On Generalizations of Network Design Problems with Degree Bounds

Iterative rounding and relaxation have arguably become the method of choice in dealing with unconstrained and constrained network design problems. In this paper we extend the scope of the iterative relaxation method in two directions: (1) by handling more complex degree constraints in the minimum spanning tree problem (namely, laminar crossing spanning tree), and (2) by incorporating `degree bounds' in other combinatorial optimization problems such as matroid intersection and lattice polyhedra. We give new or improved approximation algorithms, hardness results, and integrality gaps for these problems.Comment: v2, 24 pages, 4 figure

Approximating Source Location and Star Survivable Network Problems

In Source Location (SL) problems the goal is to select a mini-mum cost source set $S \subseteq V$ such that the connectivity (or flow) $\psi(S,v)$ from $S$ to any node $v$ is at least the demand $d_v$ of $v$. In many SL problems $\psi(S,v)=d_v$ if $v \in S$, namely, the demand of nodes selected to $S$ is completely satisfied. In a node-connectivity variant suggested recently by Fukunaga, every node $v$ gets a "bonus" $p_v \leq d_v$ if it is selected to $S$. Fukunaga showed that for undirected graphs one can achieve ratio $O(k \ln k)$ for his variant, where $k=\max_{v \in V}d_v$ is the maximum demand. We improve this by achieving ratio \min\{p^*\lnk,k\}\cdot O(\ln (k/q^*)) for a more general version with node capacities, where $p^*=\max_{v \in V} p_v$ is the maximum bonus and $q^*=\min_{v \in V} q_v$ is the minimum capacity. In particular, for the most natural case $p^*=1$ considered by Fukunaga, we improve the ratio from $O(k \ln k)$ to $O(\ln^2k)$. We also get ratio $O(k)$ for the edge-connectivity version, for which no ratio that depends on $k$ only was known before. To derive these results, we consider a particular case of the Survivable Network (SN) problem when all edges of positive cost form a star. We give ratio $O(\min\{\ln n,\ln^2 k\})$ for this variant, improving over the best ratio known for the general case $O(k^3 \ln n)$ of Chuzhoy and Khanna

The parsimonious property of cut covering problems and its applications

Includes bibliographical references (p. 26-27).Supported by a Presidential Young Investigator Award. DDM-91568118 Supported by Draper Laboratory.Dimitris Bertsimas and Chungpiaw Teo

Valid Inequalities and Facets for Multi-Module (Survivable) Capacitated Network Design Problem

In this dissertation, we develop new methodologies and algorithms to solve the multi-module (survivable) network design problem. Many real-world decision-making problems can be modeled as network design problems, especially on networks with capacity requirements on arcs or edges. In most cases, network design problems of this type that have been studied involve different types of capacity sizes (modules), and we call them the multi-module capacitated network design (MMND) problem. MMND problems arise in various industrial applications, such as transportation, telecommunication, power grid, data centers, and oil production, among many others. In the first part of the dissertation, we study the polyhedral structure of the MMND problem. We summarize current literature on polyhedral study of MMND, which generates the family of the so-called cutset inequalities based on the traditional mixed integer rounding (MIR). We then introduce a new family of inequalities for MMND based on the so-called n-step MIR, and show that various classes of cutset inequalities in the literature are special cases of these inequalities. We do so by studying a mixed integer set, the cutset polyhedron, which is closely related to MMND. We We also study the strength of this family of inequalities by providing some facet-defining conditions. These inequalities are then tested on MMND instances, and our computational results show that these classes of inequalities are very effective for solving MMND problems. Generalizations of these inequalities for some variants of MMND are also discussed. Network design problems have many generalizations depending on the application. In the second part of the dissertation, we study a highly applicable form of SND, referred to as multi-module SND (MM-SND), in which transmission capacities on edges can be sum of integer multiples of differently sized capacity modules. For the first time, we formulate MM-SND as a mixed integer program (MIP) using preconfigured-cycles (p-cycles) to reroute flow on failed edges. We derive several classes of valid inequalities for this MIP, and show that the valid inequalities previously developed in the literature for single-module SND are special cases of our inequalities. Furthermore, we show that our valid inequalities are facet-defining for MM-SND in many cases. Our computational results, using a heuristic separation algorithm, show that these inequalities are very effective in solving MM-SND. In particular they are more effective than compared to using single-module inequalities alone. Lastly, we generalize the inequalities for MMND for other mixed integer sets relaxed from MMND and the cutset polyhedron. These inequalities also generalize several valid inequalities in the literature. We conclude the dissertation by summarizing the work and pointing out potential directions for future research

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

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