Pub Date : 2025-01-13DOI: 10.1016/j.ipl.2025.106557
Tamanna Chhabra , Sukhpal Singh Ghuman , Jorma Tarhio
We present new algorithms for the k mismatches version of approximate string matching. Our algorithms utilize the SIMD (Single Instruction Multiple Data) instruction set extensions, particularly AVX2 and AVX-512 instructions. Our approach is an extension of an earlier algorithm for exact string matching with SSE2 and AVX2. In addition, we modify this exact string matching algorithm to work with AVX-512. We demonstrate the competitiveness of our solutions by practical experiments. Our algorithms outperform earlier algorithms for both exact and approximate string matching on various benchmark data sets.
{"title":"String searching with mismatches using AVX2 and AVX-512 instructions","authors":"Tamanna Chhabra , Sukhpal Singh Ghuman , Jorma Tarhio","doi":"10.1016/j.ipl.2025.106557","DOIUrl":"10.1016/j.ipl.2025.106557","url":null,"abstract":"<div><div>We present new algorithms for the <em>k</em> mismatches version of approximate string matching. Our algorithms utilize the SIMD (Single Instruction Multiple Data) instruction set extensions, particularly AVX2 and AVX-512 instructions. Our approach is an extension of an earlier algorithm for exact string matching with SSE2 and AVX2. In addition, we modify this exact string matching algorithm to work with AVX-512. We demonstrate the competitiveness of our solutions by practical experiments. Our algorithms outperform earlier algorithms for both exact and approximate string matching on various benchmark data sets.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106557"},"PeriodicalIF":0.7,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099052","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-03DOI: 10.1016/j.ipl.2024.106556
Naoto Ohsaka
Given a two-prover game G and its two satisfying labelings and , the Label Cover Reconfiguration problem asks whether can be transformed into by repeatedly changing the label of a single vertex while preserving any intermediate labeling satisfying G. We consider its optimization version by relaxing the feasibility of labelings, referred to as Maxmin Label Cover Reconfiguration: We are allowed to pass through any non-satisfying labelings, but required to maximize the “soundness error,” which is defined as the minimum fraction of satisfied edges during transformation from to . Since the parallel repetition theorem of Raz (1998) [32], which implies -hardness of approximating Label Cover within any constant factor, gives strong inapproximability results for many -hard problems, one may think of using Maxmin Label Cover Reconfiguration to derive inapproximability results for reconfiguration problems. We prove the following results on Maxmin Label Cover Reconfiguration, which display different trends from those of Label Cover and the parallel repetition theorem:
•
Maxmin Label Cover Reconfiguration can be approximated within a factor of for some restricted graph classes, including biregular graphs, balanced bipartite graphs with no isolated vertices, and superconstant average degree graphs.
•
A “naive” parallel repetition of Maxmin Label Cover Reconfiguration does not decrease the soundness error for every two-prover game.
•
Label Cover Reconfiguration on projection games can be decided in polynomial time.
Our results suggest that a reconfiguration analogue of the parallel repetition theorem is unlikely.
{"title":"On approximate reconfigurability of label cover","authors":"Naoto Ohsaka","doi":"10.1016/j.ipl.2024.106556","DOIUrl":"10.1016/j.ipl.2024.106556","url":null,"abstract":"<div><div>Given a two-prover game <em>G</em> and its two satisfying labelings <span><math><msub><mrow><mi>ψ</mi></mrow><mrow><mi>ini</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>ψ</mi></mrow><mrow><mi>tar</mi></mrow></msub></math></span>, the <span>Label Cover Reconfiguration</span> problem asks whether <span><math><msub><mrow><mi>ψ</mi></mrow><mrow><mi>ini</mi></mrow></msub></math></span> can be transformed into <span><math><msub><mrow><mi>ψ</mi></mrow><mrow><mi>tar</mi></mrow></msub></math></span> by repeatedly changing the label of a single vertex while preserving any intermediate labeling satisfying <em>G</em>. We consider its optimization version by relaxing the feasibility of labelings, referred to as <span>Maxmin Label Cover Reconfiguration</span>: We are allowed to pass through any <em>non-satisfying</em> labelings, but required to maximize the “soundness error,” which is defined as the <em>minimum</em> fraction of satisfied edges during transformation from <span><math><msub><mrow><mi>ψ</mi></mrow><mrow><mi>ini</mi></mrow></msub></math></span> to <span><math><msub><mrow><mi>ψ</mi></mrow><mrow><mi>tar</mi></mrow></msub></math></span>. Since the parallel repetition theorem of Raz (1998) <span><span>[32]</span></span>, which implies <figure><img></figure>-hardness of approximating <span>Label Cover</span> within any constant factor, gives strong inapproximability results for many <figure><img></figure>-hard problems, one may think of using <span>Maxmin Label Cover Reconfiguration</span> to derive inapproximability results for reconfiguration problems. We prove the following results on <span>Maxmin Label Cover Reconfiguration</span>, which display different trends from those of <span>Label Cover</span> and the parallel repetition theorem:<ul><li><span>•</span><span><div><span>Maxmin Label Cover Reconfiguration</span> can be approximated within a factor of <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>−</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> for some restricted graph classes, including biregular graphs, balanced bipartite graphs with no isolated vertices, and superconstant average degree graphs.</div></span></li><li><span>•</span><span><div>A “naive” parallel repetition of <span>Maxmin Label Cover Reconfiguration</span> does not decrease the soundness error for <em>every</em> two-prover game.</div></span></li><li><span>•</span><span><div><span>Label Cover Reconfiguration</span> on <em>projection games</em> can be decided in polynomial time.</div></span></li></ul> Our results suggest that a reconfiguration analogue of the parallel repetition theorem is unlikely.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106556"},"PeriodicalIF":0.7,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099050","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-27DOI: 10.1016/j.ipl.2024.106554
Umang Bhaskar , A.R. Sricharan , Rohit Vaish
We study the problem of fair cake-cutting where each agent receives a connected piece of the cake. A division of the cake is deemed fair if it is equitable, which means that all agents derive the same value from their assigned piece. Prior work has established the existence of a connected equitable division for agents with nonnegative valuations using various techniques. We provide a simple proof of this result using Sperner's lemma. Our proof extends known existence results for connected equitable divisions to significantly more general classes of valuations, including nonnegative valuations with externalities, as well as several interesting subclasses of general (possibly negative) valuations.
{"title":"Connected equitable cake division via Sperner's lemma","authors":"Umang Bhaskar , A.R. Sricharan , Rohit Vaish","doi":"10.1016/j.ipl.2024.106554","DOIUrl":"10.1016/j.ipl.2024.106554","url":null,"abstract":"<div><div>We study the problem of fair cake-cutting where each agent receives a connected piece of the cake. A division of the cake is deemed fair if it is <em>equitable</em>, which means that all agents derive the same value from their assigned piece. Prior work has established the existence of a connected equitable division for agents with nonnegative valuations using various techniques. We provide a simple proof of this result using Sperner's lemma. Our proof extends known existence results for connected equitable divisions to significantly more general classes of valuations, including nonnegative valuations with externalities, as well as several interesting subclasses of general (possibly negative) valuations.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106554"},"PeriodicalIF":0.7,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099049","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}
Pub Date : 2024-12-20DOI: 10.1016/j.ipl.2024.106555
Hassene Aissi , Mourad Baiou , Francisco Barahona
Consider an undirected graph with positive integer edge weights. Subramanian [11] established an upper bound of on the number of minimum weight cycles. We present a new algorithm to enumerate all minimum weight cycles with a complexity of . Using this algorithm, we derive the following upper bounds for the number of minimum weight cycles: if the minimum weight is even, the bound is , and if it is odd, the bound is . Notably, we improve Subramanian's bound by an order of magnitude when the minimum weight of a cycle is odd. Additionally, we demonstrate that these bounds are asymptotically tight.
{"title":"New bounds for the number of lightest cycles in undirected graphs","authors":"Hassene Aissi , Mourad Baiou , Francisco Barahona","doi":"10.1016/j.ipl.2024.106555","DOIUrl":"10.1016/j.ipl.2024.106555","url":null,"abstract":"<div><div>Consider an undirected graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> with positive integer edge weights. Subramanian <span><span>[11]</span></span> established an upper bound of <span><math><mo>|</mo><mi>V</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>4</mn></mrow></msup><mo>/</mo><mn>6</mn></math></span> on the number of minimum weight cycles. We present a new algorithm to enumerate all minimum weight cycles with a complexity of <span><math><mi>O</mi><mo>(</mo><mo>|</mo><mi>V</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>3</mn></mrow></msup><mo>(</mo><mo>|</mo><mi>E</mi><mo>|</mo><mo>+</mo><mo>|</mo><mi>V</mi><mo>|</mo><mi>log</mi><mo></mo><mo>|</mo><mi>V</mi><mo>|</mo><mo>)</mo><mo>)</mo></math></span>. Using this algorithm, we derive the following upper bounds for the number of minimum weight cycles: if the minimum weight is even, the bound is <span><math><mo>|</mo><mi>V</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>4</mn></mrow></msup><mo>/</mo><mn>4</mn></math></span>, and if it is odd, the bound is <span><math><mo>|</mo><mi>V</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>3</mn></mrow></msup><mo>/</mo><mn>2</mn></math></span>. Notably, we improve Subramanian's bound by an order of magnitude when the minimum weight of a cycle is odd. Additionally, we demonstrate that these bounds are asymptotically tight.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106555"},"PeriodicalIF":0.7,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099048","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}
Pub Date : 2024-12-18DOI: 10.1016/j.ipl.2024.106553
Shravas Rao
A matrix satisfies the restricted isometry property if is approximately equal to for all k-sparse vectors x. We give a construction of RIP matrices with the optimal rows using bits of randomness. The main technical ingredient is an extension of the Hanson-Wright inequality to ε-biased distributions.
{"title":"Satisfying the restricted isometry property with the optimal number of rows and slightly less randomness","authors":"Shravas Rao","doi":"10.1016/j.ipl.2024.106553","DOIUrl":"10.1016/j.ipl.2024.106553","url":null,"abstract":"<div><div>A matrix <span><math><mi>Φ</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>Q</mi><mo>×</mo><mi>N</mi></mrow></msup></math></span> satisfies the restricted isometry property if <span><math><msubsup><mrow><mo>‖</mo><mi>Φ</mi><mi>x</mi><mo>‖</mo></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math></span> is approximately equal to <span><math><msubsup><mrow><mo>‖</mo><mi>x</mi><mo>‖</mo></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math></span> for all <em>k</em>-sparse vectors <em>x</em>. We give a construction of RIP matrices with the optimal <span><math><mi>Q</mi><mo>=</mo><mi>O</mi><mo>(</mo><mi>k</mi><mi>log</mi><mo></mo><mo>(</mo><mi>N</mi><mo>/</mo><mi>k</mi><mo>)</mo><mo>)</mo></math></span> rows using <span><math><mi>O</mi><mo>(</mo><mi>k</mi><mi>log</mi><mo></mo><mo>(</mo><mi>N</mi><mo>/</mo><mi>k</mi><mo>)</mo><mi>log</mi><mo></mo><mo>(</mo><mi>k</mi><mo>)</mo><mo>)</mo></math></span> bits of randomness. The main technical ingredient is an extension of the Hanson-Wright inequality to <em>ε</em>-biased distributions.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106553"},"PeriodicalIF":0.7,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099047","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}
Pub Date : 2024-12-17DOI: 10.1016/j.ipl.2024.106552
Arka Ray , Sai Sandeep
The Geometric Bin Packing (GBP) problem is a generalization of Bin Packing where the input is a set of d-dimensional rectangles, and the goal is to pack them into d-dimensional unit cubes efficiently. It is NP-hard to obtain a PTAS for the problem, even when . For general d, the best-known approximation algorithm has an approximation guarantee that is exponential in d. In contrast, the best hardness of approximation is still a small constant inapproximability from the case when . In this paper, we show that the problem cannot be approximated within a factor unless .
Recently, d-dimensional Vector Bin Packing, a problem closely related to the GBP, was shown to be hard to approximate within a factor when d is a fixed constant, using a notion of Packing Dimension of set families. In this paper, we introduce a geometric analog of it, the Geometric Packing Dimension of set families. While we fall short of obtaining similar inapproximability results for the Geometric Bin Packing problem when d is fixed, we prove a couple of key properties of the Geometric Packing Dimension which highlight fundamental differences between Geometric Bin Packing and Vector Bin Packing.
{"title":"Improved hardness of approximation for Geometric Bin Packing","authors":"Arka Ray , Sai Sandeep","doi":"10.1016/j.ipl.2024.106552","DOIUrl":"10.1016/j.ipl.2024.106552","url":null,"abstract":"<div><div>The Geometric Bin Packing (GBP) problem is a generalization of Bin Packing where the input is a set of <em>d</em>-dimensional rectangles, and the goal is to pack them into <em>d</em>-dimensional unit cubes efficiently. It is NP-hard to obtain a PTAS for the problem, even when <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>. For general <em>d</em>, the best-known approximation algorithm has an approximation guarantee that is exponential in <em>d</em>. In contrast, the best hardness of approximation is still a small constant inapproximability from the case when <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>. In this paper, we show that the problem cannot be approximated within a <span><math><msup><mrow><mi>d</mi></mrow><mrow><mn>1</mn><mo>−</mo><mi>ϵ</mi></mrow></msup></math></span> factor unless <span><math><mtext>NP</mtext><mo>=</mo><mtext>P</mtext></math></span>.</div><div>Recently, <em>d</em>-dimensional Vector Bin Packing, a problem closely related to the GBP, was shown to be hard to approximate within a <span><math><mi>Ω</mi><mo>(</mo><mi>log</mi><mo></mo><mi>d</mi><mo>)</mo></math></span> factor when <em>d</em> is a fixed constant, using a notion of Packing Dimension of set families. In this paper, we introduce a geometric analog of it, the Geometric Packing Dimension of set families. While we fall short of obtaining similar inapproximability results for the Geometric Bin Packing problem when <em>d</em> is fixed, we prove a couple of key properties of the Geometric Packing Dimension which highlight fundamental differences between Geometric Bin Packing and Vector Bin Packing.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106552"},"PeriodicalIF":0.7,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099046","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}
Pub Date : 2024-12-10DOI: 10.1016/j.ipl.2024.106551
David Flores-Peñaloza , Mario A. Lopez , Nestaly Marín , David Orden
Let P be a k-colored set of n points in the plane, . We study the problem of deciding if P contains a subset of four points of different colors such that its Rectilinear Convex Hull has positive area. We show this problem to be equivalent to deciding if there exists a point c in the plane such that each of the open quadrants defined by c contains a point of P, each of them having a different color. We provide an -time algorithm for this problem, where the hidden constant does not depend on k; then, we prove that this problem has time complexity in the algebraic computation tree model. No general position assumptions for P are required.
{"title":"An efficient algorithm for identifying rainbow ortho-convex 4-sets in k-colored point sets","authors":"David Flores-Peñaloza , Mario A. Lopez , Nestaly Marín , David Orden","doi":"10.1016/j.ipl.2024.106551","DOIUrl":"10.1016/j.ipl.2024.106551","url":null,"abstract":"<div><div>Let <em>P</em> be a <em>k</em>-colored set of <em>n</em> points in the plane, <span><math><mn>4</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></math></span>. We study the problem of deciding if <em>P</em> contains a subset of four points of different colors such that its Rectilinear Convex Hull has positive area. We show this problem to be equivalent to deciding if there exists a point <em>c</em> in the plane such that each of the open quadrants defined by <em>c</em> contains a point of <em>P</em>, each of them having a different color. We provide an <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span>-time algorithm for this problem, where the hidden constant does not depend on <em>k</em>; then, we prove that this problem has time complexity <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> in the algebraic computation tree model. No general position assumptions for <em>P</em> are required.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106551"},"PeriodicalIF":0.7,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099044","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}
Pub Date : 2024-12-09DOI: 10.1016/j.ipl.2024.106550
Adam J. Przeździecki
We present a very simple algorithm for computing Pfaffians which uses no division operations. Essentially, it amounts to iterating matrix multiplication and truncation.
Its complexity, for a matrix, is , where is the cost of matrix multiplication. In case of a sparse matrix, is the cost of the dense-sparse matrix multiplication.
The algorithm is an adaptation of the Bird algorithm for determinants. We show how to extract, with practically no additional work, the characteristic polynomial and the Pfaffian characteristic polynomial from these algorithms.
{"title":"A simple division-free algorithm for computing Pfaffians","authors":"Adam J. Przeździecki","doi":"10.1016/j.ipl.2024.106550","DOIUrl":"10.1016/j.ipl.2024.106550","url":null,"abstract":"<div><div>We present a very simple algorithm for computing Pfaffians which uses no division operations. Essentially, it amounts to iterating matrix multiplication and truncation.</div><div>Its complexity, for a <span><math><mn>2</mn><mi>n</mi><mo>×</mo><mn>2</mn><mi>n</mi></math></span> matrix, is <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>M</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></math></span>, where <span><math><mi>M</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is the cost of matrix multiplication. In case of a sparse matrix, <span><math><mi>M</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is the cost of the dense-sparse matrix multiplication.</div><div>The algorithm is an adaptation of the Bird algorithm for determinants. We show how to extract, with practically no additional work, the characteristic polynomial and the Pfaffian characteristic polynomial from these algorithms.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106550"},"PeriodicalIF":0.7,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099045","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}
Pub Date : 2024-11-27DOI: 10.1016/j.ipl.2024.106549
Naoyuki Kamiyama
The topic of this paper is the stable matching problem in a bipartite graph. Super-stability is one of the stability concepts in the stable matching problem with ties. It is known that there may not exist a super-stable matching, and the existence of a super-stable matching can be checked in polynomial time. In this paper, we consider the problem of modifying an instance of the stable matching problem with ties by deleting some bounded number of agents in such a way that there exists a super-stable matching in the modified instance. First, we consider the setting where we are allowed to delete agents on only one side. We prove that, in this setting, our problem can be solved in polynomial time. Interestingly, this result is obtained by carefully observing the existing algorithm for checking the existence of a super-stable matching. Next, we consider the setting where we are given an upper bound on the number of deleted agents for each side, and we are allowed to delete agents on both sides. We prove that, in this setting, our problem is NP-complete.
{"title":"Modifying an instance of the super-stable matching problem","authors":"Naoyuki Kamiyama","doi":"10.1016/j.ipl.2024.106549","DOIUrl":"10.1016/j.ipl.2024.106549","url":null,"abstract":"<div><div>The topic of this paper is the stable matching problem in a bipartite graph. Super-stability is one of the stability concepts in the stable matching problem with ties. It is known that there may not exist a super-stable matching, and the existence of a super-stable matching can be checked in polynomial time. In this paper, we consider the problem of modifying an instance of the stable matching problem with ties by deleting some bounded number of agents in such a way that there exists a super-stable matching in the modified instance. First, we consider the setting where we are allowed to delete agents on only one side. We prove that, in this setting, our problem can be solved in polynomial time. Interestingly, this result is obtained by carefully observing the existing algorithm for checking the existence of a super-stable matching. Next, we consider the setting where we are given an upper bound on the number of deleted agents for each side, and we are allowed to delete agents on both sides. We prove that, in this setting, our problem is NP-complete.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106549"},"PeriodicalIF":0.7,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142748594","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}
Pub Date : 2024-11-22DOI: 10.1016/j.ipl.2024.106542
Chris R. Giannella
Because many dissimilarity functions behave differently in low versus high-dimensional spaces, the behavior of high-dimensional nearest neighbor search has been studied extensively. One line of research involves the characterization of nearest neighbor queries as unstable if their query points have nearly identical dissimilarity with most points in the dataset. This research has shown that, for various data distributions and dissimilarity functions, the probability of query instability approaches one. Previous work in Information Processing Letters by C. Giannella in 2021 explicated this phenomenon for centered Gaussian data and Euclidean distance. This paper addresses the problem of characterizing query instability behavior over centered Gaussian data and a fundamentally different dissimilarity function, cosine dissimilarity. Conditions are provided on the covariance matrices and dataset size function guaranteeing that the probability of query instability goes to one. Furthermore, conditions are provided under which the instability probability is bounded away from one.
{"title":"Instability results for cosine-dissimilarity-based nearest neighbor search on high dimensional Gaussian data","authors":"Chris R. Giannella","doi":"10.1016/j.ipl.2024.106542","DOIUrl":"10.1016/j.ipl.2024.106542","url":null,"abstract":"<div><div>Because many dissimilarity functions behave differently in low versus high-dimensional spaces, the behavior of high-dimensional nearest neighbor search has been studied extensively. One line of research involves the characterization of nearest neighbor queries as unstable if their query points have nearly identical dissimilarity with most points in the dataset. This research has shown that, for various data distributions and dissimilarity functions, the probability of query instability approaches one. Previous work in <em>Information Processing Letters</em> by C. Giannella in 2021 explicated this phenomenon for centered Gaussian data and Euclidean distance. This paper addresses the problem of characterizing query instability behavior over centered Gaussian data and a fundamentally different dissimilarity function, cosine dissimilarity. Conditions are provided on the covariance matrices and dataset size function guaranteeing that the probability of query instability goes to one. Furthermore, conditions are provided under which the instability probability is bounded away from one.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106542"},"PeriodicalIF":0.7,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698622","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}
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