IA Scholar Query: Exact exponential-time algorithms for finding bicliques.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgTue, 22 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Spurious Valleys, NP-hardness, and Tractability of Sparse Matrix Factorization With Fixed Support
https://scholar.archive.org/work/fgqienr5bres3artsiywxjls5e
The problem of approximating a dense matrix by a product of sparse factors is a fundamental problem for many signal processing and machine learning tasks. It can be decomposed into two subproblems: finding the position of the non-zero coefficients in the sparse factors, and determining their values. While the first step is usually seen as the most challenging one due to its combinatorial nature, this paper focuses on the second step, referred to as sparse matrix approximation with fixed support. First, we show its NP-hardness, while also presenting a nontrivial family of supports making the problem practically tractable with a dedicated algorithm. Then, we investigate the landscape of its natural optimization formulation, proving the absence of spurious local valleys and spurious local minima, whose presence could prevent local optimization methods to achieve global optimality. The advantages of the proposed algorithm over state-of-the-art first-order optimization methods are discussed.Quoc-Tung Lework_fgqienr5bres3artsiywxjls5eTue, 22 Nov 2022 00:00:00 GMTDagstuhl Reports, Volume 12, Issue 4, April 2022, Complete Issue
https://scholar.archive.org/work/oiijemxg5zhmzehjc3gy2mvkhm
Dagstuhl Reports, Volume 12, Issue 4, April 2022, Complete Issuework_oiijemxg5zhmzehjc3gy2mvkhmMon, 14 Nov 2022 00:00:00 GMTTight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs
https://scholar.archive.org/work/3npf6q457nbwja2f7l5kkusn3a
We investigate how efficiently a well-studied family of domination-type problems can be solved on bounded-treewidth graphs. For sets σ,ρ of non-negative integers, a (σ,ρ)-set of a graph G is a set S of vertices such that |N(u)∩ S|∈σ for every u∈ S, and |N(v)∩ S|∈ρ for every v∉S. The problem of finding a (σ,ρ)-set (of a certain size) unifies standard problems such as Independent Set, Dominating Set, Independent Dominating Set, and many others. For all pairs of finite or cofinite sets (σ,ρ), we determine (under standard complexity assumptions) the best possible value c_σ,ρ such that there is an algorithm that counts (σ,ρ)-sets in time c_σ,ρ^ tw· n^O(1) (if a tree decomposition of width tw is given in the input). For example, for the Exact Independent Dominating Set problem (also known as Perfect Code) corresponding to σ={0} and ρ={1}, we improve the 3^ tw· n^O(1) algorithm of [van Rooij, 2020] to 2^ tw· n^O(1). Despite the unusually delicate definition of c_σ,ρ, we show that our algorithms are most likely optimal, i.e., for any pair (σ, ρ) of finite or cofinite sets where the problem is non-trivial, and any ε>0, a (c_σ,ρ-ε)^ tw· n^O(1)-algorithm counting the number of (σ,ρ)-sets would violate the Counting Strong Exponential-Time Hypothesis (#SETH). For finite sets σ and ρ, our lower bounds also extend to the decision version, showing that our algorithms are optimal in this setting as well. In contrast, for many cofinite sets, we show that further significant improvements for the decision and optimization versions are possible using the technique of representative sets.Jacob Focke, Dániel Marx, Fionn Mc Inerney, Daniel Neuen, Govind S. Sankar, Philipp Schepper, Philip Wellnitzwork_3npf6q457nbwja2f7l5kkusn3aTue, 08 Nov 2022 00:00:00 GMTComputing Square Colorings on Bounded-Treewidth and Planar Graphs
https://scholar.archive.org/work/w22mnomipbepxidtc2ocabttsa
A square coloring of a graph G is a coloring of the square G^2 of G, that is, a coloring of the vertices of G such that any two vertices that are at distance at most 2 in G receive different colors. We investigate the complexity of finding a square coloring with a given number of q colors. We show that the problem is polynomial-time solvable on graphs of bounded treewidth by presenting an algorithm with running time n^2^tw + 4+O(1) for graphs of treewidth at most tw. The somewhat unusual exponent 2^tw in the running time is essentially optimal: we show that for any ϵ>0, there is no algorithm with running time f(tw)n^(2-ϵ)^tw unless the Exponential-Time Hypothesis (ETH) fails. We also show that the square coloring problem is NP-hard on planar graphs for any fixed number q ≥ 4 of colors. Our main algorithmic result is showing that the problem (when the number of colors q is part of the input) can be solved in subexponential time 2^O(n^2/3log n) on planar graphs. The result follows from the combination of two algorithms. If the number q of colors is small (≤ n^1/3), then we can exploit a treewidth bound on the square of the graph to solve the problem in time 2^O(√(qn)log n). If the number of colors is large (≥ n^1/3), then an algorithm based on protrusion decompositions and building on our result for the bounded-treewidth case solves the problem in time 2^O(nlog n/q).Akanksha Agrawal, Dániel Marx, Daniel Neuen, Jasper Slusallekwork_w22mnomipbepxidtc2ocabttsaTue, 08 Nov 2022 00:00:00 GMTTwin-width and Transductions of Proper k-Mixed-Thin Graphs
https://scholar.archive.org/work/6vvriat5qrezfgsgo3eeoq3a4m
The new graph parameter twin-width, introduced by Bonnet, Kim, Thomass e and Watrigant in 2020, allows for an FPT algorithm for testing all FO properties of graphs. This makes classes of efficiently bounded twin-width attractive from the algorithmic point of view. In particular, classes of efficiently bounded twin-width include proper interval graphs, and (as digraphs) posets of width k. Inspired by an existing generalization of interval graphs into so-called k-thin graphs, we define a new class of proper k-mixed-thin graphs which largely generalizes proper interval graphs. We prove that proper k-mixed-thin graphs have twin-width linear in k, and that a slight subclass of k-mixed-thin graphs is transduction-equivalent to posets of width k' such that there is a quadratic-polynomial relation between k and k'. In addition to that, we also give an abstract overview of the so-called red potential method which we use to prove our twin-width bounds.Jakub Balabán, Petr Hliněný, Jan Jedelskýwork_6vvriat5qrezfgsgo3eeoq3a4mSun, 06 Nov 2022 00:00:00 GMTImproved Inapproximability of VC Dimension and Littlestone's Dimension via (Unbalanced) Biclique
https://scholar.archive.org/work/cwawx24p7fap5czq5awbljruvm
We study the complexity of computing (and approximating) VC Dimension and Littlestone's Dimension when we are given the concept class explicitly. We give a simple reduction from Maximum (Unbalanced) Biclique problem to approximating VC Dimension and Littlestone's Dimension. With this connection, we derive a range of hardness of approximation results and running time lower bounds. For example, under the (randomized) Gap-Exponential Time Hypothesis or the Strongish Planted Clique Hypothesis, we show a tight inapproximability result: both dimensions are hard to approximate to within a factor of o(log n) in polynomial-time. These improve upon constant-factor inapproximability results from [Manurangsi and Rubinstein, COLT 2017].Pasin Manurangsiwork_cwawx24p7fap5czq5awbljruvmWed, 02 Nov 2022 00:00:00 GMTPlanted Dense Subgraphs in Dense Random Graphs Can Be Recovered using Graph-based Machine Learning
https://scholar.archive.org/work/6nvtfl25ujcc5mqyshicvirosi
Multiple methods of finding the vertices belonging to a planted dense subgraph in a random dense G(n, p) graph have been proposed, with an emphasis on planted cliques. Such methods can identify the planted subgraph in polynomial time, but are all limited to several subgraph structures. Here, we present PYGON, a graph neural network-based algorithm, which is insensitive to the structure of the planted subgraph. This is the first algorithm that uses learning tools for recovering dense subgraphs. We show that PYGON can recover cliques of sizes Θ (√ n), where n is the size of the background graph, comparable with the state of the art. We also show that the same algorithm can recover multiple other planted subgraphs of size Θ (√ n), in both directed and undirected graphs. We suggest a conjecture that no polynomial time PAC-learning algorithm can detect planted dense subgraphs with size smaller than O ( √ n), even if in principle one could find dense subgraphs of logarithmic size.Itay Levinas, Yoram Louzounwork_6nvtfl25ujcc5mqyshicvirosiSun, 16 Oct 2022 00:00:00 GMTExploiting ideal-sparsity in the generalized moment problem with application to matrix factorization ranks
https://scholar.archive.org/work/m5gsvmt2ovf4zex6eyfssquczi
We explore a new type of sparsity for the generalized moment problem (GMP) that we call ideal-sparsity. This sparsity exploits the presence of equality constraints requiring the measure to be supported on the variety of an ideal generated by bilinear monomials modeled by an associated graph. We show that this enables an equivalent sparse reformulation of the GMP, where the single (high dimensional) measure variable is replaced by several (lower-dimensional) measure variables supported on the maximal cliques of the graph. We explore the resulting hierarchies of moment-based relaxations for the original dense formulation of GMP and this new, equivalent ideal-sparse reformulation, when applied to the problem of bounding nonnegative- and completely positive matrix factorization ranks. We show that the ideal-sparse hierarchies provide bounds that are at least as good (and often tighter) as those obtained from the dense hierarchy. This is in sharp contrast to the situation when exploiting correlative sparsity, as is most common in the literature, where the resulting bounds are weaker than the dense bounds. Moreover, while correlative sparsity requires the underlying graph to be chordal, no such assumption is needed for ideal-sparsity. Numerical results show that the ideal-sparse bounds are often tighter and much faster to compute than their dense analogs.Milan Korda and Monique Laurent and Victor Magron and Andries Steenkampwork_m5gsvmt2ovf4zex6eyfssqucziTue, 20 Sep 2022 00:00:00 GMTOn (Random-order) Online Contention Resolution Schemes for the Matching Polytope of (Bipartite) Graphs
https://scholar.archive.org/work/6vabli7i4vcvhiyguewmcgbsra
We present new results for online contention resolution schemes for the matching polytope of graphs, in the random-order (RCRS) and adversarial (OCRS) arrival models. Our results include improved selectability guarantees (i.e., lower bounds), as well as new impossibility results (i.e., upper bounds). By well-known reductions to the prophet (secretary) matching problem, a c-selectable OCRS (RCRS) implies a c-competitive algorithm for adversarial (random order) edge arrivals. Similar reductions are also known for the query-commit matching problem. For the adversarial arrival model, we present a new analysis of the OCRS of Ezra et al. (EC, 2020). We show that this scheme is 0.344-selectable for general graphs and 0.349-selectable for bipartite graphs, improving on the previous 0.337 selectability result for this algorithm. We also show that the selectability of this scheme cannot be greater than 0.361 for general graphs and 0.382 for bipartite graphs. We further show that no OCRS can achieve a selectability greater than 0.4 for general graphs, and 0.433 for bipartite graphs. For random-order arrivals, we present two attenuation-based schemes which use new attenuation functions. Our first RCRS is 0.474-selectable for general graphs, and our second is 0.476-selectable for bipartite graphs. These results improve upon the recent 0.45 (and 0.456) selectability results for general graphs (respectively, bipartite graphs) due to Pollner et al. (EC, 2022). On general graphs, our 0.474-selectable RCRS provides the best known positive result even for offline contention resolution, and also for the correlation gap. We conclude by proving a fundamental upper bound of 0.5 on the selectability of RCRS, using bipartite graphs.Calum MacRury, Will Ma, Nathaniel Grammelwork_6vabli7i4vcvhiyguewmcgbsraThu, 15 Sep 2022 00:00:00 GMTMaximum k-Biplex Search on Bipartite Graphs: A Symmetric-BK Branching Approach
https://scholar.archive.org/work/bgba2sxtzberzj3aondatm7h2m
Enumerating maximal k-biplexes (MBPs) of a bipartite graph has been used for applications such as fraud detection. Nevertheless, there usually exists an exponential number of MBPs, which brings up two issues when enumerating MBPs, namely the effectiveness issue (many MBPs are of low values) and the efficiency issue (enumerating all MBPs is not affordable on large graphs). Existing proposals of tackling this problem impose constraints on the number of vertices of each MBP to be enumerated, yet they are still not sufficient (e.g., they require to specify the constraints, which is often not user-friendly, and cannot control the number of MBPs to be enumerated directly). Therefore, in this paper, we study the problem of finding K MBPs with the most edges called MaxBPs, where K is a positive integral user parameter. The new proposal well avoids the drawbacks of existing proposals. We formally prove the NP-hardness of the problem. We then design two branch-and-bound algorithms, among which, the better one called FastBB improves the worst-case time complexity to O^*(γ_k^ n), where O^* suppresses the polynomials, γ_k is a real number that relies on k and is strictly smaller than 2, and n is the number of vertices in the graph. For example, for k=1, γ_k is equal to 1.754. We further introduce three techniques for boosting the performance of the branch-and-bound algorithms, among which, the best one called PBIE can further improve the time complexity to O^*(γ_k^d^3) for large sparse graphs, where d is the maximum degree of the graph. We conduct extensive experiments on both real and synthetic datasets, and the results show that our algorithm is up to four orders of magnitude faster than all baselines and finding MaxBPs works better than finding all MBPs for a fraud detection application.Kaiqiang Yu, Cheng Longwork_bgba2sxtzberzj3aondatm7h2mSun, 28 Aug 2022 00:00:00 GMTAntiBenford Subgraphs: Unsupervised Anomaly Detection in Financial Networks
https://scholar.archive.org/work/73nq2n4tcjeuxglrr673kb5hmu
Benford's law describes the distribution of the first digit of numbers appearing in a wide variety of numerical data, including tax records, and election outcomes, and has been used to raise "red flags" about potential anomalies in the data such as tax evasion. In this work, we ask the following novel question: Given a large transaction or financial graph, how do we find a set of nodes that perform many transactions among each other that also deviate significantly from Benford's law? We propose the AntiBenford subgraph framework that is founded on well-established statistical principles. Furthermore, we design an efficient algorithm that finds AntiBenford subgraphs in nearlinear time on real data. We evaluate our framework on both real and synthetic data against a variety of competitors. We show empirically that our proposed framework enables the detection of anomalous subgraphs in cryptocurrency transaction networks that go undetected by state-of-the-art graph-based anomaly detection methods. Our empirical findings show that our AntiBenford framework is able to mine anomalous subgraphs, and provide novel insights into financial transaction data. The code and the datasets are available at https://github.com/ tsourakakis-lab/antibenford-subgraphs. CCS CONCEPTS • Mathematics of computing → Graph algorithms; • Computing methodologies → Anomaly detection.Tianyi Chen, Charalampos Tsourakakiswork_73nq2n4tcjeuxglrr673kb5hmuSun, 14 Aug 2022 00:00:00 GMTSubexponential Parameterized Directed Steiner Network Problems on Planar Graphs: a Complete Classification
https://scholar.archive.org/work/vj7js77zljgmrbulcuqbl3nfbi
In the Directed Steiner Network problem, the input is a directed graph G, a subset T of k vertices of G called the terminals, and a demand graph D on T. The task is to find a subgraph H of G with the minimum number of edges such that for every edge (s,t) in D, the solution H contains a directed s to t path. In this paper we investigate how the complexity of the problem depends on the demand pattern when G is planar. Formally, if 𝒟 is a class of directed graphs closed under identification of vertices, then the 𝒟-Steiner Network (𝒟-SN) problem is the special case where the demand graph D is restricted to be from 𝒟. For general graphs, Feldmann and Marx [ICALP 2016] characterized those families of demand graphs where the problem is fixed-parameter tractable (FPT) parameterized by the number k of terminals. They showed that if 𝒟 is a superset of one of the five hard families, then 𝒟-SN is W[1]-hard parameterized by k, otherwise it can be solved in time f(k)n^O(1). For planar graphs an interesting question is whether the W[1]-hard cases can be solved by subexponential parameterized algorithms. Chitnis et al. [SICOMP 2020] showed that, assuming the ETH, there is no f(k)n^o(k) time algorithm for the general 𝒟-SN problem on planar graphs, but the special case called Strongly Connected Steiner Subgraph can be solved in time f(k) n^O(√(k)) on planar graphs. We present a far-reaching generalization and unification of these two results: we give a complete characterization of the behavior of every 𝒟-SN problem on planar graphs. We show that assuming ETH, either the problem is (1) solvable in time 2^O(k)n^O(1), and not in time 2^o(k)n^O(1), or (2) solvable in time f(k)n^O(√(k)), but not in time f(k)n^o(√(k)), or (3) solvable in time f(k)n^O(k), but not in time f(k)n^o(k).Esther Galby, Sandor Kisfaludi-Bak, Daniel Marx, Roohani Sharmawork_vj7js77zljgmrbulcuqbl3nfbiThu, 11 Aug 2022 00:00:00 GMTMakespan Scheduling of Unit Jobs with Precedence Constraints in O(1.995^n) time
https://scholar.archive.org/work/jjyjuhkpzjd4vi4np24iwrrdye
In a classical scheduling problem, we are given a set of n jobs of unit length along with precedence constraints and the goal is to find a schedule of these jobs on m identical machines that minimizes the makespan. This problem is well-known to be NP-hard for an unbounded number of machines. Using standard 3-field notation, it is known as P|prec, p_j=1|C_max. We present an algorithm for this problem that runs in O(1.995^n) time. Before our work, even for m=3 machines the best known algorithms ran in O^∗(2^n) time. In contrast, our algorithm works when the number of machines m is unbounded. A crucial ingredient of our approach is an algorithm with a runtime that is only single-exponential in the vertex cover of the comparability graph of the precedence constraint graph. This heavily relies on insights from a classical result by Dolev and Warmuth (Journal of Algorithms 1984) for precedence graphs without long chains.Jesper Nederlof, Céline M. F. Swennenhuis, Karol Węgrzyckiwork_jjyjuhkpzjd4vi4np24iwrrdyeThu, 04 Aug 2022 00:00:00 GMTComputing Tree Decompositions with Small Independence Number
https://scholar.archive.org/work/y3rikipw6vcopcrrgumde24lxe
The independence number of a tree decomposition is the maximum of the independence numbers of the subgraphs induced by its bags. The tree-independence number of a graph is the minimum independence number of a tree decomposition of it. Several NP-hard graph problems, like maximum weight independent set, can be solved in time n^O(k) if the input graph is given with a tree decomposition of independence number at most k. However, it was an open problem if tree-independence number could be computed or approximated in n^f(k) time, for some function f, and in particular it was not known if maximum weight independent set could be solved in polynomial time on graphs of bounded tree-independence number. In this paper, we resolve the main open problems about the computation of tree-independence number. First, we give an algorithm that given an n-vertex graph G and an integer k, in time 2^O(k^2) n^O(k) either outputs a tree decomposition of G with independence number at most 8k, or determines that the tree-independence number of G is larger than k. This implies 2^O(k^2) n^O(k) time algorithms for various problems, like maximum weight independent set, parameterized by tree-independence number k without needing the decomposition as an input. Then, we show that the exact computing of tree-independence number is para-NP-hard, in particular, that for every constant k ≥ 4 it is NP-hard to decide if a given graph has tree-independence number at most k.Clément Dallard, Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, Martin Milaničwork_y3rikipw6vcopcrrgumde24lxeWed, 20 Jul 2022 00:00:00 GMTImproved Parameterized Complexity of Happy Set Problems
https://scholar.archive.org/work/iqzr543oxvhbtnc7eua6lcibpm
We present fixed-parameter tractable (FPT) algorithms for two problems, Maximum Happy Set (MaxHS) and Maximum Edge Happy Set (MaxEHS)–also known as Densest k-Subgraph. Given a graph G and an integer k, MaxHS asks for a set S of k vertices such that the number of happy vertices with respect to S is maximized, where a vertex v is happy if v and all its neighbors are in S. We show that MaxHS can be solved in time 𝒪(2^·· k^2 · |V(G)|) and 𝒪(8^· k^2 · |V(G)|), where and denote the modular-width and the clique-width of G, respectively. This resolves the open questions posed in literature. The MaxEHS problem is an edge-variant of MaxHS, where we maximize the number of happy edges, the edges whose endpoints are in S. In this paper we show that MaxEHS can be solved in time f()·|V(G)|^𝒪(1) and 𝒪(2^· k^2 · |V(G)|), where and denote the neighborhood diversity and the cluster deletion number of G, respectively, and f is some computable function. This result implies that MaxEHS is also fixed-parameter tractable by twin cover number.Yosuke Mizutani, Blair D. Sullivanwork_iqzr543oxvhbtnc7eua6lcibpmThu, 14 Jul 2022 00:00:00 GMTA polynomial-time approximation to a minimum dominating set in a graph
https://scholar.archive.org/work/h2sy75zraffdxlwdezwi6oib3y
A dominating set of a graph G=(V,E) is a subset of vertices S⊆ V such that every vertex v∈ V∖ S has at least one neighbor in S. Finding a dominating set with the minimum cardinality in a connected graph G=(V,E) is known to be NP-hard. A polynomial-time approximation algorithm for this problem, described here, works in two stages. At the first stage a dominant set is generated by a greedy algorithm, and at the second stage this dominating set is purified (reduced). The reduction is achieved by the analysis of the flowchart of the algorithm of the first stage and a special kind of clustering of the dominating set generated at the first stage. The clustering of the dominating set naturally leads to a special kind of a spanning forest of graph G, which serves as a basis for the second purification stage. We expose some types of graphs for which the algorithm of the first stage already delivers an optimal solution and derive sufficient conditions when the overall algorithm constructs an optimal solution. We give three alternative approximation ratios for the algorithm of the first stage, two of which are expressed in terms of solely invariant problem instance parameters, and we also give one additional approximation ratio for the overall two-stage algorithm. The greedy algorithm of the first stage turned out to be essentially the same as the earlier known state-of-the-art algorithms for the set cover and dominating set problem Chvátal and Parekh . The second purification stage results in a significant reduction of the dominant set created at the first stage, in practice. The practical behavior of both stages was verified for randomly generated problem instances. The computational experiments emphasize the gap between a solution of Stage 1 and a solution of Stage 2.Frank Hernandez, Ernesto Parra, Jose Maria Sigarreta, Nodari Vakhaniawork_h2sy75zraffdxlwdezwi6oib3ySun, 10 Jul 2022 00:00:00 GMTThe Complexity of Proportionality Degree in Committee Elections
https://scholar.archive.org/work/iegx5z5e5jayvbaky6xazqmc2y
Over the last few years, researchers have put significant effort into understanding of the notion of proportional representation in committee election. In particular, recently they have proposed the notion of proportionality degree. We study the complexity of computing committees with a given proportionality degree and of testing if a given committee provides a particular one. This way, we complement recent studies that mostly focused on the notion of (extended) justified representation. We also study the problems of testing if a cohesive group of a given size exists and of counting such groups.Łukasz Janeczko, Piotr Faliszewskiwork_iegx5z5e5jayvbaky6xazqmc2yThu, 07 Jul 2022 00:00:00 GMTPulse propagation, graph cover, and packet forwarding
https://scholar.archive.org/work/5fo2l5smorf7zfd7wtvqsc5rzu
Thank you, Christoph Lenzen, for supervising my years-long journey to my graduation, and providing so many opportunities to learn and explore. Especially when I took the scenic route down some rabbit hole 1 . And last but not least, thank you for organizing so many and great Board Games Nights! I would also like to thank Antonios Antoniadis, for having an open ear, giving great feedback, and introducing me to the topic and fine points of Packet Scheduling. Thank you so much Attila Kinali for introducing me to the world of Swiss chocolatewithout you I would weigh significantly less. Also, I thoroughly enjoyed talking with you about all the (un)important things in life; including but not limited to technical details of IRC, transistors, Quartz oscillators, GPS, the Charly and Dorothy effect, and several ski huts. I'm very grateful to Saeed Amiri. It was pure joy to develop and analyze our distance-r minimum dominating set algorithm and lower bound.ane I'm also thankful to Matthias Függer, for all the productive discussions, his unique perspective on life, and excellent recommendations on Viennese cuisine. I would like to thank Will Rosenbaum, for demonstrating that not all which is JavaScript is evil, and introducing me to the topic of Packet Forwarding. I am also thankful to him (and Christoph and Attila and Matthias) for fruitful discussions about the Metastability-Containing Frequency Adaption Module, which greatly expanded my (still limited) understanding of electrical engineering. Also, thank you Corinna Coupette, Nick Fischer, Johannes Bund, Cosmina Croitoru, André Nusser, and Bhaskar Ray Chaudhury for all the wonderful discussions on Computer Science, Law, Jazz, Pizza, Jam sessions, hats, and cake. Next, I would like to thank our Servicedesk, especially Maik Muschter and Andreas Alzano, for keeping all the infrastructure running, handling the countless tickets I opened, and keeping calm even after I found the umpteenth way to crash our computing cluster or bringing in a laptop battery that was about to explode. Außerdem will ich mich besonders bei Tim, Doris und Wilm bedanken, für all die Unterstützung, Hilfestellungen, und auch die sprichwörtlichen Arschtritte. Ohne euch wäre ich nie so weit gekommen. I would also like to thank Julia, Ferdinand, Max, Nora, and Pascal; it has been a Long Road with you. I am grateful to Christian, who helped us Find our Path. Finally, a special thank you to Julius, Andreas, Vladislav, and the silk painting youth group, who each showed me wonderful aspects of life that I would never want to miss. 1 Exercise for the reader: Let X ∼ N(µ, σ 2 ), show that lim σ→∞ Var(⌈X⌋) = σ 2 + 1/12, where ⌈ • ⌋ is the function that rounds to the nearest integer.Ben Wiederhake, Universität Des Saarlandeswork_5fo2l5smorf7zfd7wtvqsc5rzuThu, 07 Jul 2022 00:00:00 GMTThe Complexity of Proportionality Degree in Committee Elections
https://scholar.archive.org/work/naikgrflofcglfypszq2ui3foi
Over the last few years, researchers have put significant effort into understanding of the notion of proportional representation in committee election. In particular, recently they have proposed the notion of proportionality degree. We study the complexity of computing committees with a given proportionality degree and of testing if a given committee provides a particular one. This way, we complement recent studies that mostly focused on the notion of (extended) justified representation. We also study the problems of testing if a cohesive group of a given size exists and of counting such groups.Łukasz Janeczko, Piotr Faliszewskiwork_naikgrflofcglfypszq2ui3foiTue, 28 Jun 2022 00:00:00 GMTTwin-width II: small classes
https://scholar.archive.org/work/ixsguyspnfeatnxmqardqadu7q
The recently introduced twin-width of a graph G is the minimum integer d such that G has a d-contraction sequence, that is, a sequence of |V (G)| − 1 iterated vertex identifications for which the overall maximum number of red edges incident to a single vertex is at most d, where a red edge appears between two sets of identified vertices if they are not homogeneous in G (not fully adjacent nor fully non-adjacent). We show that if a graph admits a d-contraction sequence, then it also has a linear-arity tree of f (d)-contractions, for some function f . Informally if we accept to worsen the twin-width bound, we can choose the next contraction from a set of Θ(|V (G)|) pairwise disjoint pairs of vertices. This has two main consequences. First it permits to show that every bounded twin-width class is small, i.e., has at most n!c n graphs labeled by [n], for some constant c. This unifies and extends the same result for bounded treewidth graphs [Beineke and Pippert, JCT '69], proper subclasses of permutations graphs [Marcus and Tardos, JCTA '04], and proper minor-free classes [Norine et al., JCTB '06]. It implies in turn that bounded-degree graphs, interval graphs, and unit disk graphs have unbounded twin-width. The second consequence is an O(log n)-adjacency labeling scheme for bounded twin-width graphs, confirming several cases of the implicit graph conjecture. We then explore the small conjecture that, conversely, every small hereditary class has bounded twin-width. The conjecture passes many tests. Inspired by sorting networks of logarithmic depth, we show that log Θ(log log d) n-subdivisions of K n (a small class when d is constant) have twin-width at most d. We obtain a rather sharp converse with a surprisingly direct proof: the log d+1 n-subdivision of K n has twin-width at least d. Secondly graphs with bounded stack or queue number (also small classes) have bounded twin-width. These sparse classes are surprisingly rich since they contain certain (small) classes of expanders. Thirdly we show that cubic expanders obtained by iterated random 2-lifts from K 4 [Bilu and Linial, Combinatorica '06] also have bounded twin-width. These graphs are related * All the authors were supported by the ANR projects (French National Research Agency) TWIN-WIDTH (ANR-21-CE48-0014-01) and Digraphs (ANR-19-CE48-0013-01). † supported by the ANR project ASSK (ANR-18-CE40-0025-01). Édouard Bonnet et al. to so-called separable permutations and also form a small class. We suggest a promising connection between the small conjecture and group theory. Finally we define a robust notion of sparse twin-width. We show that for a hereditary class C of bounded twin-width the five following conditions are equivalent: every graph in C (1) has no K t,t subgraph for some fixed t, (2) has an adjacency matrix without a d-by-d division with a 1 entry in each of the d 2 cells for some fixed d, (3) has at most linearly many edges, (4) the subgraph closure of C has bounded twin-width, and (5) C has bounded expansion. We discuss how sparse classes with similar behavior with respect to clique subdivisions compare to bounded sparse twin-width.Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, Rémi Watrigantwork_ixsguyspnfeatnxmqardqadu7qTue, 28 Jun 2022 00:00:00 GMT