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We study rectangle stabbing problems in which we are given n axis-aligned rectangles in the plane that we want to stab, that is, we want to select line segments such that for each given rectangle there is a line segment that intersects two opposite edges of it. In the horizontal rectangle stabbing problem (Stabbing), the goal is to find a set of horizontal line segments of minimum total length such that all rectangles are stabbed. In the horizontal–vertical stabbing problem (HV-Stabbing), the goal is to find a set of rectilinear (that is, either vertical or horizontal) line segments of minimum total length such that all rectangles are stabbed. Both variants are NP-hard. Chan et al. (ISAAC, 2018) initiated the study of these problems by providing constant approximation algorithms. Recently, Eisenbrand et al. (A QPTAS for stabbing rectangles, 2021) have presented a QPTAS and a polynomial-time 8-approximation algorithm for Stabbing, but it was open whether the problem admits a PTAS. In this paper, we obtain a PTAS for Stabbing, settling this question. For HV-Stabbing, we obtain a (2 + ε) -approximation. We also obtain PTASs for special cases of HV-Stabbing: (i) when all rectangles are squares, (ii) when each rectangle's width is at most its height, and (iii) when all rectangles are δ -large, that is, have at least one edge whose length is at least δ , while all edge lengths are at most 1. Our result also implies improved approximations for other problems such as generalized minimum Manhattan network. [ABSTRACT FROM AUTHOR]

In generalized malleable scheduling, jobs can be allocated and processed simultaneously on multiple machines so as to reduce the overall makespan of the schedule. The required processing time for each job is determined by the joint processing speed of the allocated machines. We study the case that processing speeds are job-dependent M ♮ -concave functions and provide a constant-factor approximation for this setting, significantly expanding the realm of functions for which such an approximation is possible. Further, we explore the connection between malleable scheduling and the problem of fairly allocating items to a set of agents with distinct utility functions, devising a black-box reduction that allows to obtain resource-augmented approximation algorithms for the latter. [ABSTRACT FROM AUTHOR]

In the non-uniform sparsest cut problem, we are given a supply graph G and a demand graph D, both with the same set of nodes V. The goal is to find a cut of V that minimizes the ratio of the total capacity on the edges of G crossing the cut over the total demand of the crossing edges of D. In this work, we study the non-uniform sparsest cut problem for supply graphs with bounded treewidth k. For this case, Gupta et al. (ACM STOC, 2013) obtained a 2-approximation with polynomial running time for fixed k, and it remained open the question of whether there exists a c-approximation algorithm for a constant c independent of k, that runs in FPT time. We answer this question in the affirmative. We design a 2-approximation algorithm for the non-uniform sparsest cut with bounded treewidth supply graphs that runs in FPT time, when parameterized by the treewidth. Our algorithm is based on rounding the optimal solution of a linear programming relaxation inspired by the Sherali-Adams hierarchy. In contrast to the classic Sherali-Adams approach, we construct a relaxation driven by a tree decomposition of the supply graph by including a carefully chosen set of lifting variables and constraints to encode information of subsets of nodes with super-constant size, and at the same time we have a sufficiently small linear program that can be solved in FPT time. [ABSTRACT FROM AUTHOR]

We consider the class of disjoint bilinear programs max { x T y ∣ x ∈ X , y ∈ Y } where X and Y are packing polytopes. We present an O (log log m 1 log m 1 log log m 2 log m 2 ) -approximation algorithm for this problem where m 1 and m 2 are the number of packing constraints in X and Y respectively. In particular, we show that there exists a near-optimal solution (x ~ , y ~) such that x ~ and y ~ are "near-integral". We give an LP relaxation of the problem from which we obtain the near-optimal near-integral solution via randomized rounding. We show that our relaxation is tightly related to the widely used reformulation linearization technique. As an application of our techniques, we present a tight approximation for the two-stage adjustable robust optimization problem with covering constraints and right-hand side uncertainty where the separation problem is a bilinear optimization problem. In particular, based on the ideas above, we give an LP restriction of the two-stage problem that is an O (log n log log n log L log log L) -approximation where L is the number of constraints in the uncertainty set. This significantly improves over state-of-the-art approximation bounds known for this problem. Furthermore, we show that our LP restriction gives a feasible affine policy for the two-stage robust problem with the same (or better) objective value. As a consequence, affine policies give an O (log n log log n log L log log L) -approximation of the two-stage problem, significantly generalizing the previously known bounds on their performance. [ABSTRACT FROM AUTHOR]

We present approximation algorithms for several network design problems in the model of flexible graph connectivity (Adjiashvili et al., in: IPCO, pp 13–26, 2020, Math Program 1–33, 2021). Let k ≥ 1 , p ≥ 1 and q ≥ 0 be integers. In an instance of the (p, q)-Flexible Graph Connectivity problem, denoted (p , q) -FGC , we have an undirected connected graph G = (V , E) , a partition of E into a set of safe edges S and a set of unsafe edges U , and nonnegative costs c : E → R ≥ 0 on the edges. A subset F ⊆ E of edges is feasible for the (p , q) -FGC problem if for any set F ′ ⊆ U with | F ′ | ≤ q , the subgraph (V , F \ F ′) is p-edge connected. The algorithmic goal is to find a feasible solution F that minimizes c (F) = ∑ e ∈ F c e . We present a simple 2-approximation algorithm for the (1 , 1) -FGC problem via a reduction to the minimum-cost rooted 2-arborescence problem. This improves on the 2.527-approximation algorithm of Adjiashvili et al. Our 2-approximation algorithm for the (1 , 1) -FGC problem extends to a (k + 1) -approximation algorithm for the (1 , k) -FGC problem. We present a 4-approximation algorithm for the (k , 1) -FGC problem, and an O (q log | V |) -approximation algorithm for the (p , q) -FGC problem. Finally, we improve on the result of Adjiashvili et al. for the unweighted (1 , 1) -FGC problem by presenting a 16/11-approximation algorithm. The (p , q) -FGC problem is related to the well-known Capacitatedk-Connected Subgraph problem (denoted Cap- k -ECSS ) that arises in the area of Capacitated Network Design. We give a min (k , 2 u max) -approximation algorithm for the Cap- k -ECSS problem, where u max denotes the maximum capacity of an edge. [ABSTRACT FROM AUTHOR]

We initiate the work on maximin share (MMS) fair allocation of m indivisible chores to n agents using only their ordinal preferences, from both algorithmic and mechanism design perspectives. The previous best-known approximation ratio using ordinal preferences is 2 - 1 / n by Aziz et al. [AAAI 2017]. We improve this result by giving a deterministic 5/3-approximation algorithm that determines an allocation sequence of agents, according to which items are allocated one by one. By a tighter analysis, we show that for n = 2 and 3, our algorithm achieves better approximation ratios, and is actually optimal. We also consider the setting with strategic agents, where agents may misreport their preferences to manipulate the outcome. We first provide a strategyproof O (log (m / n)) -approximation consecutive picking algorithm, and then improve the approximation ratio to O (log n) by a randomized algorithm. Both algorithms only use the ordinal preferences of agents. Our results uncover some interesting contrasts between the approximation ratios achieved for chores versus goods. [ABSTRACT FROM AUTHOR]

This paper considers approximation algorithms for generalized k-median problems. These problems can be informally described as k-median with a constant number of extra constraints, and includes k-median with outliers, and knapsack median. Our first contribution is a pseudo-approximation algorithm for generalized k-median that outputs a 6.387-approximate solution, with a constant number of fractional variables. The algorithm builds on the iterative rounding framework introduced by Krishnaswamy, Li, and Sandeep for k-median with outliers as reported (Krishnaswamy et al. in: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, 2018). The main technical innovation is allowing richer constraint sets in the iterative rounding and using the structure of the resulting extreme points. Using our pseudo-approximation algorithm, we give improved approximation algorithms for k-median with outliers and knapsack median. This involves combining our pseudo-approximation with pre- and post-processing steps to round a constant number of fractional variables at a small increase in cost. Our algorithms achieve approximation ratios 6.994+ϵ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$6.994 + \epsilon $$\end{document} and 6.387+ϵ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$6.387 + \epsilon $$\end{document} for k-median with outliers and knapsack median, respectively. These improve on the best-known approximation ratio 7.081+ϵ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$7.081 + \epsilon $$\end{document} for both problems as reported (Krishnaswamy et al. in: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, 2018). [ABSTRACT FROM AUTHOR]

In the unsplittable capacitated vehicle routing problem (UCVRP) on trees, we are given a rooted tree with edge weights and a subset of vertices of the tree called terminals. Each terminal is associated with a positive demand between 0 and 1. The goal is to find a minimum length collection of tours starting and ending at the root of the tree such that the demand of each terminal is covered by a single tour (i.e., the demand cannot be split), and the total demand of the terminals in each tour does not exceed the capacity of 1.For the special case when all terminals have equal demands, a long line of research culminated in a quasi-polynomial time approximation scheme [Jayaprakash and Salavatipour, TALG 2023] and a polynomial time approximation scheme [Mathieu and Zhou, TALG 2023].In this work, we study the general case when the terminals have arbitrary demands. Our main contribution is a polynomial time (1.5+ϵ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(1.5+\epsilon )$$\end{document}-approximation algorithm for the UCVRP on trees. This is the first improvement upon the 2-approximation algorithm more than 30 years ago. Our approximation ratio is essentially best possible, since it is NP-hard to approximate the UCVRP on trees to better than a 1.5 factor. [ABSTRACT FROM AUTHOR]

The Connectivity Augmentation Problem (CAP) together with a well-known special case thereof known as the Tree Augmentation Problem (TAP) are among the most basic Network Design problems. There has been a surge of interest recently to find approximation algorithms with guarantees below 2 for both TAP and CAP, culminating in the currently best approximation factor for both problems of 1.393 through quite sophisticated techniques. We present a new and arguably simple matching-based method for the well-known special case of leaf-to-leaf instances. Combining our work with prior techniques, we readily obtain a ( 4 / 3 + ε ) -approximation for Leaf-to-Leaf CAP by returning the better of our solution and one of an existing method. Prior to our work, a 4 / 3 -guarantee was only known for Leaf-to-Leaf TAP instances on trees of height 2. Moreover, when combining our technique with a recently introduced stack analysis approach, which is part of the above-mentioned 1.393-approximation, we can further improve the approximation factor to 1.29, obtaining for the first time a factor below 4 3 for a nontrivial class of TAP/CAP instances., (© The Author(s) 2023.)