Theoretical Computer Science最新文献

{"title":"Graph realization of distance sets","authors":"Amotz Bar-Noy , David Peleg , Mor Perry , Dror Rawitz","doi":"10.1016/j.tcs.2024.114810","DOIUrl":"10.1016/j.tcs.2024.114810","url":null,"abstract":"<div><p>The <span>Distance Realization</span> problem is defined as follows. Given an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrix <em>D</em> of nonnegative integers, interpreted as inter-vertex distances, find an <em>n</em>-vertex weighted or unweighted graph <em>G</em> realizing <em>D</em>, i.e., whose inter-vertex distances satisfy <span><math><mi>d</mi><mi>i</mi><mi>s</mi><msub><mrow><mi>t</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></math></span> for every <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo><</mo><mi>j</mi><mo>≤</mo><mi>n</mi></math></span>, or decide that no such realizing graph exists. The problem was studied for general weighted and unweighted graphs, as well as for cases where the realizing graph is restricted to a specific family of graphs (e.g., trees or bipartite graphs). An extension of <span>Distance Realization</span> that was studied in the past is where each entry in the matrix <em>D</em> may contain a <em>range</em> of consecutive permissible values. We refer to this extension as <span>Range Distance Realization</span> (or <span>Range-DR</span>). Restricting each range to at most <em>k</em> values yields the problem <em>k</em>-<span>Range Distance Realization</span> (or <em>k</em>-<span>Range-DR</span>). The current paper introduces a new extension of <span>Distance Realization</span>, in which each entry <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></math></span> of the matrix may contain an arbitrary set of acceptable values for the distance between <em>i</em> and <em>j</em>, for every <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo><</mo><mi>j</mi><mo>≤</mo><mi>n</mi></math></span>. We refer to this extension as <span>Set Distance Realization</span> (<span>Set-DR</span>), and to the restricted problem where each entry may contain at most <em>k</em> values as <em>k</em>-<span>Set Distance Realization</span> (or <em>k</em>-<span>Set-DR</span>).</p><p>We first show that 2-<span>Range-DR</span> is NP-hard for unweighted graphs (implying the same for 2-<span>Set-DR</span>). Next we prove that 2-<span>Set-DR</span> is NP-hard for unweighted and weighted trees.</p><p>Finally, we explore <span>Set-DR</span> where the realization is restricted to the families of stars, paths, cycles, or caterpillars. For the weighted case, our positive results are that there exist polynomial time algorithms for the 2-<span>Set-DR</span> problem on stars, paths and cycles, and for the 1-<span>Set-DR</span> problem on caterpillars. On the hardness side, we prove that 6-<span>Set-DR</span> is NP-hard for stars and 5-<span>Set-DR</span> is NP-hard for paths, cycles and caterpillars. For the unweighted case, our results are the same, except for the case of unweighted stars, for which <em>k</em>-<span>Set-DR</span> is polynomially solvable for a","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1019 ","pages":"Article 114810"},"PeriodicalIF":0.9,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142158431","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 submodular prophet inequalities and correlation gap","authors":"Chandra Chekuri,&nbsp;Vasilis Livanos","doi":"10.1016/j.tcs.2024.114814","DOIUrl":"10.1016/j.tcs.2024.114814","url":null,"abstract":"<div><p>Prophet inequalities and secretary problems have been extensively studied in recent years due to their elegance, connections to online algorithms, stochastic optimization, and mechanism design problems in game theoretic settings. Rubinstein and Singla <span><span>[31]</span></span> developed a notion of <em>combinatorial</em> prophet inequalities in order to generalize the standard prophet inequality setting to combinatorial valuation functions such as submodular and subadditive functions. For non-negative submodular functions they demonstrated a constant factor prophet inequality for matroid constraints. Along the way they showed a variant of the correlation gap for non-negative submodular functions.</p><p>In this paper we revisit their notion of correlation gap as well as the standard notion of correlation gap and prove much tighter and cleaner bounds. Via these bounds and other insights we obtain substantially improved constant factor combinatorial prophet inequalities for both monotone and non-monotone submodular functions over any constraint that admits an Online Contention Resolution Scheme. In addition to improved bounds we describe efficient polynomial-time algorithms that achieve these bounds.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1019 ","pages":"Article 114814"},"PeriodicalIF":0.9,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142158430","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":"The diameter of sum basic equilibria games","authors":"Aida Abiad ,&nbsp;Carme Àlvarez ,&nbsp;Arnau Messegué","doi":"10.1016/j.tcs.2024.114807","DOIUrl":"10.1016/j.tcs.2024.114807","url":null,"abstract":"<div><p>We study the sum basic network creation game introduced in 2010 by Alon, Demaine, Hajiaghai and Leighton. In this game, an undirected and unweighted graph <em>G</em> is said to be a <em>sum basic equilibrium</em> if and only if, for every edge <em>uv</em> and any vertex <span><math><msup><mrow><mi>v</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> in <em>G</em>, swapping edge <em>uv</em> with edge <span><math><mi>u</mi><msup><mrow><mi>v</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> does not decrease the total sum of the distances from <em>u</em> to all the other vertices. This concept lies at the heart of the network creation games, where the central problem is to understand the structure of the resulting equilibrium graphs, and in particular, how well they globally minimize the diameter. In this sense, in 2013 Alon et al. showed an upper bound of <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><msqrt><mrow><mi>log</mi><mo>⁡</mo><mi>n</mi></mrow></msqrt><mo>)</mo></mrow></msup></math></span> on the diameter of sum basic equilibria, and they also proved that if a sum basic equilibrium graph is a tree, then it has diameter at most 2. In this paper, we prove that the upper bound of 2 also holds for bipartite graphs and even for some non-bipartite classes like block graphs and cactus graphs.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1018 ","pages":"Article 114807"},"PeriodicalIF":0.9,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0304397524004249/pdfft?md5=ab3c3a76dbdf1ff573755d82085be616&pid=1-s2.0-S0304397524004249-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142099008","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}

{"title":"Towards zero knowledge argument for double discrete logarithm with constant cost","authors":"Diya Krishnan ,&nbsp;Xiang Fu","doi":"10.1016/j.tcs.2024.114799","DOIUrl":"10.1016/j.tcs.2024.114799","url":null,"abstract":"<div><p>Given that the Schnorr's protocol for Discrete Logarithm (DLOG) exchanges three messages, it is an interesting problem whether a constant round zero-knowledge protocol exists for the Double Discrete Logarithm problem (DDLOG), i.e., to demonstrate the knowledge of a secret witness <em>x</em> in <span><math><msup><mrow><mi>g</mi></mrow><mrow><msup><mrow><mi>h</mi></mrow><mrow><mi>x</mi></mrow></msup></mrow></msup></math></span>. In this paper, we show that it exists for a fragment of DDLOG with two restrictions: (1) The outer group of DDLOG supports bilinear pairing, and it needs a trusted set-up for common reference string (CRS). (2) <span><math><mi>x</mi><mo>&lt;</mo><mi>t</mi></math></span> where <em>t</em> is the size of KZG commitment key in CRS. The protocol is zero knowledge and constant round, with <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> complexity for prover and verifier, regardless of the desired security strength. The contributions of the work are mainly theoretical due to its restrictions and concrete performance.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1018 ","pages":"Article 114799"},"PeriodicalIF":0.9,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142098954","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":"Parameterised approximation of the fixation probability of the dominant mutation in the multi-type Moran process","authors":"Leslie Ann Goldberg ,&nbsp;Marc Roth ,&nbsp;Tassilo Schwarz","doi":"10.1016/j.tcs.2024.114785","DOIUrl":"10.1016/j.tcs.2024.114785","url":null,"abstract":"<div><p>The multi-type Moran process is an evolutionary process on a connected graph <em>G</em> in which each vertex has one of <em>k</em> types and, in each step, a vertex <em>v</em> is chosen to reproduce its type to one of its neighbours. The probability of a vertex <em>v</em> being chosen for reproduction is proportional to the fitness of the type of <em>v</em>. So far, the literature was almost solely concerned with the 2-type Moran process in which each vertex is either healthy (type 0) or a mutant (type 1), and the main problem of interest has been the (approximate) computation of the so-called <em>fixation probability</em>, i.e., the probability that eventually all vertices are mutants.</p><p>In this work we initiate the study of approximating fixation probabilities in the multi-type Moran process on general graphs. Our main result is an FPTRAS (fixed-parameter tractable randomised approximation scheme) for computing the fixation probability of the dominant mutation; the parameter is the number of types and their fitnesses. In the course of our studies we also provide novel upper bounds on the expected <em>absorption time</em>, i.e., the time that it takes the multi-type Moran process to reach a state in which each vertex has the same type.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1016 ","pages":"Article 114785"},"PeriodicalIF":0.9,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142041213","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":"Stand-up indulgent gathering on lines","authors":"Quentin Bramas ,&nbsp;Sayaka Kamei ,&nbsp;Anissa Lamani ,&nbsp;Sébastien Tixeuil","doi":"10.1016/j.tcs.2024.114796","DOIUrl":"10.1016/j.tcs.2024.114796","url":null,"abstract":"<div><p>We consider a variant of the crash-fault gathering problem called stand-up indulgent gathering (SUIG). In this problem, a group of mobile robots must eventually gather at a single location, which is not known in advance. If no robots crash, they must all meet at the same location. However, if one or more robots crash at a single location, all non-crashed robots must eventually gather at that location. The SUIG problem was first introduced for robots operating in a two-dimensional continuous Euclidean space, with most solutions relying on the ability of robots to move a prescribed (real) distance at each time instant.</p><p>In this paper, we investigate the SUIG problem for robots operating in a discrete universe (i.e., a graph) where they can only move one unit of distance (i.e., to an adjacent node) at each time instant. Specifically, we focus on line-shaped networks and characterize the solvability of the SUIG problem for oblivious robots without multiplicity detection.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1016 ","pages":"Article 114796"},"PeriodicalIF":0.9,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142050029","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 accelerated deterministic algorithm for maximizing monotone submodular minus modular function with cardinality constraint","authors":"Shufang Gong,&nbsp;Bin Liu,&nbsp;Qizhi Fang","doi":"10.1016/j.tcs.2024.114798","DOIUrl":"10.1016/j.tcs.2024.114798","url":null,"abstract":"<div><p>Submodular optimization not only covers some classical combinatorial optimization problems, but also has a wide range of applications in fields such as machine learning and artificial intelligence. For submodular maximization problems with constraints, some work has been done including the design of approximation algorithms, the measurement of approximation algorithms in terms of quality and efficiency, etc. In this paper, we consider the problem of maximizing a non-negative monotone submodular function minus a non-negative modular function with the cardinality constraint. This model has been applied to many scenarios, such as team formation problem, influence maximization problem, recommender systems problem, etc. We propose a threshold algorithm that achieve a <span><math><mo>(</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>−</mo><mi>O</mi><mo>(</mo><mi>ε</mi><mo>)</mo><mo>,</mo><mn>2</mn><mo>)</mo></math></span>-bicriteria approximation ratio and query complexity <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span>. Our algorithm makes a small sacrifice in the approximation ratio but improves the best query complexity result of existing deterministic algorithms from <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> to <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> in the worst case.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1016 ","pages":"Article 114798"},"PeriodicalIF":0.9,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142050030","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 the connectedness of arithmetic hyperplanes","authors":"Bastien Laboureix,&nbsp;Eric Domenjoud","doi":"10.1016/j.tcs.2024.114797","DOIUrl":"10.1016/j.tcs.2024.114797","url":null,"abstract":"<div><p>Discrete geometry is a geometry specific to computers that studies <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> structures. It appears naturally in image analysis or 3D printing. Our goal is to find efficient algorithms to characterise these geometric structures and their properties.</p><p>We are interested in a fundamental structure of discrete geometry, the arithmetic hyperplanes, and more particularly in their connectedness. Many works have studied a connectedness defined from the neighbourhood by faces and have allowed to observe a percolation phenomenon. These studies have also allowed to decide the connectedness of a plane in an efficient way. We propose an extension of these results in the case of connectivity defined from general neighbourhoods.</p><p>Beyond the new concepts that this extension requires, the main contribution of the paper lies in the use of analysis to solve this arithmetic problem and in the design of an algorithm that decides the general connectedness problem. The study of the thickness of connectedness as a function reveals discontinuities at each rational point. However, a much more regular underlying structure appears in the irrational case. Thus, the consideration of irrational vectors allows a simpler approach to the connectedness of arithmetic hyperplanes.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1017 ","pages":"Article 114797"},"PeriodicalIF":0.9,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142087845","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":"Process-commutative distributed objects: From cryptocurrencies to Byzantine-Fault-Tolerant CRDTs","authors":"Davide Frey ,&nbsp;Lucie Guillou ,&nbsp;Michel Raynal ,&nbsp;François Taïani","doi":"10.1016/j.tcs.2024.114794","DOIUrl":"10.1016/j.tcs.2024.114794","url":null,"abstract":"<div><p>This paper explores the territory that lies between best-effort Byzantine-Fault-Tolerant Conflict-free Replicated Data Types (BFT CRDTs) and totally ordered distributed ledgers, such as those implemented by Blockchains. It formally characterizes a novel class of distributed objects that only requires a First In First Out (FIFO) order on the object operations from each process (taken individually). The formalization leverages <em>Mazurkiewicz traces</em> to define legal sequences of operations and ensure both Strong Eventual Consistency (SEC) and Pipleline Consistency (PC). The paper presents a generic algorithm that implements this novel class of distributed objects both in a crash and Byzantine setting. It also illustrates the practical interest of the proposed approach using four instances of this class of objects, namely money transfer, Petri nets, multi-sets, and concurrent work stealing dequeues.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1017 ","pages":"Article 114794"},"PeriodicalIF":0.9,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142098016","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":"Transformations of probability distributions","authors":"Fabian Frei ,&nbsp;Peter Rossmanith","doi":"10.1016/j.tcs.2024.114786","DOIUrl":"10.1016/j.tcs.2024.114786","url":null,"abstract":"<div><p>Almost all of computer science is concerned with transformations of information in the form of strings. We initiate the study of a neglected transformation type, namely transformations between probability distributions. We begin by examining the deceivingly simple-looking case of Bernoulli distributions and procedures to transform them.</p><p>A <em>p</em>-coin is a coin that, whenever tossed, lands heads up with probability <em>p</em> and tails up with probability <span><math><mn>1</mn><mo>−</mo><mi>p</mi></math></span>. A neat trick due to von Neumann allows us to simulate a 1/2-coin with any <em>p</em>-coin for an arbitrary unknown <span><math><mi>p</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. We show how to apply this trick to simulate a <em>q</em>-coin for an arbitrary computable <span><math><mi>q</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. In contrast, it is impossible to simulate a <em>q</em>-coin with a noncomputable <em>q</em>.</p><p>More generally, we are interested in what transformations between probability distributions are feasible. Is it possible to simulate a <span><math><mi>p</mi><mo>/</mo><mn>2</mn></math></span>-coin with a <em>p</em>-coin for unknown <em>p</em>? How about 2<em>p</em> or <span><math><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>? We attempt to characterize the feasible transformations. For example, we show how to transform a <em>p</em>-coin into an <span><math><mi>f</mi><mo>(</mo><mi>p</mi><mo>)</mo></math></span>-coin where any finite number of pairs <span><math><mo>(</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>f</mi><mo>(</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>)</mo></math></span> of non-zero, non-one probabilities is prescribed, and that it is impossible to do the same for an infinite number of pairs. We also examine which probability distributions are feasible to approximate to arbitrary precision, showing that this is impossible for discontinuous ones but feasible for most but not all remaining ones.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1018 ","pages":"Article 114786"},"PeriodicalIF":0.9,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142136055","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}