construction heuristics. Our experimental results disclosed some interesting properties of the BAP model, different from those of comparable models. This is the first thorough experimental analysis of algorithms on BAP. We have also introduced benchmark test instances that can be used for future experiments on exact and heuristic algorithms for the problem.

cs.DS↗

arXiv2017

The generalized vertex cover problem and some variations

In this paper we study the generalized vertex cover problem (GVC), which is a generalization of various well studied combinatorial optimization problems. GVC is shown to be equivalent to the unconstrained binary quadratic programming problem and also equivalent to some other variations of the general GVC. Some solvable cases are identified and approximation algorithms are suggested for special cases. We also study GVC on bipartite graphs and identify some polynomially solvable cases. We show that GVC on bipartite graphs is equivalent to the bipartite unconstrained 0-1 quadratic programming problem. Integer programming formulations of GVC and related problems are presented and establish half-integrality property on some variables for the corresponding linear programming relaxations. We also discuss special cases of GVC where all feasible solutions are independent sets or vertex covers. These problems are observed to be equivalent to the maximum weight independent set problem or minimum weight vertex cover problem along with some algorithmic results.

cs.CC↗

arXiv2017

Combinatorial Optimization Problems with Interaction Costs: Complexity and Solvable Cases

We introduce and study the combinatorial optimization problem with interaction costs (COPIC). COPIC is the problem of finding two combinatorial structures, one from each of two given families, such that the sum of their independent linear costs and the interaction costs between elements of the two selected structures is minimized. COPIC generalizes the quadratic assignment problem and many other well studied combinatorial optimization problems, and hence covers many real world applications. We show how various topics from different areas in the literature can be formulated as special cases of COPIC. The main contributions of this paper are results on the computational complexity and approximability of COPIC for different families of combinatorial structures (e.g. spanning trees, paths, matroids), and special structures of the interaction costs. More specifically, we analyze the complexity if the interaction cost matrix is parameterized by its rank and if it is a diagonal matrix. Also, we determine the structure of the intersection cost matrix, such that COPIC is equivalent to independently solving linear optimization problems for the two given families of combinatorial structures.

math.OC↗

arXiv2016

Markov Chain methods for the bipartite Boolean quadratic programming problem

We study the Bipartite Boolean Quadratic Programming Problem (BBQP) which is an extension of the well known Boolean Quadratic Programming Problem (BQP). Applications of the BBQP include mining discrete patterns from binary data, approximating matrices by rank-one binary matrices, computing the cut-norm of a matrix, and solving optimisation problems such as maximum weight biclique, bipartite maximum weight cut, maximum weight induced sub-graph of a bipartite graph, etc. For the BBQP, we first present several algorithmic components, specifically, hill climbers and mutations, and then show how to combine them in a high-performance metaheuristic. Instead of hand-tuning a standard metaheuristic to test the efficiency of the hybrid of the components, we chose to use an automated generation of a multi-component metaheuristic to save human time, and also improve objectivity in the analysis and comparisons of components. For this we designed a new metaheuristic schema which we call Conditional Markov Chain Search (CMCS). We show that CMCS is flexible enough to model several standard metaheuristics; this flexibility is controlled by multiple numeric parameters, and so is convenient for automated generation. We study the configurations revealed by our approach and show that the best of them outperforms the previous state-of-the-art BBQP algorithm by several orders of magnitude. In our experiments we use benchmark instances introduced in the preliminary version of this paper and described here, which have already become the de facto standard in the BBQP literature.

cs.DS↗

arXiv2016

On a linearization technique for solving quadratic set covering problem and variations

In this paper we identify some inaccuracies in the paper by R.R. Saxena and S.R. Arora, A Linearization technique for solving the Quadratic Set Covering Problem, Optimization, 39 (1997) 33-42. In particular, we observe that their algorithm need not guarantee optimality, contrary to what is claimed. Experimental analysis with the algorithm has been carried out to evaluate its merit as a heuristic and compared with CPLEX. The results disclose that for some class of problems the algorithm is reasonably effective while for some other class, it's performance is very poor. we also discussion similar inaccuracies in another related paper.

cs.DM↗

arXiv2016

The Bilinear Assignment Problem: Complexity and polynomially solvable special cases

In this paper we study the {\it bilinear assignment problem} (BAP) with size parameters $m$ and $n$, $m\leq n$. BAP is a generalization of the well known quadratic assignment problem and the three dimensional assignment problem and hence NP-hard. We show that BAP cannot be approximated within a constant factor unless P=NP even if the associated quadratic cost matrix $Q$ is diagonal. Further, we show that BAP remains NP-hard if $m = O(\sqrt[r]{n})$, for some fixed $r$, but is solvable in polynomial time if $m = O(\sqrt{\log n})$. When the rank of $Q$ is fixed, BAP is observed to admit FPTAS and when this rank is one, it is solvable in polynomial time under some additional restrictions. We then provide a necessary and sufficient condition for BAP to be equivalent to two linear assignment problems. A closed form expression to compute the average of the objective function values of all solutions is presented, whereas the median of the solution values cannot be identified in polynomial time, unless P=NP. We then provide polynomial time heuristic algorithms that find a solution with objective function value no worse than that of $(m-1)!(n-1)!$ solutions. However, computing a solution whose objective function value is no worse than that of $m!n!-\lceil\frac{m}β\rceil !\lceil\frac{n}β\rceil !$ solutions is NP-hard for any fixed rational number $β>1$.

math.OC↗

arXiv2016

The Quadratic Minimum Spanning Tree Problem and its Variations

The quadratic minimum spanning tree problem and its variations such as the quadratic bottleneck spanning tree problem, the minimum spanning tree problem with conflict pair constraints, and the bottleneck spanning tree problem with conflict pair constraints are useful in modeling various real life applications. All these problems are known to be NP-hard. In this paper, we investigate these problems to obtain additional insights into the structure of the problems and to identify possible demarcation between easy and hard special cases. New polynomially solvable cases have been identified, as well as NP-hard instances on very simple graphs. As a byproduct, we have a recursive formula for counting the number of spanning trees on a $(k,n)$-accordion and a characterization of matroids in the context of a quadratic objective function.

cs.DS↗

arXiv2015

Average value of solutions of the bipartite quadratic assignment problem and linkages to domination analysis

In this paper we study the complexity and domination analysis in the context of the \emph{bipartite quadratic assignment problem}. Two variants of the problem, denoted by BQAP1 and BQAP2, are investigated. A formula for calculating the average objective function value $\mathcal{A}$ of all solutions is presented whereas computing the median objective function value is shown to be NP-hard. We show that any heuristic algorithm that produces a solution with objective function value at most $\mathcal{A}$ has the domination ratio at least $\frac{1}{mn}$. Analogous results for the standard \emph{quadratic assignment problem} is an open question. We show that computing a solution whose objective function value is no worse than that of $n^mm^n-{\lceil\frac{n}α\rceil}^{\lceil\frac{m}α\rceil}{\lceil\frac{m}α\rceil}^{\lceil\frac{n}α\rceil}$ solutions of BQAP1 or $m^mn^n-{\lceil\frac{m}α\rceil}^{\lceil\frac{m}α\rceil}{\lceil\frac{n}α\rceil}^{\lceil\frac{n}α\rceil}$ solutions of BQAP2, is NP-hard for any fixed natural numbers $a$ and $b$ such that $α=\frac{a}{b}>1$. However, a solution with the domination number $Ω(m^{n-1}n^{m-1}+m^{n+1}n+mn^{m+1})$ for BQAP1 and $Ω(m^{m-1}n^{n-1}+m^2n^{n}+m^mn^2)$ for BQAP2, can be found in $O(m^3n^3)$ time.

math.OC↗

arXiv2015

A characterization of linearizable instances of the quadratic minimum spanning tree problem

We investigate special cases of the quadratic minimum spanning tree problem (QMSTP) on a graph $G=(V,E)$ that can be solved as a linear minimum spanning tree problem. Characterization of such problems on graphs with special properties are given. This include complete graphs, complete bipartite graphs, cactuses among others. Our characterization can be verified in $O(|E|^2)$ time. In the case of complete graphs and when the cost matrix is given in factored form, we show that our characterization can be verified in $O(|E|)$ time. Related open problems are also indicated.

math.OC↗

arXiv2015

Satellite downlink scheduling problem: A case study

The synthetic aperture radar (SAR) technology enables satellites to efficiently acquire high quality images of the Earth surface. This generates significant communication traffic from the satellite to the ground stations, and, thus, image downlinking often becomes the bottleneck in the efficiency of the whole system. In this paper we address the downlink scheduling problem for Canada's Earth observing SAR satellite, RADARSAT-2. Being an applied problem, downlink scheduling is characterised with a number of constraints that make it difficult not only to optimise the schedule but even to produce a feasible solution. We propose a fast schedule generation procedure that abstracts the problem specific constraints and provides a simple interface to optimisation algorithms. By comparing empirically several standard meta-heuristics applied to the problem, we select the most suitable one and show that it is clearly superior to the approach currently in use.

math.OC↗

arXiv2014

The bipartite unconstrained 0-1 quadratic programming problem: polynomially solvable cases

We consider the bipartite unconstrained 0-1 quadratic programming problem (BQP01) which is a generalization of the well studied unconstrained 0-1 quadratic programming problem (QP01). BQP01 has numerous applications and the problem is known to be MAX SNP hard. We show that if the rank of an associated $m\times n$ cost matrix $Q=(q_{ij})$ is fixed, then BQP01 can be solved in polynomial time. When $Q$ is of rank one, we provide an $O(n\log n)$ algorithm and this complexity reduces to $O(n)$ with additional assumptions. Further, if $q_{ij}=a_i+b_j$ for some $a_i$ and $b_j$, then BQP01 is shown to be solvable in $O(mn\log n)$ time. By restricting $m=O(\log n),$ we obtain yet another polynomially solvable case of BQP01 but the problem remains MAX SNP hard if $m=O(\sqrt[k]{n})$ for a fixed $k$. Finally, if the minimum number of rows and columns to be deleted from $Q$ to make the remaining matrix non-negative is $O(\log n)$ then we show that BQP01 polynomially solvable but it is NP-hard if this number is $O(\sqrt[k]{n})$ for any fixed $k$. Keywords: quadratic programming, 0-1 variables, polynomial algorithms, complexity, pseudo-Boolean programming.

math.OC↗

arXiv2013

The quadratic balanced optimization problem

We introduce the quadratic balanced optimization problem (QBOP) which can be used to model equitable distribution of resources with pairwise interaction. QBOP is strongly NP-hard even if the family of feasible solutions has a very simple structure. Several general purpose exact and heuristic algorithms are presented. Results of extensive computational experiments are reported using randomly generated quadratic knapsack problems as the test bed. These results illustrate the efficacy of our exact and heuristic algorithms. We also show that when the cost matrix is specially structured, QBOP can be solved as a sequence of linear balanced optimization problems. As a consequence, we have several polynomially solvable cases of QBOP.

math.OC↗

arXiv2013

Heuristic algorithms for the bipartite unconstrained 0-1 quadratic programming problem

We study the Bipartite Unconstrained 0-1 Quadratic Programming Problem (BQP) which is a relaxation of the Unconstrained 0-1 Quadratic Programming Problem (QP). Applications of the BQP include mining discrete patterns from binary data, approximating matrices by rank-one binary matrices, computing cut-norm of a matrix, and solving optimization problems such as maximum weight biclique, bipartite maximum weight cut, maximum weight induced subgraph of a bipartite graph, etc. We propose several classes of heuristic approaches to solve the BQP and discuss a number of construction algorithms, local search algorithms and their combinations. Results of extensive computational experiments are reported to establish the practical performance of our algorithms. For this purpose, we propose several sets of test instances based on various applications of the BQP. Our algorithms are compared with state-of-the-art heuristics for QP which can also be used to solve BQP with reformulation. We also study theoretical properties of the neighborhoods and algorithms. In particular, we establish complexity of all neighborhood search algorithms and establish tight worst-case performance ratio for the greedy algorithm.

cs.DM↗