Optimizing constrained subtrees of trees

Aghezzaf, El‐Houssaine; Magnanti, Thomas L.; Wolsey, Laurence A.

doi:10.1007/bf01585993

Cited by 11 publications

(9 citation statements)

References 8 publications

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“…When weights are assigned to the points of the convex geometry, the natural question of finding a convex set with maximum-weight arises. Particular subproblems are the closure problem [32], the maximum-weight subtree problem [1,24], the maximum-weight path-closed set [17], or in a more geometrical setting, some variants of the minimum kgons problem [11]. A more recent example is the problem of finding a maximum-weight convex set in a split graph [5].…”

Section: Previous Workmentioning

confidence: 99%

“…Here is our main result. The well-known problem of finding a maximum-weight connected subtree in a tree can be solved by selecting a vertex as "root", finding a maximum-weight subtree that contains the root, and iterating this procedure for all possible roots (see Wolsey et al [1]). In order to use a similar approach to solve Problem 1, we define a notion of root.…”

Section: The Problemsmentioning

confidence: 99%

“…We give a polynomial-time algorithm to solve the problem. Until now, only the special cases of trees [1,24] and split graphs [5] were known to be polynomial-time solvable.…”

Section: Introductionmentioning

confidence: 99%

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Finding a Maximum-Weight Convex Set in a Chordal Graph

Cardinal¹,

Doignon²,

Merckx³

2019

JGAA

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We consider a natural combinatorial optimization problem on chordal graphs, the class of graphs with no induced cycle of length four or more. A subset of vertices of a chordal graph is (monophonically) convex if it contains the vertices of all chordless paths between any two vertices of the set. The problem is to find a maximum-weight convex subset of a given vertex-weighted chordal graph. It generalizes previously studied special cases in trees and split graphs. It also happens to be closely related to the closure problem in partially ordered sets and directed graphs. We give the first polynomial-time algorithm for the problem. PreliminariesWe review here some basic notation and results for graphs and convex geometries, we also formally define the problems we investigate.

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Section: Previous Workmentioning

confidence: 99%

Section: The Problemsmentioning

confidence: 99%

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Finding a Maximum-Weight Convex Set in a Chordal Graph

Cardinal¹,

Doignon²,

Merckx³

2019

JGAA

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“…Park and Park [14] and van de Leensel et al [15] describe inequalities obtained by lifting the cover inequalities for the knapsack problem. Aghezzaf et al [1] study the related problem of packing subtrees with cardinality constraints. The second related set is the (fixed-charge) flow set, which is the relaxation of for A = .…”

Section: Motivating Examplesmentioning

confidence: 99%

The Flow Set with Partial Order

Atamtürk

Zhang

2008

Mathematics of OR

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The flow set with partial order is a mixed-integer set described by a budget on total flow and a partial order on the arcs that may carry positive flow. This set is a common substructure of resource allocation and scheduling problems with precedence constraints and robust network flow problems under demand/capacity uncertainty. We give a polyhedral analysis of the convex hull of the flow set with partial order. Unlike for the flow set without partial order, cover-type inequalities based on partial order structure are a function of a lifting sequence. We study the lifting sequences and describe structural results on the lifting coefficients for general and simpler special cases. We show that all lifting coefficients can be computed in polynomial time by solving maximum weight closure problems in general. For the special case of induced-minimal covers, we give a sequence-dependent characterization of the lifting coefficients. We prove, however, if the partial order is defined by an arborescence, then lifting is sequence-independent and all lifting coefficients can be computed in linear time. Moreover, if the partial order is defined by a path (total order), then the coefficients can be expressed explicitly. We also give a complete polyhedral description of the flow set with partial order for the polynomially-solvable total order case. We show that finding an optimal lifting order for a given induced-minimal cover and a given fractional solution is a submodular optimization problem, which is solved greedily. Finally, we present preliminary computational results with a cutting-plane algorithm based on the lifting and separation results.

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“…Two other examples are the intersection of forest and cover polyhedra (Gamble and Pulleyblank [ 1989]) and the intersection of tree and matching polyhedra (Hall and Magnanti [1992]). As another example, Barany, Edmonds, and Wolsey [1986] and Aghezzaf, Magnanti, and Wolsey [1992] have shown that the packing of certain polyhedra (to model rooted trees) produces an integer polyhedra. As we show in this paper, by combining integer polyhedra via forcing constraints as in the problem [P], we do not always create an integer polyhedron.…”

Section: Problem Definition and Overviewmentioning

confidence: 99%

Heuristics, LPs, and Trees on Trees: Network Design Analyses

Balakrishnan

Magnanti

Mirchandani

1996

Operations Research

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We study a class of models, known as overlay optimization problems, with a "base" subproblem and an "overlay" subproblem, linked by the requirement that the overlay solution be contained in the base solution. In some telecommunication settings, a feasible base solution is a spanning tree and the overlay solution is an embedded Steiner tree (or an embedded path). For the general overlay optimization problem, we describe a heuristic solution procedure that selects the better of two feasible solutions obtained by independently solving the base and overlay subproblems, and establish worst-case performance guarantees on both this heuristic and a LP relaxation of the model. These guarantees depend upon worst-case bounds for the heuristics and LP relaxations of the unlinked base and overlay problems. Under certain assumptions about the cost structure and the optimality of the subproblem solutions, both the heuristic and the LP relaxation of the combined overlay optimization model have performance guarantees of 4/3. We extend this analysis to multiple overlays on the same base solution, producing the first known worst-case bounds (approximately proportional to the square root of the number of commodities) for the uncapacitated multicommodity network design problem. In a companion paper, we develop heuristic performance guarantees for various new multi-tier. survivable network design models that incorporate both multiple facility types or technologies and differential node connectivity levels. IntroductionThis paper considers a general class of models, which we call overlay optimization problems, that combines two sets of decisions: the choice of activity levels to provide a basic level of service to all customers, and decisions regarding which of these activities to enhance to meet more stringent service requirements for subsets of important customers. The activity levels might represent facility installation and sizing decisions, with the basic and enhanced activities representing two levels of technology that differ in speed, capacity. or functionality. We treat the installation of higher grade facilities as "overlaying" or upgrading the base facilities at extra cost. Overlay optimization has potential applications in logistics and infrastructure planning including the design of telecommunications, transportation, electric power, and pipeline networks. For instance, in transportation planning, customers correspond to cities and towns. Basic service represents providing access to every town via a paved road, but certain important cities must be interconnected by, say, all-weather highways. Likewise, in telecommunications planning, the base decisions model the installation of switching and transmission facilities that can accommodate basic voice and data services (e.g., DS 1 services), while overlay variables represent upgrading certain facilities to high capacity, broadband (e.g., fiber optic) system,, between selected locations.We analyze the worst-case performance of a generic heuristic strategy for solving o...

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Optimizing constrained subtrees of trees

Cited by 11 publications

References 8 publications

Finding a Maximum-Weight Convex Set in a Chordal Graph

Finding a Maximum-Weight Convex Set in a Chordal Graph

The Flow Set with Partial Order

Heuristics, LPs, and Trees on Trees: Network Design Analyses

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