This comprehensive textbook presents a clean and coherent account of most fundamental tools and techniques in Parameterized Algorithms and is a self-contained guide to the area. The book covers many of the recent developments of the field, including application of important separators, branching based on linear programming, Cut & Count to obtain faster algorithms on tree decompositions, algorithms based on representative families of matroids, and use of the Strong Exponential Time Hypothesis. A number of older results are revisited and explained in a modern and didactic way. The book provides a toolbox of algorithmic techniques. Part I is an overview of basic techniques, each chapter discussing a certain algorithmic paradigm. The material covered in this part can be used for an introductory course on fixed-parameter tractability. Part II discusses more advanced and specialized algorithmic ideas, bringing the reader to the cutting edge of current research. Part III presents complexity results and lower bounds, giving negative evidence by way of W[1]-hardness, the Exponential Time Hypothesis, and kernelization lower bounds. All the results and concepts are introduced at a level accessible to graduate students and advanced undergraduate students. Every chapter is accompanied by exercises, many with hints, while the bibliographic notes point to original publications and related work.

Parameterized Algorithms

Journal of the ACM

Ingredients of Locational Analysis (J. Krarup & P. Pruzan). The p-Median Problem and Generalizations (P. Mirchandani). The Uncapacitated Facility Location Problem (G. Cornuejols, et al.). Multiperiod Capacitated Location Models (S. Jacobsen). Decomposition Methods for Facility Location Problems (T. Magnanti & R. Wong). Covering Problems (A. Kolen & A. Tamir). p-Center Problems (G. Handler). Duality: Covering and Constraining p-Center Problems on Trees (B. Tansel, et al.). Locations with Spatial Interactions: The Quadratic Assignment Problem (R. Burkard). Locations with Spatial Interactions: Competitive Locations and Games (S. Hakimi). Equilibrium Analysis for Voting and Competitive Location Problems (P. Hansen, et al.). Location of Mobile Units in a Stochastic Environment (O. Berman, et al.). Index.

https://link.springer.com/content/pdf/10.1057%2Fjors.1991.208.pdf

Discrete location theory

We investigate the problem of computing a minimum set of solutions that approximates within a specified accuracy i¾?the Pareto curve of a multiobjective optimization problem. We show that for a broad class of bi-objective problems (containing many important widely studied problems such as shortest paths, spanning tree, and many others), we can compute in polynomial time an i¾?-Pareto set that contains at most twice as many solutions as the minimum such set. Furthermore we show that the factor of 2 is tight for these problems, i.e., it is NP-hard to do better. We present further results for three or more objectives, as well as for the dual problem of computing a specified number kof solutions which provide a good approximation to the Pareto curve.

Small Approximate Pareto Sets for Bi-objective Shortest Paths and Other Problems

Regression test prioritization is often performed in a time constrained execution environment in which testing only occurs for a fixed time period. For example, many organizations rely upon nightly building and regression testing of their applications every time source code changes are committed to a version control repository. This paper presents a regression test prioritization technique that uses a genetic algorithm to reorder test suites in light of testing time constraints. Experiment results indicate that our prioritization approach frequently yields higher average percentage of faults detected (APFD) values, for two case study applications, when basic block level coverage is used instead of method level coverage. The experiments also reveal fundamental trade offs in the performance of time-aware prioritization. This paper shows that our prioritization technique is appropriate for many regression testing environments and explains how the baseline approach can be extended to operate in additional time constrained testing circumstances.

/pdf/timeaware-test-suite-prioritization-4dx9v2z1i4.pdf

TimeAware test suite prioritization

One of the key results in Robertson and Seymour's seminal work on graph minors is the Grid-Minor Theorem (also called the Excluded Grid Theorem). The theorem states that for every fixed-size grid H, every graph whose treewidth is large enough, contains H as a minor. This theorem has found many applications in graph theory and algorithms. Let f(k) denote the largest value, such that every graph of treewidth k contains a grid minor of size f(k) × f(k). The best current quantitative bound, due to recent work of Kawarabayashi and Kobayashi [15], and Leaf and Seymour [18], shows that f(k) = Ω(√logk/loglogk). In contrast, the best known upper bound implies that f(k) = O(√k/logk) [22]. In this paper we obtain the first polynomial relationship between treewidth and grid-minor size by showing that f(k) = Ω(kδ) for some fixed constant δ > 0, and describe an algorithm, whose running time is polynomial in |V (G)| and k, that finds a model of such a grid-minor in G.

Polynomial bounds for the grid-minor theorem

We study the Maximum Independent Set of Rectangles (MISR) problem: given a collection R of n axis-parallel rectangles, find a maximum-cardinality subset of disjoint rectangles. MISR is a special case of the classical Maximum Independent Set problem, where the input is restricted to intersection graphs of axis-parallel rectangles. Due to its many applications, ranging from map labeling to data mining, MISR has received a significant amount of attention from various research communities. Since the problem is NP-hard, the main focus has been on the design of approximation algorithms. Several groups of researches have independently suggested O(log n)-approximation algorithms for MISR, and this remained the best currently known approximation factor for the problem. The main result of our paper is an O(log log n)-approximation algorithm for MISR. Our algorithm combines existing approaches for solving special cases of the problem, in which the input set of rectangles is restricted to containing specific intersection types, with new insights into the combinatorial structure of sets of intersecting rectangles in the plane.We also consider a generalization of MISR to higher dimensions, where rectangles are replaced by d-dimensional hyper-rectangles. Our results for MISR imply an O((log n)d−2 log log n)-approximation algorithm for this problem, improving upon the best previously known O((log n)d−1)-approximation.

https://epubs.siam.org/doi/pdf/10.1137/1.9781611973068.97

Maximum independent set of rectangles

We consider the Max-Min Allocation problem: given a set of m agents and a set of n items, where agent A has utility u(A, i) for item i, our goal is to allocate items to agents so as to maximize fairness. Specifically, the utility of an agent is the sum of its utilities for the items it receives, and we seek to maximize the minimum utility of any agent. While this problem has received much attention recently, its approximability has not been well-understood thus far. The best known approximation algorithm achieves a roughly O(\sqrt m}-approximation, and in contrast, the best known hardness of approximation stands at 2. Our main result is an algorithm that achieves a \tilde{O}(n^{\eps})-approximation in time n^{O(1/\eps)} for any \eps=\Omega(log log n/log n). In particular, we obtain a poly-logarithmic approximation in quasi-polynomial time, and for every constant \eps ≫ 0, we obtain an n^{\eps}-approximation in polynomial time. Our algorithm also yields a quasi-polynomial time m^{\eps}-approximation algorithm for any constant \eps ≫ 0. An interesting technical aspect of our algorithm is that we use as a building block a linear program whose integrality gap is \Omega(\sqrt m). We bypass this obstacle by iteratively using the solutions produced by the LP to construct new instances with significantly smaller integrality gaps, eventually obtaining the desired approximation. We also investigate a special case of the problem, where every item has a non-zero utility for at most two agents. This problem is hard to approximate to within any factor better than 2. We give a factor 2-approximation algorithm.

/pdf/on-allocating-goods-to-maximize-fairness-25pqgx1prm.pdf

On Allocating Goods to Maximize Fairness

We consider the classical vertex cover and set cover problems with the addition of hard capacity constraints. This means that a set (vertex) can only cover a limited number of its elements (adjacent edges) and the number of available copies of each set (vertex) is bounded. This is a natural generalization of the classical problems that also captures resource limitations in practical scenarios. We obtain the following results. For the unweighted vertex cover problem with hard capacities we give a 3-approximation algorithm which is based on randomized rounding with alterations. We prove that the weighted version is at least as hard as the set cover problem. This is an interesting separation between the approximability of weighted and unweighted versions of a "natural" graph problem. A logarithmic approximation factor for both the set cover and the weighted vertex cover problem with hard capacities follows from the work of Wolsey (1982) on submodular set cover. We provide in this paper a simple and intuitive proof for this bound.

/pdf/covering-problems-with-hard-capacities-49t7elfhw0.pdf

Covering problems with hard capacities

The authors consider the job interval selection problem (JISP), a simple scheduling model with a rich history and numerous applications. Special cases of this problem include the so-called real-time scheduling problem (also known as the throughput maximization problem) in single and multiple machine environments. In these special cases we have to maximize the number of jobs scheduled between their release date and deadline (preemption is not allowed). Even the single machine case is NP-hard. The unrelated machines case, as well as other special cases of JISP, are MAX SNP-hard. A simple greedy algorithm gives a 2-approximation for JISP. Despite many efforts, this was the best approximation guarantee known, even for throughput maximization on a single machine. The authors break this barrier and show an approximation guarantee of less than 1.582 for arbitrary instances of JISP. For some special cases, we show better results. Our methods can be used to give improved bounds for some related resource allocation problems that were considered recently in the literature.

/pdf/approximation-algorithms-for-the-job-interval-selection-29oekewxgh.pdf

Julia Chuzhoy

Papers

Polynomial bounds for the grid-minor theorem

Maximum independent set of rectangles

On Allocating Goods to Maximize Fairness

Covering problems with hard capacities

Approximation algorithms for the job interval selection problem and related scheduling problems