Polynomial Number Of Constraints Research Articles

Partitioning a graph into balanced connected classes: Formulations, separation and experiments

This work addresses the balanced connectedk-partition problem (BCPk), which is formally defined as follows. Given a connected graph G=(V,E) with nonnegative weights on the vertices, find a partition {Vi}i=1k of V such that each class Vi induces a connected subgraph of G, and the weight of a class with the minimum weight is as large as possible. This problem, known to be NP-hard, has been largely investigated under different approaches and perspectives: exact algorithms, approximation algorithms for some values of k or special classes of graphs, and inapproximability results. On the practical side, BCPk is used to model many applications arising in image processing, cluster analysis, operating systems and robotics. We propose three linear programming formulations for BCPk. The first one contains only binary variables and a potentially large number of constraints that can be separated in polynomial time in the corresponding linear relaxation. We introduce new valid inequalities and design polynomial-time separation algorithms for them. The other two formulations are based on flows and have a polynomial number of constraints and variables. Our computational experiments show that the exact algorithms based on the proposed formulations outperform the other exact approaches presented in the literature.

Simultaneous node and link districting in transportation networks: Model, algorithms and railway application

Partitioning a large network into multiple subnetworks is adopted extensively in various practical applications involving the architecture of distributed management. In this study, we consider strategic simultaneous node and link districting to partition the nodes and links of a given transportation network into a fixed number of mutually disjoint districts, based on the districting criteria of integrity, contiguity, balance, and independence. We first formulate the resulting problem as a mixed integer linear programming model to minimize the weighted sum of the total size deviation of districts and total cooperation between districts. An improved network flow-based technique is proposed to incorporate the complicated contiguity criterion by using a polynomial number of constraints. Valid inequalities, which break the symmetry within and between districts, are designed to strengthen the model. To solve this challenge problem, we reformulate it as a binary linear programming model, and develop a column generation-based algorithm to find tight lower bounds and good-quality solutions. Then, an iterative search algorithm is designed to obtain good-quality solutions rapidly. Furthermore, a more efficient hybrid algorithm is proposed by using the results of the iterative search algorithm to initialize the column generation-based algorithm. We assess the proposed model and algorithms by using various scales of instances derived from the train dispatcher desk districting problem, which is a practical application of the investigated problem in the context of railway. The computational results reveal that our approaches outperform existing approaches and a commercial solver, and our best algorithm can solve almost all the investigated instances to optimality within a considerably short average computation time. The districting solutions of our approaches are also better than the empirical solutions designed by railway managers mainly based on experience.

Integer linear programming models for the weighted total domination problem

A total dominating set of a graph G=(V,E) is a subset D of V such that every vertex in V (including the vertices from D) has at least one neighbour in D. Suppose that every vertex v ∈ V has an integer weight w(v) ≥ 0 and every edge e ∈ E has an integer weight w(e) ≥ 0. Then the weighted total domination (WTD) problem is to find a total dominating set D which minimizes the cost f(D):=∑u∈Dw(u)+∑e∈E[D]w(e)+∑v∈V∖Dmin{w(uv)|u∈N(v)∩D}. In this paper, we put forward three integer linear programming (ILP) models with a polynomial number of constraints, and present some numerical results implemented on random graphs for WTD problem.

Affine reductions for LPs and SDPs

We define a reduction mechanism for LP and SDP formulations that degrades approximation factors in a controlled fashion. Our reduction mechanism is a minor restriction of classical hardness reductions requiring an additional independence assumption and it allows for reusing many hardness reductions that have been used to show inapproximability in the context of PCP theorems. As a consequence we establish strong linear programming inapproximability (for LPs with a polynomial number of constraints) for many problems. In particular we obtain a $$\frac{3}{2}-\varepsilon $$ inapproximability for answering an open question in Chan et al. (Proceedings of FOCS, pp. 350–359, 2013, https://doi.org/10.1109/FOCS.2013.45 ) and prove an inapproximability factor of $$\frac{1}{2}+\varepsilon $$ for bounded degree . In the case of SDPs, we obtain inapproximability results for these problems relative to the SDP-inapproximability of $${\textsf {MaxCUT}}_{}$$ . Moreover, using our reduction framework we are able to reproduce various results for CSPs from Chan et al. (Proceedings of FOCS, pp. 350–359, 2013, https://doi.org/10.1109/FOCS.2013.45 ) via simple reductions from Max- $$2$$ -XOR.

Two extended formulations for cardinality maximum flow network interdiction problem

We consider the maximum flow network interdiction problem in its cardinality case. There is an integer programming model for this problem by Wood (Math Comput Model 17 (1993), 1–18). Two types of valid inequalities have also been proposed (Altner et al., Oper Res Lett 38 (2010), 33–38 and Wood, Math Comput Model 17 (1993), 1–18) to strengthen the LP relaxation of the integer model. However, due to their combinatorial nature, the number of these inequalities are exponential. Here, we present an equivalent reformulation (extended formulation) for this problem which has a polynomial number of constraints. We also introduce new valid inequalities, and show that the corresponding reformulation of the LP relaxation of the integer model augmented with these inequalities, significantly decreases the integrality gap for a class of network interdiction problems with proven large integrality gaps. Numerical results for some benchmark as well as randomly generated instances are also reported. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 69(4), 367–377 2017

An extended formulation of the convex recoloring problem on a tree

We introduce a strong extended formulation of the convex recoloring problem on a tree, which has an application in analyzing phylogenetic trees. The extended formulation has only a polynomial number of constraints, but dominates the conventional formulation and the exponentially many valid inequalities introduced by Campelo et al. (Math Progr 156:303–330, 2016). We show that all valid inequalities introduced by Campelo et al. can be derived from the extended formulation. We also show that the natural restriction of the extended formulation provides a complete inequality description of the polytope of subtrees of a tree. The solution time using the extended formulation is much smaller than that with the conventional formulation. Moreover the extended formulation solves all the problem instances attempted in Campelo et al. (2016) and larger sized instances at the root node of the branch-and-bound tree without branching.

A new mixed integer linear programming formulation for the maximum degree bounded connected subgraph problem

We give a new mixed integer linear programming (MILP) formulation for Maximum Degree Bounded Connected Subgraph Problem (MDBCSP). The proposed MILP formulation is the first in literature with polynomial number of constraints. Therefore, it will be possible to solve optimally much more instances before in a reasonable time.

Deriving compact extended formulations via LP-based separation techniques

The best formulations for some combinatorial optimization problems are integer linear programming models with an exponential number of rows and/or columns, which are solved incrementally by generating missing rows and columns only when needed. As an alternative to row generation, some exponential formulations can be rewritten in a compact extended form, which have only a polynomial number of constraints and a polynomial, although larger, number of variables. As an alternative to column generation, there are compact extended formulations for the dual problems, which lead to compact equivalent primal formulations, again with only a polynomial number of constraints and variables. In this this paper we introduce a tool to derive compact extended formulations and survey many combinatorial optimization problems for which it can be applied. The tool is based on the possibility of formulating the separation procedure by an LP model. It can be seen as one further method to generate compact extended formulations besides other tools of geometric and combinatorial nature present in the literature.

Deriving compact extended formulations via LP-based separation techniques

The best formulations for some combinatorial optimization problems are integer linear programming models with an exponential number of rows and/or columns, which are solved incrementally by generating missing rows and columns only when needed. As an alternative to row generation, some exponential formulations can be rewritten in a compact extended form, which have only a polynomial number of constraints and a polynomial, although larger, number of variables. As an alternative to column generation, there are compact extended formulations for the dual problems, which lead to compact equivalent primal formulations, again with only a polynomial number of constraints and variables. In this this paper we introduce a tool to derive compact extended formulations and survey many combinatorial optimization problems for which it can be applied. The tool is based on the possibility of formulating the separation procedure by an LP model. It can be seen as one further method to generate compact extended formulations besides other tools of geometric and combinatorial nature present in the literature.

Production lot sizing and scheduling with non-triangular sequence-dependent setup times

This paper considers a production lot sizing and scheduling problem with sequence-dependent setup times that are not triangular. Consider, for example, a product that contaminates some other product unless either a decontamination occurs as part of a substantial setup time or there is a third product that can absorb ’s contamination. When setup times are triangular then and there is always an optimal lot sequence with at most one lot per product per period (AM1L). However, product ’s ability to absorb ’s contamination presents a shortcut opportunity and could result in shorter non-triangular setup times such that . This implies that it can sometimes be optimal for a shortcut product such as to be produced in more than one lot within the same period, breaking the AM1L assumption in much research. This paper formulates and explains a new optimal model that not only permits multiple setups and lots per product in a period (ML), but also prohibits subtours using a polynomial number of constraints rather than an exponential number. Computational tests demonstrate the effectiveness of the ML model, even in the presence of just one decontaminating shortcut product, and its fast speed of solution compared to the equivalent AM1L model.

The coastal seaspace patrol sector design and allocation problem

In this paper, we model and solve the problem of designing and allocating coastal seaspace sectors for steady-state patrolling operations by the vessels of a maritime protection agency. The problem addressed involves optimizing a multi-criteria objective function that minimizes a weighted combination of proportional measures of the vessels’ distances between home ports and patrol sectors, the sector workload, and the sector span. We initially assure contiguity of each patrol sector in our mixed-integer programming formulation via an exponential number of subtour elimination constraints, and then propose three alternative solution methods, two of which are based on reformulations that suitably replace the original contiguity representation with a polynomial number of constraints, and a third approach that employs an iterative cut generation procedure based on identifying violated subtour elimination constraints. We further enhance these reformulations with symmetry defeating constraints, either in isolation or in combination with a suitable perturbation of the objective function using weighted functions based on such constraints. Computational comparisons are provided for solving the problem using the original formulation versus either of our three alternative solution approaches for a representative instance. Overall, a model formulation based on Steiner tree problem (STP) constructs and enhanced by the reformulation-linearization technique (RLT) yielded the best performance.

Compact optimization can outperform separation: A case study in structural proteomics

In Combinatorial Optimization, one is frequently faced with linear programming (LP) problems with exponentially many constraints, which can be solved either using separation or what we call compact optimization. The former technique relies on a separation algorithm, which, given a fractional solution, tries to produce a violated valid inequality. Compact optimization relies on describing the feasible region of the LP by a polynomial number of constraints, in a higher dimensional space. A commonly held belief is that compact optimization does not perform as well as separation in practice. In this paper, we report on an application in which compact optimization does in fact largely outperform separation. The problem arises in structural proteomics, and concerns the comparison of 3-dimensional protein folds. Our computational results show that compact optimization achieves an improvement of up to two orders of magnitude over separation. We discuss some reasons why compact optimization works in this case but not, e.g., for the LP relaxation of the TSP.

New Models of the Generalized Minimum Spanning Tree Problem

We consider a generalization of the Minimum Spanning Tree Problem, called the Generalized Minimum Spanning Tree Problem, denoted by GMST. It is known that the GMST problem is NP-hard. We present a stronger result regarding its complexity, namely, the GMST problem is NP-hard even on trees as well an exact exponential time algorithm for the problem based on dynamic programming. We describe new mixed integer programming models of the GMST problem, mainly containing a polynomial number of constraints. We establish relationships between the polytopes corresponding to their linear relaxations. Based on a new model of the GMST we present a solution procedure that solves the problem to optimality for graphs with nodes up to 240. We discuss the advantages of our method in comparison with earlier methods.

On the equivalence of the multistage-insertion and cycle-shrink formulations of the symmetric traveling salesman problem

Arthanari (On the traveling salesman problem, presented at the XI Symposium on Mathematical Programming held at Bonn, West Germany, 1982) proposed a Multistage-insertion (MI)-formulation of the symmetric traveling salesman problem (STSP). This formulation has a polynomial number of constraints. Carr (Polynomial separation procedures and facet determination for inequalities of the traveling salesman polytope, Ph.D. Thesis, Carneige Mellon University, 1995; Separating over classes of TSP inequalities defined by 0 node-lifting in polynomial time, preprint, 1996) proposed the Cycle-shrink relaxation of the STSP. In this paper, we show that there exists a natural transformation which establishes a one-to-one correspondence between the variables of the two formulations, say X and U, such that X is feasible for MI-formulation if and only if U is feasible for the other. The size of the Cycle-shrink formulation is larger than that of the MI-formulation, both in the number of variables and constraints. However, both are compact descriptions of the subtour elimination polytope.

On the integral dicycle packings and covers and the linear ordering polytope

The linear ordering polytope P LO n is defined as the convex hull of the incidence vectors of the acyclic tournaments on n nodes. It is known that for every facet of P LO n , there corresponds a digraph inducing it. Let D be a digraph that induces a facet-defining inequality for P LO n , that is nonequivalent to a trivial inequality or to a 3-dicycle inequality. We show that for such a digraph the following holds: the value τ of a minimum integral dicycle cover is greater than the value τ ∗ of a minimum dicycle cover. We show that τ ∗ can be found by minimizing a linear function over a polytope which is defined by a polynomial number of constraints. Let v denote the value of a maximum integral dicycle packing. We prove that if D is a certain digraph with a two-node cut satisfying τ = v in each part, then τ = v in D as well. Dridi's description of P LO 5 enables a simple derivation of the fact that τ = v for any digraph on 5 nodes. Combining these results with the theorem of Lucchesi and Younger for planar digraphs as well as Wagner's decomposition, we obtain that τ = v in K 3.3-free digraphs. This last result was proved recently by Barahona et al. (1990) using polyhedral techniques while our proof is based mainly on combinatorial tools.

Compact systems for T-join and perfect matching polyhedra of graphs with bounded genus

We extend a result of Barahona, saying that T-join and perfect matching problems for planar graphs can be formulated as linear programming problems using only a polynomial number of constraints and variables, to graphs embeddable on an arbitrary, but fixed, surface.