Approximating Minimum Cost Connectivity Orientation and Augmentation

We investigate problems addressing combined connectivity augmentation and orientations settings. We give a polynomial-time 6-approximation algorithm for finding a minimum cost subgraph of an undirected graph $G$ that admits an orientation covering a nonnegative crossing $G$-supermodular demand function, as defined by Frank. An important example is $(k,\ell)$-edge-connectivity, a common generalization of global and rooted edge-connectivity. Our algorithm is based on a non-standard application of the iterative rounding method. We observe that the standard linear program with cut constraints is not amenable and use an alternative linear program with partition and co-partition constraints instead. The proof requires a new type of uncrossing technique on partitions and co-partitions. We also consider the problem setting when the cost of an edge can be different for the two possible orientations. The problem becomes substantially more difficult already for the simpler requirement of $k$-edge-connectivity. Khanna, Naor, and Shepherd showed that the integrality gap of the natural linear program is at most $4$ when $k=1$ and conjectured that it is constant for all fixed $k$. We disprove this conjecture by showing an $\Omega(|V|)$ integrality gap even when $k=2$

Changing Bases: Multistage Optimization for Matroids and Matchings

This paper is motivated by the fact that many systems need to be maintained continually while the underlying costs change over time. The challenge is to continually maintain near-optimal solutions to the underlying optimization problems, without creating too much churn in the solution itself. We model this as a multistage combinatorial optimization problem where the input is a sequence of cost functions (one for each time step); while we can change the solution from step to step, we incur an additional cost for every such change. We study the multistage matroid maintenance problem, where we need to maintain a base of a matroid in each time step under the changing cost functions and acquisition costs for adding new elements. The online version of this problem generalizes online paging. E.g., given a graph, we need to maintain a spanning tree $T_t$ at each step: we pay $c_t(T_t)$ for the cost of the tree at time $t$, and also $| T_t\setminus T_{t-1} |$ for the number of edges changed at this step. Our main result is an $O(\log m \log r)$-approximation, where $m$ is the number of elements/edges and $r$ is the rank of the matroid. We also give an $O(\log m)$ approximation for the offline version of the problem. These bounds hold when the acquisition costs are non-uniform, in which caseboth these results are the best possible unless P=NP. We also study the perfect matching version of the problem, where we must maintain a perfect matching at each step under changing cost functions and costs for adding new elements. Surprisingly, the hardness drastically increases: for any constant $\epsilon>0$, there is no $O(n^{1-\epsilon})$-approximation to the multistage matching maintenance problem, even in the offline case

LP-Relaxations for Tree Augmentation

In the Tree Augmentation Problem (TAP) the goal is to augment a tree T by a minimum size edge set F from a given edge set E such that T+F is 2-edge-connected. The best approximation ratio known for TAP is 1.5. In the more general Weighted TAP problem, F should be of minimum weight. Weighted TAP admits several 2-approximation algorithms w.r.t. the standard cut-LP relaxation. The problem is equivalent to the problem of covering a laminar set family. Laminar set families play an important role in the design of approximation algorithms for connectivity network design problems. In fact, Weighted TAP is the simplest connectivity network design problem for which a ratio better than 2 is not known. Improving this "natural" ratio is a major open problem, which may have implications on many other network design problems. It seems that achieving this goal requires finding an LP-relaxation with integrality gap better than 2, which is an old open problem even for TAP. In this paper we introduce two different LP-relaxations, and for each of them give a simple algorithm that computes a feasible solution for TAP of size at most 7/4 times the optimal LP value. This gives some hope to break the ratio 2 for the weighted case

Building Networks in the Face of Uncertainty

The subject of this thesis is to study approximation algorithms for some network design problems in face of uncertainty. We consider two widely studied models of handling uncertainties - Robust Optimization and Stochastic Optimization. We study a robust version of the well studied Uncapacitated Facility Location Problem (UFLP). In this version, once the set of facilities to be opened is decided, an adversary may close at most β facilities. The clients must then be assigned to the remaining open facilities. The performance of a solution is measured by the worst possible set of facilities that the adversary may close. We introduce a novel LP for the problem, and provide an LP rounding algorithm when all facilities have same opening costs. We also study the 2-stage Stochastic version of the Steiner Tree Problem. In this version, the set of terminals to be covered is not known in advance. Instead, a probability distribution over the possible sets of terminals is known. One is allowed to build a partial solution in the first stage a low cost, and when the exact scenario to be covered becomes known in the second stage, one is allowed to extend the solution by building a recourse network, albeit at higher cost. The aim is to construct a solution of low cost in expectation. We provide an LP rounding algorithm for this problem that beats the current best known LP rounding based approximation algorithm

Connectivity-splitting models for survivable network design

"January 2000." Title from cover.Includes bibliographical references (p. 24-25).by T.L. Magnanti, A. Balakrishnan, P. Mirchandani

Larval morphology of Meruidae (Coleoptera: Adephaga) and its phylogenetic implications

This is the publisher's version, also available electronically from http://www.ingentaconnect.com/content/esa/aesa. This article is the copyright property of the Entomological Society of America and may not be used for any commercial or other private purpose without specific written permission of the Entomological Society of AmericaMeruidae, or comb-clawed cascade beetles, are a recently discovered monotypic family of Adephaga endemic to Venezuela. The larvae of Meruidae are described for the first time, based on material of Meru phyllisae Spangler & Steiner, 2005, collected together with adults in southern Venezuela. External morphological features, including chaetotaxy, are reported for the mature larva and an assessment made of the polarity of larval characters of phylogenetic utility in Adephaga. Larvae of Meruidae possess a mixture of primitive and derived character states, and they are unique within the Adephaga in that here the mandibles are asymmetrical, the respiratory system is comprised of only two pairs of spiracles (=oligopneustic), the claws are pectinate, and the abdominal sternite VIII is prolonged overlapping the abdominal sternite IX. A parsimony analysis based on 18 informative larval characteristics was conducted with the program PAUP*. The most parsimonious trees confirm Meruidae as a relatively basal lineage within the Dytiscoidea. Both Meru Spangler & Steiner and Noteridae are hypothesized to have diverged anterior to Amphizoidae, Aspidytidae, Hygrobiidae, and Dytiscidae

Iterative Rounding Approximation Algorithms in Network Design

Iterative rounding has been an increasingly popular approach to solving network design optimization problems ever since Jain introduced the concept in his revolutionary 2-approximation for the Survivable Network Design Problem (SNDP). This paper looks at several important iterative rounding approximation algorithms and makes improvements to some of their proofs. We generalize a matrix restatement of Nagarajan et al.'s token argument, which we can use to simplify the proofs of Jain's 2-approximation for SNDP and Fleischer et al.'s 2-approximation for the Element Connectivity (ELC) problem. Lau et al. show how one can construct a (2,2B + 3)-approximation for the degree bounded ELC problem, and this thesis provides the proof. We provide some structural results for basic feasible solutions of the Prize-Collecting Steiner Tree problem, and introduce a new problem that arises, which we call the Prize-Collecting Generalized Steiner Tree problem

Doubling or Splitting: Strategies for Modeling and Analyzing Survivable Network Design Problems

Survivability is becoming an increasingly important criterion in network design. This paper studies formulations, heuristic worst-case performance, and linear programming relaxations for two classes of survivable network design problems: the low connectivity Steiner (LCS) problem for graphs containing nodes with connectivity requirement of 0, 1, or 2, and a more general multi-connected network with branches (MNB) that requires connectivities of two or more for certain (critical) nodes and single connectivity for other secondary nodes. We consider both unitary and nonunitary MNB problems that respectively require a connected design or permit multiple components. Using a doubling argument, we first show how to generalize heuristic bounds of the Steiner tree and traveling salesman problems to LCS problems. We then develop a disaggregate formulation for the MNB problem that uses fractional edge selection variables to split the total connectivity requirement across each critical cutset into two separate requirements. This model, which is tighter than the usual cutset formulation, has three special cases: a "secondary-peeling" version that peels off the lowest connectivity level, a "connectivity-dividing" version that divides the connectivity requirements for all the critical cutsets, and a "secondarycompletion" version that attempts to separate the design decisions for the multi-connected network from those for the branches. We examine the tightness of the linear programming relaxations for these extended formulations, and then use them to analyze heuristics for the LCS and MNB problems. Our analysis strengthens some previously known heuristic-to-IP worst-case performance ratios for LCS and MNB problems by showing that the same bounds apply to the heuristic-to-LP ratios using our stronger formulations

Fire-Derived Charcoal Along an Ecological Gradient in the Colorado Front Range

Terrestrial ecosystems are shaped by natural disturbances such as wildland fire. In the intermountain western United States, forests, shrub and grasslands adapt to repeated fires. An important long-term legacy of wildland fires is black C (BC) commonly referred to as char or charcoal. Black C is a recalcitrant C form that has been long known to influence soil physical, chemical, and biological processes that they vary across landscapes and over time. The objective of this research is to address two key areas in the emerging field of ecosystem BC research; 1) how much BC as charcoal C is formed per fire at a watershed scale and 2) how much charcoal C and total soil organic C are in mineral soil pools in the predominant Colorado Front Range vegetation types. For the former, we combined fire model results for fuel consumption with published charcoal conversion constants to create maps of predicted charcoal C per fire. These maps represent the first spatial estimates shown at a watershed scale. For the latter, we measured charcoal C pools in surface soils (0-10 cm) at mid-slope positions on east facing aspects in five continuous shrublands and forests from grassland to tundra. We found a significant statistical effect of vegetation type on soil charcoal C pools along this ecological gradient, but not a linear pattern of increasing charcoal C amounts with elevation gain. This study yielded the largest collection of soil samples analyzed for charcoal C in the United States. The geospatial data and thermo-chemical analysis methods developed here are an advance in the framework for evaluating the two critical phases in ecosystem black C cycling. Future modeling and field-based efforts are called for after revealing a landscape-pattern of SOC and charcoal C pools