Information Processing Letters最新文献

{"title":"Fast primal-dual update against local weight update in linear assignment problem and its application","authors":"Kohei Morita ,&nbsp;Shinya Shiroshita ,&nbsp;Yutaro Yamaguchi ,&nbsp;Yu Yokoi","doi":"10.1016/j.ipl.2023.106432","DOIUrl":"10.1016/j.ipl.2023.106432","url":null,"abstract":"<div><p><span>We consider a dynamic situation in the weighted bipartite matching<span> problem: edge weights in the input graph are repeatedly updated and we are asked to maintain an optimal matching at any moment. A trivial approach is to compute an optimal matching from scratch each time an update occurs. In this paper, we show that if each update occurs locally around a single vertex, then a single execution of Dijkstra's algorithm is sufficient to preserve optimality with the aid of a dual solution. As an application of our result, we provide a faster implementation of the envy-cycle procedure for finding an envy-free allocation of indivisible items. Our algorithm runs in </span></span><span><math><mi>O</mi><mo>(</mo><mi>m</mi><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> time, while the known bound of the original one is <span><math><mi>O</mi><mo>(</mo><mi>m</mi><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span>, where <em>n</em> and <em>m</em> denote the numbers of agents and items, respectively.</p></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"183 ","pages":"Article 106432"},"PeriodicalIF":0.5,"publicationDate":"2023-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43169566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Tight FPT Approximation for Socially Fair Clustering","authors":"Dishant Goyal,&nbsp;Ragesh Jaiswal","doi":"10.1016/j.ipl.2023.106383","DOIUrl":"https://doi.org/10.1016/j.ipl.2023.106383","url":null,"abstract":"&lt;div&gt;&lt;p&gt;In this work, we study the &lt;em&gt;socially fair k-median/k-means problem&lt;/em&gt;. We are given a set of points &lt;em&gt;P&lt;/em&gt; in a metric space &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; with a distance function &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. There are &lt;em&gt;ℓ&lt;/em&gt; groups: &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. We are also given a set &lt;em&gt;F&lt;/em&gt; of feasible centers in &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. The goal in the socially fair &lt;em&gt;k&lt;/em&gt;-median problem is to find a set &lt;span&gt;&lt;math&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of &lt;em&gt;k&lt;/em&gt; centers that minimizes the maximum average cost over all the groups. That is, find &lt;em&gt;C&lt;/em&gt; that minimizes the objective function &lt;span&gt;&lt;math&gt;&lt;mi&gt;Φ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≡&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;max&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is the distance of &lt;em&gt;x&lt;/em&gt; to the closest center in &lt;em&gt;C&lt;/em&gt;. The socially fair &lt;em&gt;k&lt;/em&gt;-means problem is defined similarly by using squared distances, i.e., &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; instead of &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;. The current best approximation guarantee for both of the problems is &lt;/span&gt;&lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; due to Makarychev and Vakilian (COLT 2021). In this work, we study the fixed-parameter tractability of the problems with respect to parameter &lt;em&gt;k&lt;/em&gt;. We design &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt; approximation algorithms for the socially fair &lt;/span&gt;&lt;em&gt;k&lt;/em&gt;-median and &lt;em&gt;k&lt;/em&gt;-means problems, respectively, in FPT (fixed-parameter tractable) time &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/m","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"182 ","pages":"Article 106383"},"PeriodicalIF":0.5,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50191233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A sharper lower bound on Rankin's constant","authors":"Kang Li ,&nbsp;Fengjun Xiao ,&nbsp;Bingpeng Zhou ,&nbsp;Jinming Wen","doi":"10.1016/j.ipl.2023.106379","DOIUrl":"https://doi.org/10.1016/j.ipl.2023.106379","url":null,"abstract":"<div><p><span>Rankin's constant is an important lattice constant which has applications in many fields including cryptography and communications. In spite of its importance, few of exact values of Rankin's constant are known. In this paper, we develop a lower bound on Rankin's constant </span><span><math><msub><mrow><mi>γ</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span> which corresponds to the <em>half volume problem</em>. Compared with the previous best lower bound developed by Wen et al., ours is more than <span><math><mfrac><mrow><msqrt><mrow><mi>k</mi></mrow></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> times better. This improved lower bound on Rankin's constant directly leads to a sharper lower bound on Schnorr's constant and helps to better understand the intrinsic limitations of the 2<em>k</em>-block-Rankin reduction.</p></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"182 ","pages":"Article 106379"},"PeriodicalIF":0.5,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50191237","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On optimization problems in acyclic hypergraphs","authors":"Naoyuki Kamiyama","doi":"10.1016/j.ipl.2023.106390","DOIUrl":"https://doi.org/10.1016/j.ipl.2023.106390","url":null,"abstract":"<div><p>Binary polynomial optimization (BPO) is the problem of maximizing a polynomial function on the Boolean domain. This problem can be formulated by using a hypergraph, and various properties of the input hypergraph have been investigated from the viewpoint of polynomial-time solvability. In this paper, we especially focus on <em>β</em>-acyclic hypergraphs. For BPO over <em>β</em><span>-acyclic hypergraphs, Del Pia and Di Gregorio proposed a polynomial-time algorithm. We prove that the algorithm proposed by Del Pia and Di Gregorio can be extended to a more general optimization problem in </span><em>β</em>-acyclic hypergraphs.</p></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"182 ","pages":"Article 106390"},"PeriodicalIF":0.5,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50190916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Polynomial invariants for cactuses","authors":"Leo van Iersel ,&nbsp;Vincent Moulton ,&nbsp;Yukihiro Murakami","doi":"10.1016/j.ipl.2023.106394","DOIUrl":"https://doi.org/10.1016/j.ipl.2023.106394","url":null,"abstract":"<div><p>Graph invariants are a useful tool in graph theory. Not only do they encode useful information about the graphs to which they are associated, but complete invariants can be used to distinguish between non-isomorphic graphs. Polynomial invariants for graphs such as the well-known Tutte polynomial have been studied for several years, and recently there has been interest to also define such invariants for phylogenetic networks, a special type of graph that arises in the area of evolutionary biology. Recently Liu gave a complete invariant for (phylogenetic) trees. However, the polynomial invariants defined thus far for phylogenetic networks that are not trees require vertex labels and either contain a large number of variables, or they have exponentially many terms in the number of reticulations. This can make it difficult to compute these polynomials and to use them to analyse unlabelled networks. In this paper, we shall show how to circumvent some of these difficulties for rooted cactuses and cactuses. As well as being important in other areas such as operations research, rooted cactuses contain some common classes of phylogenetic networks such phylogenetic trees and level-1 networks. More specifically, we define a polynomial <em>F</em> that is a complete invariant for the class of rooted cactuses without vertices of indegree 1 and outdegree 1 that has 5 variables, and a polynomial <em>Q</em> that is a complete invariant for the class of rooted cactuses that has 6 variables whose degree can be bounded linearly in terms of the size of the rooted cactus. We also explain how to extend the <em>Q</em> polynomial to define a complete invariant for leaf-labelled rooted cactuses as well as (unrooted) cactuses.</p></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"182 ","pages":"Article 106394"},"PeriodicalIF":0.5,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50190923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Connectivity, super connectivity and generalized 3-connectivity of folded divide-and-swap cubes","authors":"Shu-Li Zhao ,&nbsp;Jou-Ming Chang","doi":"10.1016/j.ipl.2023.106377","DOIUrl":"https://doi.org/10.1016/j.ipl.2023.106377","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Let &lt;em&gt;G&lt;/em&gt;&lt;span&gt; be a connected graph and &lt;/span&gt;&lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. A tree &lt;em&gt;T&lt;/em&gt; in &lt;em&gt;G&lt;/em&gt; is called an &lt;em&gt;S&lt;/em&gt;-tree if &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Two &lt;em&gt;S&lt;/em&gt;-trees &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are internally disjoint if &lt;span&gt;&lt;math&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;∅&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. For an integer &lt;span&gt;&lt;math&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, the &lt;em&gt;generalized r-connectivity&lt;/em&gt; of a graph &lt;em&gt;G&lt;/em&gt;, denoted by &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, is defined as &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;min&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; denotes the maximum number of pairwise internally disjoint &lt;em&gt;S&lt;/em&gt;-trees in &lt;em&gt;G&lt;/em&gt;. The folded divide-and-swap cube, denoted by &lt;span&gt;&lt;math&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;, is a variant of the hypercube. &lt;/span&gt;&lt;span&gt;&lt;math&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt; has better network cost measured by the product of degree and diameter than the hypercube and folded hypercube. Connectivity and super connectivity are two important parameters to evaluate the reliability of an interconnection network. In addition, as a generalization of traditional connectivity, generalized connectivity can more accurately assess the reliability of an interconnection network. In this paper, we first acquire the (edge) connectivity and super (edge) connectivity of &lt;/span&gt;&lt;span&gt;&lt;math&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and then obtain the generalized 3-connectivit","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"182 ","pages":"Article 106377"},"PeriodicalIF":0.5,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50191229","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On maximum bipartite matching with separation","authors":"Pasin Manurangsi ,&nbsp;Erel Segal-Halevi ,&nbsp;Warut Suksompong","doi":"10.1016/j.ipl.2023.106388","DOIUrl":"https://doi.org/10.1016/j.ipl.2023.106388","url":null,"abstract":"<div><p>Maximum bipartite matching is a fundamental algorithmic problem which can be solved in polynomial time. We consider a natural variant in which there is a separation constraint: the vertices on one side lie on a path or a grid, and two vertices that are close to each other are not allowed to be matched simultaneously. We show that the problem is hard to approximate even for paths, and provide constant-factor approximation algorithms for both paths and grids.</p></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"182 ","pages":"Article 106388"},"PeriodicalIF":0.5,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50190922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Exact PPS sampling with bounded sample size","authors":"Brian Hentschel ,&nbsp;Peter J. Haas ,&nbsp;Yuanyuan Tian","doi":"10.1016/j.ipl.2023.106382","DOIUrl":"https://doi.org/10.1016/j.ipl.2023.106382","url":null,"abstract":"<div><p>Probability proportional to size (PPS) sampling schemes with a target sample size aim to produce a sample comprising a specified number <em>n</em> of items while ensuring that each item in the population appears in the sample with a probability proportional to its specified “weight” (also called its “size”). These two objectives, however, cannot always be achieved simultaneously. Existing PPS schemes prioritize control of the sample size, violating the PPS property if necessary. We provide a new PPS scheme, called EB-PPS, that allows a different trade-off: EB-PPS enforces the PPS property at all times while ensuring that the sample size never exceeds the target value <em>n</em>. The sample size is exactly equal to <em>n</em> if possible, and otherwise has maximal expected value and minimal variance. Thus we bound the sample size, thereby avoiding storage overflows and helping to control the time required for analytics over the sample, while allowing the user complete control over the sample contents. In the context of training classifiers at scale under imbalanced loss functions, we show that such control yields superior classifiers. The method is both simple to implement and efficient, being a one-pass streaming algorithm with an amortized processing time of <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> per item, which makes it computationally preferable even in cases where both EB-PPS and prior algorithms can ensure the PPS property and a target sample size simultaneously.</p></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"182 ","pages":"Article 106382"},"PeriodicalIF":0.5,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50191234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An improved kernel for the flip distance problem on simple convex polygons","authors":"Miguel Bosch-Calvo ,&nbsp;Steven Kelk","doi":"10.1016/j.ipl.2023.106381","DOIUrl":"https://doi.org/10.1016/j.ipl.2023.106381","url":null,"abstract":"<div><p>The complexity of computing the flip distance between two triangulations of a simple convex polygon is unknown. Here we approach the problem from a parameterized complexity perspective and improve upon the 2<em>k</em> kernel of Lucas <span>[12]</span>. Specifically, we describe a kernel of size <span><math><mfrac><mrow><mn>4</mn><mi>k</mi></mrow><mrow><mn>3</mn></mrow></mfrac></math></span> and then show how it can be improved to <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>ϵ</mi><mo>)</mo><mi>k</mi></math></span> for every constant <span><math><mi>ϵ</mi><mo>&gt;</mo><mn>0</mn></math></span>. By ensuring that the kernel consists of a single instance our result yields a kernel of the same magnitude (up to additive terms) for the almost equivalent rotation distance problem on rooted, ordered binary trees. The earlier work of Lucas left the kernel as a disjoint set of instances, potentially allowing very minor differences in the definition of the size of instances to accumulate, causing a constant-factor distortion in the kernel size when switching between flip distance and rotation distance formulations. Our approach avoids this sensitivity. We have also undertaken experiments to understand how much reduction is achieved by our kernel in practice.</p></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"182 ","pages":"Article 106381"},"PeriodicalIF":0.5,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50191235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Comparing approximate and probabilistic differential privacy parameters","authors":"Vincent Guingona ,&nbsp;Alexei Kolesnikov ,&nbsp;Julianne Nierwinski ,&nbsp;Avery Schweitzer","doi":"10.1016/j.ipl.2023.106380","DOIUrl":"https://doi.org/10.1016/j.ipl.2023.106380","url":null,"abstract":"<div><p><span>This paper compares two notions of differential privacy: approximate differential privacy (ADP) and probabilistic differential privacy (PrDP). It is well-known that the PrDP implies the ADP; and it was established in </span><span>[7]</span> that the ADP implies the PrDP, after a penalty on the parameters <em>ε</em> and <em>δ</em> that are used in the definitions of both properties. We show that the condition found in <span>[7]</span><span> is tight: if it fails, we construct a randomized algorithm that has ADP but not PrDP.</span></p></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"182 ","pages":"Article 106380"},"PeriodicalIF":0.5,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50191236","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}