New Class Of Valid Inequalities Research Articles

Rapid Influence Maximization on Social Networks: The Positive Influence Dominating Set Problem

Motivated by applications arising on social networks, we study a generalization of the celebrated dominating set problem called the Positive Influence Dominating Set (PIDS). Given a graph G with a set V of nodes and a set E of edges, each node i in V has a weight bi, and a threshold requirement gi. We seek a minimum weight subset T of V, so that every node i not in T is adjacent to at least gi members of T. When gi is one for all nodes, we obtain the weighted dominating set problem. First, we propose a strong and compact extended formulation for the PIDS problem. We then project the extended formulation onto the space of the natural node-selection variables to obtain an equivalent formulation with an exponential number of valid inequalities. Restricting our attention to trees, we show that the extended formulation is the strongest possible formulation, and its projection (onto the space of the node variables) gives a complete description of the PIDS polytope on trees. We derive the necessary and sufficient facet-dening conditions for the valid inequalities in the projection and discuss their polynomial time separation. We embed this (exponential size) formulation in a branch-and-cut framework and conduct computational experiments using real-world graph instances, with up to approximately 2.5 million nodes and 8 million edges. On a test-bed of 100 real-world graph instances, our approach finds solutions that are on average 0.2% from optimality and solves 51 out of the 100 instances to optimality. Summary of Contribution: In influence maximization problems, a decision maker wants to target individuals strategically to cause a cascade at a minimum cost over a social network. These problems have attracted significant attention as their applications can be found in many different domains including epidemiology, healthcare, marketing, and politics. However, computationally solving large-scale influence maximization problems to near optimality remains a substantial challenge for the computing community, which thus represent significant opportunities for the development of operations-research based models, algorithms, and analysis in this interface. This paper studies the positive influence dominating set (PIDS) problem, an influence maximization problem on social networks that generalizes the celebrated dominating set problem. It focuses on developing exact methods for solving large instances to near optimality. In other words, the approach results in strong bounds, which then provide meaningful comparative benchmarks for heuristic approaches. The paper first shows that straightforward generalizations of well-known formulations for the dominating set problem do not yield strong (i.e., computationally viable) formulations for the PIDS problem. It then strengthens these formulations by proposing a compact extended formulation and derives its projection onto the space on the natural node-selection variables, resulting in two equivalent (stronger) formulations for the PIDS problem. The projected formulation on the natural node-variables contains a new class of valid inequalities that are shown to be facet-defining for the PIDS problem. These theoretical results are complemented by in-depth computational experiments using a branch-and-cut framework, on a testbed of 100 real-world graph instances, with up to approximately 2.5 million nodes and 8 million edges. They demonstrate the effectiveness of the proposed formulation in solving large scale problems finding solutions that are on average 0.2% from optimality and solving 51 of the 100 instances to optimality.

Joint chance-constrained programs and the intersection of mixing sets through a submodularity lens

A particularly important substructure in modeling joint linear chance-constrained programs with random right-hand sides and finite sample space is the intersection of mixing sets with common binary variables (and possibly a knapsack constraint). In this paper, we first revisit basic mixing sets by establishing a strong and previously unrecognized connection to submodularity. In particular, we show that mixing inequalities with binary variables are nothing but the polymatroid inequalities associated with a specific submodular function. This submodularity viewpoint enables us to unify and extend existing results on valid inequalities and convex hulls of the intersection of multiple mixing sets with common binary variables. Then, we study such intersections under an additional linking constraint lower bounding a linear function of the continuous variables. This is motivated from the desire to exploit the information encoded in the knapsack constraint arising in joint linear CCPs via the quantile cuts. We propose a new class of valid inequalities and characterize when this new class along with the mixing inequalities are sufficient to describe the convex hull.

Weak flow cover inequalities for the capacitated facility location problem

The Capacitated Facility Location Problem calls for opening a set of facilities with capacity constraints, with the aim of satisfying at the minimum cost the demands of a set of customers. We present a new class of valid inequalities, the Weak Flow Cover inequalities. We show that Weak Flow Cover inequalities can be separated in polynomial time and turned into violated Flow Cover inequalities. In this way, we are able to provide a polynomial separation heuristic for the latter. Embedding the separation procedure into a cut-and-branch approach, we get results significantly better than those reported in the recent literature both for the lower and the upper bounds.

Improving physician schedules by leveraging equalization: Cases from hospitals in U.S.

In this paper, we consider physician scheduling problems originating from a medical staff scheduling service provider based in the United States. Creating a physician schedule is a complex task. It must balance several goals, including adequately staffing required assignments to ensure quality patient care; adhering to rules that vary by hospital and medical specialty, and helping physicians to maintain work/life balance. We study various types of physician and hospital requirements with different priorities, focusing on achieving an equal distribution of the workload among physicians to produce a schedule that they perceive as fair, while all other group requirements are met. We observe that as the number of such equalization constraints increases, the physician scheduling optimization problem becomes more complex and it requires a longer time to find an optimal schedule. We begin by constructing mathematical models to formulate the physician scheduling problem requirements, and then for the relaxation of the problem that includes equalization constraints we derive a new class of valid inequalities, along with a polynomial separation algorithm for them. A branch-and-cut algorithm using these valid inequalities notably improves the quality of the schedules with respect to the soft constraints. We discuss in detail an example problem from a hospitalist department and present improvements to other schedules representing different specialties and hospitals.

The min-up/min-down unit commitment polytope

The min-up/min-down unit commitment problem (MUCP) is to find a minimum-cost production plan on a discrete time horizon for a set of fossil-fuel units for electricity production. At each time period, the total production has to meet a forecast demand. Each unit must satisfy minimum up-time and down-time constraints besides featuring production and start-up costs. A full polyhedral characterization of the MUCP with only one production unit is provided by Rajan and Takriti (Minimum up/down polytopes of the unit commitment problem with start-up costs. IBM Research Report, 2005). In this article, we analyze polyhedral aspects of the MUCP with n production units. We first translate the classical extended cover inequalities of the knapsack polytope to obtain the so-called up-set inequalities for the MUCP polytope. We introduce the interval up-set inequalities as a new class of valid inequalities, which generalizes both up-set inequalities and minimum up-time inequalities. We provide a characterization of the cases when interval up-set inequalities are valid and not dominated by other inequalities. We devise an efficient Branch and Cut algorithm, using up-set and interval up-set inequalities.

On intersection of two mixing sets with applications to joint chance-constrained programs

We study the polyhedral structure of a generalization of a mixing set described by the intersection of two mixing sets with two shared continuous variables, where one continuous variable has a positive coefficient in one mixing set, and a negative coefficient in the other. Our developments are motivated from a key substructure of linear joint chance-constrained programs (CCPs) with random right hand sides from a finite probability space. The CCPs of interest immediately admit a mixed-integer programming reformulation. Nevertheless, such standard reformulations are difficult to solve at large-scale due to the weakness of their linear programming relaxations. In this paper, we initiate a systemic polyhedral study of such joint CCPs by explicitly analyzing the system obtained from simultaneously considering two linear constraints inside the chance constraint. We carry out our study on this particular intersection of two mixing sets under a nonnegativity assumption on data. Mixing inequalities are immediately applicable to our set, yet they are not sufficient. Therefore, we propose a new class of valid inequalities in addition to the mixing inequalities, and establish conditions under which these inequalities are facet defining. Moreover, under certain additional assumptions, we prove that these new valid inequalities along with the classical mixing inequalities are sufficient in terms of providing the closed convex hull description of our set. We also show that linear optimization over our set is polynomial-time, and we independently give a (high-order) polynomial-time separation algorithm for the new inequalities. We complement our theoretical results with a computational study on the strength of the proposed inequalities. Our preliminary computational experiments with a fast heuristic separation approach demonstrate that our proposed inequalities are practically effective as well.

A comparison of different routing schemes for the robust network loading problem: polyhedral results and computation

We consider the capacity formulation of the Robust Network Loading Problem. The aim of the paper is to study what happens from the theoretical and from the computational point of view when the routing policy (or scheme) changes. The theoretical results consider static, volume, affine and dynamic routing, along with splittable and unsplittable flows. Our polyhedral study provides evidence that some well-known valid inequalities (the robust cutset inequalities) are facets for all the considered routing/flows policies under the same assumptions. We also introduce a new class of valid inequalities, the robust 3-partition inequalities, showing that, instead, they are facets in some settings, but not in others. A branch-and-cut algorithm is also proposed and tested. The computational experiments refer to the problem with splittable flows and the budgeted uncertainty set. We report results on several instances coming from real-life networks, also including historical traffic data, as well as on randomly generated instances. Our results show that the problem with static and volume routing can be solved quite efficiently in practice and that, in many cases, volume routing is cheaper than static routing, thus possibly representing the best compromise between cost and computing time. Moreover, unlikely from what one may expect, the problem with dynamic routing is easier to solve than the one with affine routing, which is hardly tractable, even using decomposition methods.

A class of valid inequalities for multilinear 0–1 optimization problems

This paper investigates the polytope associated with the classical standard linearization technique for the unconstrained optimization of multilinear polynomials in 0–1 variables. A new class of valid inequalities, called 2-links, is introduced to strengthen the LP relaxation of the standard linearization. The addition of the 2-links to the standard linearization inequalities provides a complete description of the convex hull of integer solutions for the case of functions consisting of at most two nonlinear monomials. For the general case, various computational experiments show that the 2-links improve both the standard linearization bound and the computational performance of exact branch & cut methods. The improvements are especially significant for a class of instances inspired from the image restoration problem in computer vision. The magnitude of this effect is rather surprising in that the 2-links are in relatively small number (quadratic in the number of terms of the objective function).

Facets of the axial three-index assignment polytope

We revisit the facial structure of the axial 3-index assignment polytope. After reviewing known classes of facet-defining inequalities, we present a new class of valid inequalities, and show that they define facets of this polytope. This answers a question posed by Qi and Sun (2000). Moreover, we show that we can separate these inequalities in polynomial time. Finally, we assess the computational relevance of the new inequalities by performing (limited) computational experiments.

A branch-and-cut framework for the consistent traveling salesman problem

We develop an exact solution framework for the Consistent Traveling Salesman Problem. This problem calls for identifying the minimum-cost set of routes that a single vehicle should follow during the multiple time periods of a planning horizon, in order to provide consistent service to a given set of customers. Each customer may require service in one or multiple time periods and the requirement for consistent service applies at each customer location that requires service in more than one time period. This requirement corresponds to restricting the difference between the earliest and latest vehicle arrival-times, across the multiple periods, to not exceed some given allowable limit. We present three mixed-integer linear programming formulations for this problem and introduce a new class of valid inequalities to strengthen these formulations. The new inequalities are used in conjunction with traditional traveling salesman inequalities in a branch-and-cut framework. We test our framework on a comprehensive set of benchmark instances, which we compiled by extending traveling salesman instances from the well-known TSPLIB library into multiple periods, and show that instances with up to 50 customers, requiring service over a 5-period horizon, can be solved to guaranteed optimality. Our computational experience suggests that enforcing arrival-time consistency in a multi-period setting can be achieved with merely a small increase in total routing costs.

Valid inequalities for the topology optimization problem in gas network design

One quarter of Europe's energy demand is provided by natural gas distributed through a vast pipeline network covering the whole of Europe. At a cost of 1 million Euro per km extending the European pipeline network is already a multi-billion Euro business. Therefore, automatic planning tools that support the decision process are desired. Unfortunately, current mathematical methods are not capable of solving the arising network design problems due to their size and complexity. In this article, we will show how to apply optimization methods that can converge to a proven global optimal solution. By introducing a new class of valid inequalities that improve the relaxation of our mixed-integer nonlinear programming model, we are able to speed up the necessary computations substantially.

An Integer Programming Approach for Fault-Tolerant Connected Dominating Sets

This paper considers the minimum k-connected d-dominating set problem, which is a fault-tolerant generalization of the minimum connected dominating set (MCDS) problem. Three integer programming formulations based on vertex cuts are proposed (depending on whether d < k, d = k, or d > k) and their integer hulls are studied. The separation problem for the vertex-cut inequalities is a weighted vertex-connectivity problem and is polytime solvable, meaning that the LP relaxation can be solved in polytime despite having exponentially many constraints. A new class of valid inequalities—r-robust vertex-cut inequalities—is introduced and is shown to induce exponentially many facets. Finally, a lazy-constraint approach is shown to compare favorably with existing approaches for the MCDS problem (the case k = d = 1), and is in fact the fastest in literature for standard test instances. A key subroutine is an algorithm for finding an inclusion-wise minimal vertex cut in linear time. Computational results for (k, d) = (2,1), (2,2), (3,3), (4,4) are provided as well.

Merging valid inequalities over the multiple knapsack polyhedron

This paper provides the theoretical foundations for generating a new class of valid inequalities for integer programming problems through inequality merging. The inequality merging technique combines two low dimensional inequalities of a multiple knapsack problem, potentially yielding a valid inequality of higher dimension. The paper describes theoretical conditions for validity of the merged inequality and shows that the validity of a merged cover inequality may be verified in quadratic time. Conditions under which a valid merged inequality is facet defining are also presented. The technique is demonstrated through a multiple knapsack example. The example also demonstrates that inequality merging yields a new class of valid inequalities that are fundamentally different from other known techniques.

A note on a model for quay crane scheduling with non-crossing constraints

This article studies the quay crane scheduling problem with non-crossing constraints, which is an operational problem that arises in container terminals. An enhancement to a mixed integer programming model for the problem is proposed and a new class of valid inequalities is introduced. Computational results show the effectiveness of these enhancements in solving the problem to optimality.

In This Issue

In This Issue

Recoverable robust knapsacks: the discrete scenario case

The knapsack problem is one of the basic problems in combinatorial optimization. In real-world applications it is often part of a more complex problem. Examples are machine capacities in production planning or bandwidth restrictions in telecommunication network design. Due to unpredictable future settings or erroneous data, parameters of such a subproblem are subject to uncertainties. In high risk situations a robust approach should be chosen to deal with these uncertainties. Unfortunately, classical robust optimization outputs solutions with little profit by prohibiting any adaption of the solution when the actual realization of the uncertain parameters is known. This ignores the fact that in most settings minor changes to a previously determined solution are possible. To overcome these drawbacks we allow a limited recovery of a previously fixed item set as soon as the data are known by deleting at most k items and adding up to l new items. We consider the complexity status of this recoverable robust knapsack problem and extend the classical concept of cover inequalities to obtain stronger polyhedral descriptions. Finally, we present two extensive computational studies to investigate the influence of parameters k and l to the objective and evaluate the effectiveness of our new class of valid inequalities.

Row family inequalities for the set covering polyhedron

Abstract From the polyhedral point of view the set covering polyhedron (SCP) and the set packing polyhedron (SPP) associated with clutter matrices (i.e. 0, 1 matrix without dominating rows and zero columns) have strong similarities. Thus, many key concepts have been transferred from SPP to SCP by Balas and Ng (1989), Cornuejols and Sassano (1989), Nobili and Sassano (1989) and Sassano (1989). In this work, we present a new class of valid inequalities for the SCP associated with general clutter matrices, called row family inequalities, that can be viewed as the counterpart of the valid clique family inequalities introduced by Oriolo (2004) for SPP. We show that many valid inequalities for SCP belong to this class. In particular, all the inequalities with coefficients in { 0 , 1 , 2 } studied by Balas and Ng (1986).

Generalized Domino-Parity Inequalities for the Symmetric Traveling Salesman Problem

We extend the work of Letchford [Letchford, A. N. 2000. Separating a superclass of comb inequalities in planar graphs. Math. Oper. Res. 25 443–454] by introducing a new class of valid inequalities for the traveling salesman problem, called the generalized domino-parity (GDP) constraints. Just as Letchford's domino-parity constraints generalize comb inequalities, GDP constraints generalize the most well-known multiple-handle constraints, including clique-tree, bipartition, path, and star inequalities. Furthermore, we show that a subset of GDP constraints containing all of the clique-tree inequalities can be separated in polynomial time, provided that the support graph G* is planar, and provided that we bound the number of handles by a fixed constant h.

Disjunctive Decomposition for Two-Stage Stochastic Mixed-Binary Programs with Random Recourse

This paper introduces disjunctive decomposition for two-stage mixed 0-1 stochastic integer programs (SIPs) with random recourse. Disjunctive decomposition allows for cutting planes based on disjunctive programming to be generated for each scenario subproblem under a temporal decomposition setting of the SIP problem. A new class of valid inequalities for mixed 0-1 SIP with random recourse is presented. In particular, we derive valid inequalities that allow for scenario subproblems for SIP with random recourse but deterministic technology matrix and right-hand side vector to share cut coefficients. The valid inequalities are used to derive a disjunctive decomposition method whose derivation has been motivated by real-life stochastic server location problems with random recourse, which find many applications in operations research. Computational results with large-scale instances to demonstrate the potential of the method are reported.

Valid inequalities for the single-item capacitated lot sizing problem with step-wise costs

This paper presents a new class of valid inequalities for the single-item capacitated lot sizing problem with step-wise production costs (LS-SW). We first provide a survey of different optimization methods proposed to solve LS-SW. Then, flow cover and flow cover inequalities derived from the single node flow set are described in order to generate the new class of valid inequalities. The single node flow set can be seen as a generalization of some valid relaxations of LS-SW. A new class of valid inequalities we call mixed flow cover, is derived from the integer flow cover inequalities by a lifting procedure. The lifting coefficients are sequence independent when the batch sizes (V) and the production capacities (P) are constant and if V divides P. When the restriction of the divisibility is removed, the lifting coefficients are shown to be sequence independent. We identify some cases where the mixed flow cover inequalities are facet defining. A cutting plane algorithm is proposed for these three classes of valid inequalities. The exact separation algorithm proposed for the constant capacitated case runs in polynomial time. Finally, some computational results are given to compare the performance of the different optimization methods including the new class of valid inequalities.