A polyhedral study of triplet formulation for single row facility layout problem

AbstractThe single row facility layout problem (SRFLP) is the problem of arranging n departments with given lengths on a straight line so as to minimize the total weighted distance between all department pairs. We present a polyhedral study of the triplet formulation of the SRFLP introduced by Amaral [A.R.S. Amaral, A new lower bound for the single row facility layout problem, Discrete Applied Mathematics 157 (1) (2009) 183–190]. For any number of departments n, we prove that the dimension of the triplet polytope is n(n−1)(n−2)/3 (this is also true for the projections of this polytope presented by Amaral). We then prove that several valid inequalities presented by Amaral for this polytope are facet-defining. These results provide theoretical support for the fact that the linear program solved over these valid inequalities gives the optimal solution for all instances studied by Amaral

Mixed n-step MIR inequalities: Facets for the n-mixing set

AbstractGünlük and Pochet [O. Günlük, Y. Pochet, Mixing mixed integer inequalities. Mathematical Programming 90 (2001) 429–457] proposed a procedure to mix mixed integer rounding (MIR) inequalities. The mixed MIR inequalities define the convex hull of the mixing set {(y1,…,ym,v)∈Zm×R+:α1yi+v≥βi,i=1,…,m} and can also be used to generate valid inequalities for general as well as several special mixed integer programs (MIPs). In another direction, Kianfar and Fathi [K. Kianfar, Y. Fathi, Generalized mixed integer rounding inequalities: facets for infinite group polyhedra. Mathematical Programming 120 (2009) 313–346] introduced the n-step MIR inequalities for the mixed integer knapsack set through a generalization of MIR. In this paper, we generalize the mixing procedure to the n-step MIR inequalities and introduce the mixed n-step MIR inequalities. We prove that these inequalities define facets for a generalization of the mixing set with n integer variables in each row (which we refer to as the n-mixing set), i.e. {(y1,…,ym,v)∈(Z×Z+n−1)m×R+:∑j=1nαjyji+v≥βi,i=1,…,m}. The mixed MIR inequalities are simply the special case of n=1. We also show that mixed n-step MIR can generate valid inequalities based on multiple constraints for general MIPs. Moreover, we introduce generalizations of the capacitated lot-sizing and facility location problems, which we refer to as the multi-module problems, and show that mixed n-step MIR can be used to generate valid inequalities for these generalizations. Our computational results on small MIPLIB instances as well as a set of multi-module lot-sizing instances justify the effectiveness of the mixed n-step MIR inequalities

Optimum Search Schemes for Approximate String Matching Using Bidirectional FM-Index

Finding approximate occurrences of a pattern in a text using a full-text index is a central problem in bioinformatics and has been extensively researched. Bidirectional indices have opened new possibilities in this regard allowing the search to start from anywhere within the pattern and extend in both directions. In particular, use of search schemes (partitioning the pattern and searching the pieces in certain orders with given bounds on errors) can yield significant speed-ups. However, finding optimal search schemes is a difficult combinatorial optimization problem. Here for the first time, we propose a mixed integer program (MIP) capable to solve this optimization problem for Hamming distance with given number of pieces. Our experiments show that the optimal search schemes found by our MIP significantly improve the performance of search in bidirectional FM-index upon previous ad-hoc solutions. For example, approximate matching of 101-bp Illumina reads (with two errors) becomes 35 times faster than standard backtracking. Moreover, despite being performed purely in the index, the running time of search using our optimal schemes (for up to two errors) is comparable to the best state-of-the-art aligners, which benefit from combining search in index with in-text verification using dynamic programming. As a result, we anticipate a full-fledged aligner that employs an intelligent combination of search in the bidirectional FM-index using our optimal search schemes and in-text verification using dynamic programming outperforms today's best aligners. The development of such an aligner, called FAMOUS (Fast Approximate string Matching using OptimUm search Schemes), is ongoing as our future work

Generalized Mixed Integer Rounding Valid Inequalities

We present new families of valid inequalities for (mixed) integer programming (MIP) problems. These valid inequalities are based on a generalization of the 2-step mixed integer rounding (MIR), proposed by Dash and Günlük (2006). We prove that for any positive integer n, n facets of a certain (n + 1)-dimensional mixed integer set can be obtained through a process which includes n consecutive applications of MIR. We then show that for any n, the last of these facets, the n-step MIR facet, can be used to generate a family of valid inequalities for the feasible set of a general (mixed) IP constraint, the n-step MIR inequalities. The Gomory Mixed Integer Cut and the 2-step MIR inequality of Dash and Günlük (2006) are simply the first two families corresponding to n = 1, 2, respectively. The n-step MIR inequalities are easily produced using closed-form periodic functions, which we refer to as the n-step MIR functions. None of these functions dominates the other on its whole period. Moreover, for any n, the n-step MIR inequalities define new families of two-slope facets for the finite and the infinite group problems.

Research report (Southwest Region University Transportation Center (U.S.))

Report on a study which proposed an optimization framework to determine the most efficient strategies to reduce emissions of vehicles and equipment in a large fleet