arXiv - CS - Discrete Mathematics最新文献_第2页

Upward Pointset Embeddings of Planar st-Graphs 平面 st 图的向上点集嵌入

arXiv - CS - Discrete Mathematics Pub Date : 2024-08-30 DOI: arxiv-2408.17369

Carlos Alegria, Susanna Caroppo, Giordano Da Lozzo, Marco D'Elia, Giuseppe Di Battista, Fabrizio Frati, Fabrizio Grosso, Maurizio Patrignani

{"title":"Upward Pointset Embeddings of Planar st-Graphs","authors":"Carlos Alegria, Susanna Caroppo, Giordano Da Lozzo, Marco D'Elia, Giuseppe Di Battista, Fabrizio Frati, Fabrizio Grosso, Maurizio Patrignani","doi":"arxiv-2408.17369","DOIUrl":"https://doi.org/arxiv-2408.17369","url":null,"abstract":"We study upward pointset embeddings (UPSEs) of planar $st$-graphs. Let $G$ be\u0000a planar $st$-graph and let $S subset mathbb{R}^2$ be a pointset with $|S|=\u0000|V(G)|$. An UPSE of $G$ on $S$ is an upward planar straight-line drawing of $G$\u0000that maps the vertices of $G$ to the points of $S$. We consider both the\u0000problem of testing the existence of an UPSE of $G$ on $S$ (UPSE Testing) and\u0000the problem of enumerating all UPSEs of $G$ on $S$. We prove that UPSE Testing\u0000is NP-complete even for $st$-graphs that consist of a set of directed\u0000$st$-paths sharing only $s$ and $t$. On the other hand, for $n$-vertex planar\u0000$st$-graphs whose maximum $st$-cutset has size $k$, we prove that it is\u0000possible to solve UPSE Testing in $O(n^{4k})$ time with $O(n^{3k})$ space, and\u0000to enumerate all UPSEs of $G$ on $S$ with $O(n)$ worst-case delay, using $O(k\u0000n^{4k} log n)$ space, after $O(k n^{4k} log n)$ set-up time. Moreover, for an\u0000$n$-vertex $st$-graph whose underlying graph is a cycle, we provide a necessary\u0000and sufficient condition for the existence of an UPSE on a given poinset, which\u0000can be tested in $O(n log n)$ time. Related to this result, we give an\u0000algorithm that, for a set $S$ of $n$ points, enumerates all the non-crossing\u0000monotone Hamiltonian cycles on $S$ with $O(n)$ worst-case delay, using $O(n^2)$\u0000space, after $O(n^2)$ set-up time.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175147","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Approximation Algorithms for Correlated Knapsack Orienteering 相关可纳包定向的近似算法

arXiv - CS - Discrete Mathematics Pub Date : 2024-08-29 DOI: arxiv-2408.16566

David Aleman Espinosa, Chaitanya Swamy

{"title":"Approximation Algorithms for Correlated Knapsack Orienteering","authors":"David Aleman Espinosa, Chaitanya Swamy","doi":"arxiv-2408.16566","DOIUrl":"https://doi.org/arxiv-2408.16566","url":null,"abstract":"We consider the {em correlated knapsack orienteering} (CSKO) problem: we are\u0000given a travel budget $B$, processing-time budget $W$, finite metric space\u0000$(V,d)$ with root $rhoin V$, where each vertex is associated with a job with\u0000possibly correlated random size and random reward that become known only when\u0000the job completes. Random variables are independent across different vertices.\u0000The goal is to compute a $rho$-rooted path of length at most $B$, in a\u0000possibly adaptive fashion, that maximizes the reward collected from jobs that\u0000processed by time $W$. To our knowledge, CSKO has not been considered before,\u0000though prior work has considered the uncorrelated problem, {em stochastic\u0000knapsack orienteering}, and {em correlated orienteering}, which features only\u0000one budget constraint on the {em sum} of travel-time and processing-times. We show that the {em adaptivity gap of CSKO is not a constant, and is at\u0000least $Omegabigl(maxsqrt{log{B}},sqrt{loglog{W}}}bigr)$}.\u0000Complementing this, we devise {em non-adaptive} algorithms that obtain: (a)\u0000$O(loglog W)$-approximation in quasi-polytime; and (b) $O(log\u0000W)$-approximation in polytime. We obtain similar guarantees for CSKO with\u0000cancellations, wherein a job can be cancelled before its completion time,\u0000foregoing its reward. We also consider the special case of CSKO, wherein job\u0000sizes are weighted Bernoulli distributions, and more generally where the\u0000distributions are supported on at most two points (2-CSKO). Although weighted\u0000Bernoulli distributions suffice to yield an $Omega(sqrt{loglog B})$\u0000adaptivity-gap lower bound for (uncorrelated) {em stochastic orienteering}, we\u0000show that they are easy instances for CSKO. We develop non-adaptive algorithms\u0000that achieve $O(1)$-approximation in polytime for weighted Bernoulli\u0000distributions, and in $(n+log B)^{O(log W)}$-time for the more general case\u0000of 2-CSKO.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175148","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

N-Way Joint Mutual Exclusion Does Not Imply Any Pairwise Mutual Exclusion for Propositions N 向联合互斥并不意味着命题的任何成对互斥

arXiv - CS - Discrete Mathematics Pub Date : 2024-08-29 DOI: arxiv-2409.03784

Roy S. Freedman

引用次数: 0

Channel allocation revisited through 1-extendability of graphs 通过图的 1-可扩展性重新审视信道分配

arXiv - CS - Discrete Mathematics Pub Date : 2024-08-26 DOI: arxiv-2408.14633

Anthony Busson, Malory Marin, Rémi Watrigant

{"title":"Channel allocation revisited through 1-extendability of graphs","authors":"Anthony Busson, Malory Marin, Rémi Watrigant","doi":"arxiv-2408.14633","DOIUrl":"https://doi.org/arxiv-2408.14633","url":null,"abstract":"We revisit the classical problem of channel allocation for Wi-Fi access\u0000points (AP). Using mechanisms such as the CSMA/CA protocol, Wi-Fi access points\u0000which are in conflict within a same channel are still able to communicate to\u0000terminals. In graph theoretical terms, it means that it is not mandatory for\u0000the channel allocation to correspond to a proper coloring of the conflict\u0000graph. However, recent studies suggest that the structure -- rather than the\u0000number -- of conflicts plays a crucial role in the performance of each AP. More\u0000precisely, the graph induced by each channel must satisfy the so-called\u0000$1$-extendability property, which requires each vertex to be contained in an\u0000independent set of maximum cardinality. In this paper we introduce the\u00001-extendable chromatic number, which is the minimum size of a partition of the\u0000vertex set of a graph such that each part induces a 1-extendable graph. We\u0000study this parameter and the related optimization problem through different\u0000perspectives: algorithms and complexity, structure, and extremal properties. We\u0000first show how to compute this number using modular decompositions of graphs,\u0000and analyze the running time with respect to the modular width of the input\u0000graph. We also focus on the special case of cographs, and prove that the\u00001-extendable chromatic number can be computed in quasi-polynomial time in this\u0000class. Concerning extremal results, we show that the 1-extendable chromatic\u0000number of a graph with $n$ vertices is at most $2sqrt{n}$, whereas the\u0000classical chromatic number can be as large as $n$. We are also able to\u0000construct graphs whose 1-extendable chromatic number is at least logarithmic in\u0000the number of vertices.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The Parameterized Complexity Landscape of Two-Sets Cut-Uncut 双集切割-不切割的参数化复杂性景观

arXiv - CS - Discrete Mathematics Pub Date : 2024-08-24 DOI: arxiv-2408.13543

Matthias Bentert, Fedor V. Fomin, Fanny Hauser, Saket Saurabh

{"title":"The Parameterized Complexity Landscape of Two-Sets Cut-Uncut","authors":"Matthias Bentert, Fedor V. Fomin, Fanny Hauser, Saket Saurabh","doi":"arxiv-2408.13543","DOIUrl":"https://doi.org/arxiv-2408.13543","url":null,"abstract":"In Two-Sets Cut-Uncut, we are given an undirected graph $G=(V,E)$ and two\u0000terminal sets $S$ and $T$. The task is to find a minimum cut $C$ in $G$ (if\u0000there is any) separating $S$ from $T$ under the following ``uncut'' condition.\u0000In the graph $(V,E setminus C)$, the terminals in each terminal set remain in\u0000the same connected component. In spite of the superficial similarity to the\u0000classic problem Minimum $s$-$t$-Cut, Two-Sets Cut-Uncut is computationally\u0000challenging. In particular, even deciding whether such a cut of any size\u0000exists, is already NP-complete. We initiate a systematic study of Two-Sets\u0000Cut-Uncut within the context of parameterized complexity. By leveraging known\u0000relations between many well-studied graph parameters, we characterize the\u0000structural properties of input graphs that allow for polynomial kernels,\u0000fixed-parameter tractability (FPT), and slicewise polynomial algorithms (XP).\u0000Our main contribution is the near-complete establishment of the complexity of\u0000these algorithmic properties within the described hierarchy of graph\u0000parameters. On a technical level, our main results are fixed-parameter\u0000tractability for the (vertex-deletion) distance to cographs and an OR-cross\u0000composition excluding polynomial kernels for the vertex cover number of the\u0000input graph (under the standard complexity assumption NP is not contained in\u0000coNP/poly).","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Single-Machine Scheduling to Minimize the Number of Tardy Jobs with Release Dates 单机调度，最大限度减少有发布日期的延迟作业数量

arXiv - CS - Discrete Mathematics Pub Date : 2024-08-23 DOI: arxiv-2408.12967

Matthias Kaul, Matthias Mnich, Hendrik Molter

{"title":"Single-Machine Scheduling to Minimize the Number of Tardy Jobs with Release Dates","authors":"Matthias Kaul, Matthias Mnich, Hendrik Molter","doi":"arxiv-2408.12967","DOIUrl":"https://doi.org/arxiv-2408.12967","url":null,"abstract":"We study the fundamental scheduling problem $1mid r_jmidsum w_j U_j$:\u0000schedule a set of $n$ jobs with weights, processing times, release dates, and\u0000due dates on a single machine, such that each job starts after its release date\u0000and we maximize the weighted number of jobs that complete execution before\u0000their due date. Problem $1mid r_jmidsum w_j U_j$ generalizes both Knapsack\u0000and Partition, and the simplified setting without release dates was studied by\u0000Hermelin et al. [Annals of Operations Research, 2021] from a parameterized\u0000complexity viewpoint. Our main contribution is a thorough complexity analysis of $1mid r_jmidsum\u0000w_j U_j$ in terms of four key problem parameters: the number $p_#$ of\u0000processing times, the number $w_#$ of weights, the number $d_#$ of due dates,\u0000and the number $r_#$ of release dates of the jobs. $1mid r_jmidsum w_j U_j$\u0000is known to be weakly para-NP-hard even if $w_#+d_#+r_#$ is constant, and\u0000Heeger and Hermelin [ESA, 2024] recently showed (weak) W[1]-hardness\u0000parameterized by $p_#$ or $w_#$ even if $r_#$ is constant. Algorithmically, we show that $1mid r_jmidsum w_j U_j$ is fixed-parameter\u0000tractable parameterized by $p_#$ combined with any two of the remaining three\u0000parameters $w_#$, $d_#$, and $r_#$. We further provide pseudo-polynomial\u0000XP-time algorithms for parameter $r_#$ and $d_#$. To complement these\u0000algorithms, we show that $1mid r_jmidsum w_j U_j$ is (strongly) W[1]-hard\u0000when parameterized by $d_#+r_#$ even if $w_#$ is constant. Our results\u0000provide a nearly complete picture of the complexity of $1mid r_jmidsum w_j\u0000U_j$ for $p_#$, $w_#$, $d_#$, and $r_#$ as parameters, and extend those of\u0000Hermelin et al. [Annals of Operations Research, 2021] for the problem\u0000$1midmidsum w_j U_j$ without release dates.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A Constant-Approximation Algorithm for Budgeted Sweep Coverage with Mobile Sensors 移动传感器预算扫面覆盖的常量近似算法

arXiv - CS - Discrete Mathematics Pub Date : 2024-08-22 DOI: arxiv-2408.12468

Wei Liang, Shaojie Tang, Zhao Zhang

引用次数: 0

Approximately covering vertices by order-$5$ or longer paths 大约以 5$ 或更长的路径覆盖顶点

arXiv - CS - Discrete Mathematics Pub Date : 2024-08-20 DOI: arxiv-2408.11225

Mingyang Gong, Zhi-Zhong Chen, Guohui Lin, Lusheng Wang

引用次数: 0

Efficient Online Sensitivity Analysis For The Injective Bottleneck Path Problem 注入式瓶颈路径问题的高效在线敏感性分析

arXiv - CS - Discrete Mathematics Pub Date : 2024-08-18 DOI: arxiv-2408.09443

Kirill V. Kaymakov, Dmitry S. Malyshev

{"title":"Efficient Online Sensitivity Analysis For The Injective Bottleneck Path Problem","authors":"Kirill V. Kaymakov, Dmitry S. Malyshev","doi":"arxiv-2408.09443","DOIUrl":"https://doi.org/arxiv-2408.09443","url":null,"abstract":"The tolerance of an element of a combinatorial optimization problem with\u0000respect to a given optimal solution is the maximum change, i.e., decrease or\u0000increase, of its cost, such that this solution remains optimal. The bottleneck\u0000path problem, for given an edge-capacitated graph, a source, and a target, is\u0000to find the $max$-$min$ value of edge capacities on paths between the source\u0000and the target. For this problem and a network with $n$ vertices and $m$ edges,\u0000there is known the Ramaswamy-Orlin-Chakravarty's algorithm to compute all\u0000tolerances in $O(m+nlog n)$ time. In this paper, for any in advance given\u0000sample of the problem with pairwise distinct edge capacities, we present a\u0000constant-time algorithm for computing both tolerances of an arbitrary edge with\u0000a preprocessing time $Obig(m alpha(m,n)big)$, where $alpha(cdot,cdot)$ is\u0000the inverse Ackermann function. For given $k$ source-target pairs, our solution\u0000yields an $Obig((alpha(m,n)+k)mbig)$-time algorithm to find tolerances of\u0000all edges with respect to optimal paths between the sources and targets, while\u0000the known algorithm takes $Obig(k(m+nlog n)big)$ time to find them.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Parallel Repetition for $3$-Player XOR Games 3 美元玩家 XOR 游戏的并行重复

arXiv - CS - Discrete Mathematics Pub Date : 2024-08-18 DOI: arxiv-2408.09352

Amey Bhangale, Mark Braverman, Subhash Khot, Yang P. Liu, Dor Minzer

{"title":"Parallel Repetition for $3$-Player XOR Games","authors":"Amey Bhangale, Mark Braverman, Subhash Khot, Yang P. Liu, Dor Minzer","doi":"arxiv-2408.09352","DOIUrl":"https://doi.org/arxiv-2408.09352","url":null,"abstract":"In a $3$-$mathsf{XOR}$ game $mathcal{G}$, the verifier samples a challenge\u0000$(x,y,z)sim mu$ where $mu$ is a probability distribution over\u0000$SigmatimesGammatimesPhi$, and a map $tcolon\u0000SigmatimesGammatimesPhitomathcal{A}$ for a finite Abelian group\u0000$mathcal{A}$ defining a constraint. The verifier sends the questions $x$, $y$\u0000and $z$ to the players Alice, Bob and Charlie respectively, receives answers\u0000$f(x)$, $g(y)$ and $h(z)$ that are elements in $mathcal{A}$ and accepts if\u0000$f(x)+g(y)+h(z) = t(x,y,z)$. The value, $mathsf{val}(mathcal{G})$, of the\u0000game is defined to be the maximum probability the verifier accepts over all\u0000players' strategies. We show that if $mathcal{G}$ is a $3$-$mathsf{XOR}$ game with value\u0000strictly less than $1$, whose underlying distribution over questions $mu$ does\u0000not admit Abelian embeddings into $(mathbb{Z},+)$, then the value of the\u0000$n$-fold repetition of $mathcal{G}$ is exponentially decaying. That is, there\u0000exists $c=c(mathcal{G})>0$ such that $mathsf{val}(mathcal{G}^{otimes\u0000n})leq 2^{-cn}$. This extends a previous result of [Braverman-Khot-Minzer,\u0000FOCS 2023] showing exponential decay for the GHZ game. Our proof combines tools\u0000from additive combinatorics and tools from discrete Fourier analysis.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0