Non-Uniform k-Center and Greedy Clustering

In the Non-Uniform k-Center (NUkC) problem, a generalization of the famous k-center clustering problem, we want to cover the given set of points in a metric space by finding a placement of balls with specified radii. In t-NUkC, we assume that the number of distinct radii is equal to t, and we are allowed to use k_i balls of radius r_i, for 1 ≤ i ≤ t. This problem was introduced by Chakrabarty et al. [ACM Trans. Alg. 16(4):46:1-46:19], who showed that a constant approximation for t-NUkC is not possible if t is unbounded, assuming ≠ NP. On the other hand, they gave a bicriteria approximation that violates the number of allowed balls as well as the given radii by a constant factor. They also conjectured that a constant approximation for t-NUkC should be possible if t is a fixed constant. Since then, there has been steady progress towards resolving this conjecture - currently, a constant approximation for 3-NUkC is known via the results of Chakrabarty and Negahbani [IPCO 2021], and Jia et al. [SOSA 2022]. We push the horizon by giving an O(1)-approximation for the Non-Uniform k-Center for 4 distinct types of radii. Our result is obtained via a novel combination of tools and techniques from the k-center literature, which also demonstrates that the different generalizations of k-center involving non-uniform radii, and multiple coverage constraints (i.e., colorful k-center), are closely interlinked with each other. We hope that our ideas will contribute towards a deeper understanding of the t-NUkC problem, eventually bringing us closer to the resolution of the CGK conjecture.publishedVersio

New Bounds on Augmenting Steps of Block-Structured Integer Programs

Iterative augmentation has recently emerged as an overarching method for solving Integer Programs (IP) in variable dimension, in stark contrast with the volume and flatness techniques of IP in fixed dimension. Here we consider 4-block n-fold integer programs, which are the most general class considered so far. A 4-block n-fold IP has a constraint matrix which consists of n copies of small matrices A, B, and D, and one copy of C, in a specific block structure. Iterative augmentation methods rely on the so-called Graver basis of the constraint matrix, which constitutes a set of fundamental augmenting steps. All existing algorithms rely on bounding the ??- or ?_?-norm of elements of the Graver basis. Hemmecke et al. [Math. Prog. 2014] showed that 4-block n-fold IP has Graver elements of ?_?-norm at most ?_FPT(n^{2^{s_D}}), leading to an algorithm with a similar runtime; here, s_D is the number of rows of matrix D and ?_FPT hides a multiplicative factor that is only dependent on the small matrices A,B,C,D, However, it remained open whether their bounds are tight, in particular, whether they could be improved to ?_FPT(1), perhaps at least in some restricted cases. We prove that the ?_?-norm of the Graver elements of 4-block n-fold IP is upper bounded by ?_FPT(n^{s_D}), improving significantly over the previous bound ?_FPT(n^{2^{s_D}}). We also provide a matching lower bound of ?(n^{s_D}) which even holds for arbitrary non-zero lattice elements, ruling out augmenting algorithm relying on even more restricted notions of augmentation than the Graver basis. We then consider a special case of 4-block n-fold in which C is a zero matrix, called 3-block n-fold IP. We show that while the ?_?-norm of its Graver elements is ?(n^{s_D}), there exists a different decomposition into lattice elements whose ?_?-norm is bounded by ?_FPT(1), which allows us to provide improved upper bounds on the ?_?-norm of Graver elements for 3-block n-fold IP. The key difference between the respective decompositions is that a Graver basis guarantees a sign-compatible decomposition; this property is critical in applications because it guarantees each step of the decomposition to be feasible. Consequently, our improved upper bounds let us establish faster algorithms for 3-block n-fold IP and 4-block IP, and our lower bounds strongly hint at parameterized hardness of 4-block and even 3-block n-fold IP. Furthermore, we show that 3-block n-fold IP is without loss of generality in the sense that 4-block n-fold IP can be solved in FPT oracle time by taking an algorithm for 3-block n-fold IP as an oracle

The complexity landscape of decompositional parameters for ILP : programs with few global variables and constraints

Integer Linear Programming (ILP) has a broad range of applications in various areas of artificial intelligence. Yet in spite of recent advances, we still lack a thorough understanding of which structural restrictions make ILP tractable. Here we study ILP instances consisting of a small number of “global” variables and/or constraints such that the remaining part of the instance consists of small and otherwise independent components; this is captured in terms of a structural measure we call fracture backdoors which generalizes, for instance, the well-studied class of N-fold ILP instances. Our main contributions can be divided into three parts. First, we formally develop fracture backdoors and obtain exact and approximation algorithms for computing these. Second, we exploit these backdoors to develop several new parameterized algorithms for ILP; the performance of these algorithms will naturally scale based on the number of global variables or constraints in the instance. Finally, we complement the developed algorithms with matching lower bounds. Altogether, our results paint a near-complete complexity landscape of ILP with respect to fracture backdoors

Parameterized Complexity of Fair Bisection: FPT-Approximation meets Unbreakability

In the Minimum Bisection problem, input is a graph $G$ and the goal is to partition the vertex set into two parts $A$ and $B$, such that $||A|-|B|| \le 1$ and the number $k$ of edges between $A$ and $B$ is minimized. This problem can be viewed as a clustering problem where edges represent similarity, and the task is to partition the vertices into two equally sized clusters, while minimizing the number of pairs of similar objects that end up in different clusters. In this paper, we initiate the study of a fair version of Minimum Bisection. In this problem, the vertices of the graph are colored using one of $c \ge 1$ colors. The goal is to find a bisection $(A, B)$ with at most $k$ edges between the parts, such that for each color $i\in [c]$, $A$ has exactly $r_i$ vertices of color $i$. We first show that Fair Bisection is $W$[1]-hard parameterized by $c$ even when $k = 0$. On the other hand, our main technical contribution shows that is that this hardness result is simply a consequence of the very strict requirement that each color class $i$ has {\em exactly} $r_i$ vertices in $A$. In particular, we give an $f(k,c,\epsilon)n^{O(1)}$ time algorithm that finds a balanced partition $(A, B)$ with at most $k$ edges between them, such that for each color $i\in [c]$, there are at most $(1\pm \epsilon)r_i$ vertices of color $i$ in $A$. Our approximation algorithm is best viewed as a proof of concept that the technique introduced by [Lampis, ICALP '18] for obtaining FPT-approximation algorithms for problems of bounded tree-width or clique-width can be efficiently exploited even on graphs of unbounded width. The key insight is that the technique of Lampis is applicable on tree decompositions with unbreakable bags (as introduced in [Cygan et al., SIAM Journal on Computing '14]). Along the way, we also derive a combinatorial result regarding tree decompositions of graphs.Comment: Full version of ESA 2023 paper. Abstract shortened to meet the character limi

Tight Approximation Guarantees for Concave Coverage Problems

33 pages. v3 minor corrections and added FPT hardnessInternational audienceIn the maximum coverage problem, we are given subsets $T_1, \ldots, T_m$ of a universe $[n]$ along with an integer $k$ and the objective is to find a subset $S \subseteq [m]$ of size $k$ that maximizes $C(S) := \Big|\bigcup_{i \in S} T_i\Big|$. It is a classic result that the greedy algorithm for this problem achieves an optimal approximation ratio of $1-e^{-1}$. In this work we consider a generalization of this problem wherein an element $a$ can contribute by an amount that depends on the number of times it is covered. Given a concave, nondecreasing function $\varphi$, we define $C^{\varphi}(S) := \sum_{a \in [n]}w_a\varphi(|S|_a)$, where $|S|_a = |\{i \in S : a \in T_i\}|$. The standard maximum coverage problem corresponds to taking $\varphi(j) = \min\{j,1\}$. For any such $\varphi$, we provide an efficient algorithm that achieves an approximation ratio equal to the Poisson concavity ratio of $\varphi$, defined by $\alpha_{\varphi} := \min_{x \in \mathbb{N}^*} \frac{\mathbb{E}[\varphi(\text{Poi}(x))]}{\varphi(\mathbb{E}[\text{Poi}(x)])}$. Complementing this approximation guarantee, we establish a matching NP-hardness result when $\varphi$ grows in a sublinear way. As special cases, we improve the result of [Barman et al., IPCO, 2020] about maximum multi-coverage, that was based on the unique games conjecture, and we recover the result of [Dudycz et al., IJCAI, 2020] on multi-winner approval-based voting for geometrically dominant rules. Our result goes beyond these special cases and we illustrate it with applications to distributed resource allocation problems, welfare maximization problems and approval-based voting for general rules

Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Algorithmic aspects of resource allocation and multiwinner voting: theory and experiments

This thesis is concerned with investigating elements of computational social choice in the light of real-world applications. We contribute to a better understanding of the areas of fair allocation and multiwinner voting. For both areas, inspired by real-world scenarios, we propose several new notions and extensions of existing models. Then, we analyze the complexity of answering the computational questions raised by the introduced concepts. To this end, we look through the lens of parameterized complexity. We identify different parameters which describe natural features specific to the computational problems we investigate. Exploiting the parameters, we successfully develop efficient algorithms for spe- cific cases of the studied problems. We complement our analysis by showing which parameters presumably cannot be utilized for seeking efficient algorithms. Thereby, we provide comprehensive pictures of the computational complexity of the studied problems. Specifically, we concentrate on four topics that we present below, grouped by our two areas of interest. For all but one topic, we present experimental studies based on implementations of newly developed algorithms. We first focus on fair allocation of indivisible resources. In this setting, we consider a collection of indivisible resources and a group of agents. Each agent reports its utility evaluation of every resource and the task is to “fairly” allocate the resources such that each resource is allocated to at most one agent. We concentrate on the two following issues regarding this scenario. The social context in fair allocation of indivisible resources. In many fair allocation settings, it is unlikely that every agent knows all other agents. For example, consider a scenario where the agents represent employees of a large corporation. It is highly unlikely that every employee knows every other employee. Motivated by such settings, we come up with a new model of graph envy-freeness by adapting the classical envy-freeness notion to account for social relations of agents modeled as social networks. We show that if the given social network of agents is simple (for example, if it is a directed acyclic graph), then indeed we can sometimes find fair allocations efficiently. However, we contrast tractability results with showing NP-hardness for several cases, including those in which the given social network has a constant degree. Fair allocations among few agents with bounded rationality. Bounded rationality is the idea that humans, due to cognitive limitations, tend to simplify problems that they face. One of its emanations is that human agents usually tend to report simple utilities over the resources that they want to allocate; for example, agents may categorize the available resources only into two groups of desirable and undesirable ones. Applying techniques for solving integer linear programs, we show that exploiting bounded rationality leads to efficient algorithms for finding envy-free and Pareto-efficient allocations, assuming a small number of agents. Further, we demonstrate that our result actually forms a framework that can be applied to a number of different fairness concepts like envy-freeness up to one good or envy-freeness up to any good. This way, we obtain efficient algorithms for a number of fair allocation problems (assuming few agents with bounded rationality). We also empirically show that our technique is applicable in practice. Further, we study multiwinner voting, where we are given a collection of voters and their preferences over a set of candidates. The outcome of a multiwinner voting rule is a group (or a set of groups in case of ties) of candidates that reflect the voters’ preferences best according to some objective. In this context, we investigate the following themes. The robustness of election outcomes. We study how robust outcomes of multiwinner elections are against possible mistakes made by voters. Assuming that each voter casts a ballot in a form of a ranking of candidates, we represent a mistake by a swap of adjacent candidates in a ballot. We find that for rules such as SNTV, k-Approval, and k-Borda, it is computationally easy to find the minimum number of swaps resulting in a change of an outcome. This task is, however, NP-hard for STV and the Chamberlin-Courant rule. We conclude our study of robustness with experimentally studying the average number of random swaps leading to a change of an outcome for several rules. Strategic voting in multiwinner elections. We ask whether a given group of cooperating voters can manipulate an election outcome in a favorable way. We focus on the k-Approval voting rule and we show that the computational complexity of answering the posed question has a rich structure. We spot several cases for which our problem is polynomial-time solvable. However, we also identify NP-hard cases. For several of them, we show how to circumvent the hardness by fixed-parameter tractability. We also present experimental studies indicating that our algorithms are applicable in practice

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum