Farzan Byramji, Vatsal Jha, Chandrima Kayal, Rajat Mittal
In a recent result, Knop, Lovett, McGuire and Yuan (STOC 2021) proved the log-rank conjecture for communication complexity, up to log n factor, for any Boolean function composed with AND function as the inner gadget. One of the main tools in this result was the relationship between monotone analogues of well-studied Boolean complexity measures like block sensitivity and certificate complexity. The relationship between the standard measures has been a long line of research, with a landmark result by Huang (Annals of Mathematics 2019), finally showing that sensitivity is polynomially related to all other standard measures. In this article, we study the monotone analogues of standard measures like block sensitivity (mbs(f)), certificate complexity (MCC(f)) and fractional block sensitivity (fmbs(f)); and study the relationship between these measures given their connection with AND-decision tree and sparsity of a Boolean function. We show the following results: 1) Given a Boolean function $f : {0, 1}^{n} rightarrow {0, 1}$, the ratio $fmbs(f^l )/mbs(f^l )$ is bounded by a function of n (and not l). A similar result was known for the corresponding standard measures (Tal, ITCS 2013). This result allows us to extend any upper bound by a well behaved measure on monotone block sensitivity to monotone fractional block sensitivity. 2) The question of the best possible upper bound on monotone block sensitivity by the logarithm of sparsity is equivalent to the natural question of best upper bound by degree on sensitivity. One side of this relationship was used in the proof by Knop, Lovett, McGuire and Yuan (STOC 2021). 3) For two natural classes of functions, symmetric and monotone, hitting set complexity (MCC) is equal to monotone sensitivity.
{"title":"Relations between monotone complexity measures based on decision tree complexity","authors":"Farzan Byramji, Vatsal Jha, Chandrima Kayal, Rajat Mittal","doi":"arxiv-2406.07859","DOIUrl":"https://doi.org/arxiv-2406.07859","url":null,"abstract":"In a recent result, Knop, Lovett, McGuire and Yuan (STOC 2021) proved the\u0000log-rank conjecture for communication complexity, up to log n factor, for any\u0000Boolean function composed with AND function as the inner gadget. One of the\u0000main tools in this result was the relationship between monotone analogues of\u0000well-studied Boolean complexity measures like block sensitivity and certificate\u0000complexity. The relationship between the standard measures has been a long line\u0000of research, with a landmark result by Huang (Annals of Mathematics 2019),\u0000finally showing that sensitivity is polynomially related to all other standard\u0000measures. In this article, we study the monotone analogues of standard measures\u0000like block sensitivity (mbs(f)), certificate complexity (MCC(f)) and fractional\u0000block sensitivity (fmbs(f)); and study the relationship between these measures\u0000given their connection with AND-decision tree and sparsity of a Boolean\u0000function. We show the following results: 1) Given a Boolean function $f : {0,\u00001}^{n} rightarrow {0, 1}$, the ratio $fmbs(f^l )/mbs(f^l )$ is bounded by a\u0000function of n (and not l). A similar result was known for the corresponding\u0000standard measures (Tal, ITCS 2013). This result allows us to extend any upper\u0000bound by a well behaved measure on monotone block sensitivity to monotone\u0000fractional block sensitivity. 2) The question of the best possible upper bound\u0000on monotone block sensitivity by the logarithm of sparsity is equivalent to the\u0000natural question of best upper bound by degree on sensitivity. One side of this\u0000relationship was used in the proof by Knop, Lovett, McGuire and Yuan (STOC\u00002021). 3) For two natural classes of functions, symmetric and monotone, hitting\u0000set complexity (MCC) is equal to monotone sensitivity.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526097","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}
We study the classic single-item auction setting of Myerson, but under the assumption that the buyers' values for the item are distributed over finite supports. Using strong LP duality and polyhedral theory, we rederive various key results regarding the revenue-maximizing auction, including the characterization through virtual welfare maximization and the optimality of deterministic mechanisms, as well as a novel, generic equivalence between dominant-strategy and Bayesian incentive compatibility. Inspired by this, we abstract our approach to handle more general auction settings, where the feasibility space can be given by arbitrary convex constraints, and the objective is a convex combination of revenue and social welfare. We characterize the optimal auctions of such systems as generalized virtual welfare maximizers, by making use of their KKT conditions, and we present an analogue of Myerson's payment formula for general discrete single-parameter auction settings. Additionally, we prove that total unimodularity of the feasibility space is a sufficient condition to guarantee the optimality of auctions with integral allocation rules. Finally, we demonstrate this KKT approach by applying it to a setting where bidders are interested in buying feasible flows on trees with capacity constraints, and provide a combinatorial description of the (randomized, in general) optimal auction.
{"title":"Discrete Single-Parameter Optimal Auction Design","authors":"Yiannis Giannakopoulos, Johannes Hahn","doi":"arxiv-2406.08125","DOIUrl":"https://doi.org/arxiv-2406.08125","url":null,"abstract":"We study the classic single-item auction setting of Myerson, but under the\u0000assumption that the buyers' values for the item are distributed over finite\u0000supports. Using strong LP duality and polyhedral theory, we rederive various\u0000key results regarding the revenue-maximizing auction, including the\u0000characterization through virtual welfare maximization and the optimality of\u0000deterministic mechanisms, as well as a novel, generic equivalence between\u0000dominant-strategy and Bayesian incentive compatibility. Inspired by this, we abstract our approach to handle more general auction\u0000settings, where the feasibility space can be given by arbitrary convex\u0000constraints, and the objective is a convex combination of revenue and social\u0000welfare. We characterize the optimal auctions of such systems as generalized\u0000virtual welfare maximizers, by making use of their KKT conditions, and we\u0000present an analogue of Myerson's payment formula for general discrete\u0000single-parameter auction settings. Additionally, we prove that total\u0000unimodularity of the feasibility space is a sufficient condition to guarantee\u0000the optimality of auctions with integral allocation rules. Finally, we demonstrate this KKT approach by applying it to a setting where\u0000bidders are interested in buying feasible flows on trees with capacity\u0000constraints, and provide a combinatorial description of the (randomized, in\u0000general) optimal auction.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526098","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}
We propose a new deterministic algorithm called Subtree-Decomposition for the online transportation problem and show that the algorithm is $(8m-5)$-competitive, where $m$ is the number of server sites. It has long been known that the competitive ratio of any deterministic algorithm is lower bounded by $2m-1$ for this problem. On the other hand, the conjecture proposed by Kalyanasundaram and Pruhs in 1998 asking whether a deterministic $(2m-1)$-competitive algorithm exists for the online transportation problem has remained open for over two decades. The upper bound on the competitive ratio, $8m-5$, which is the result of this paper, is the first to come close to this conjecture, and is the best possible within a constant factor.
{"title":"A Nearly Optimal Deterministic Algorithm for Online Transportation Problem","authors":"Tsubasa Harada, Toshiya Itoh","doi":"arxiv-2406.03778","DOIUrl":"https://doi.org/arxiv-2406.03778","url":null,"abstract":"We propose a new deterministic algorithm called Subtree-Decomposition for the\u0000online transportation problem and show that the algorithm is\u0000$(8m-5)$-competitive, where $m$ is the number of server sites. It has long been known that the competitive ratio of any deterministic\u0000algorithm is lower bounded by $2m-1$ for this problem. On the other hand, the\u0000conjecture proposed by Kalyanasundaram and Pruhs in 1998 asking whether a\u0000deterministic $(2m-1)$-competitive algorithm exists for the online\u0000transportation problem has remained open for over two decades. The upper bound on the competitive ratio, $8m-5$, which is the result of this\u0000paper, is the first to come close to this conjecture, and is the best possible\u0000within a constant factor.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551648","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}
We investigate how elimination of variables can affect the asymptotic dynamics and phenotype control of Boolean networks. In particular, we look at the impact on minimal trap spaces, and identify a structural condition that guarantees their preservation. We examine the possible effects of variable elimination under three of the most popular approaches to control (attractor-based control, value propagation and control of minimal trap spaces), and under different update schemes (synchronous, asynchronous, generalized asynchronous). We provide some insights on the application of reduction, and an ample inventory of examples and counterexamples.
{"title":"Phenotype control and elimination of variables in Boolean networks","authors":"Elisa Tonello, Loïc Paulevé","doi":"arxiv-2406.02304","DOIUrl":"https://doi.org/arxiv-2406.02304","url":null,"abstract":"We investigate how elimination of variables can affect the asymptotic\u0000dynamics and phenotype control of Boolean networks. In particular, we look at\u0000the impact on minimal trap spaces, and identify a structural condition that\u0000guarantees their preservation. We examine the possible effects of variable\u0000elimination under three of the most popular approaches to control\u0000(attractor-based control, value propagation and control of minimal trap\u0000spaces), and under different update schemes (synchronous, asynchronous,\u0000generalized asynchronous). We provide some insights on the application of\u0000reduction, and an ample inventory of examples and counterexamples.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141258672","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}
In Terminal Monitoring Set (TMS), the input is an undirected graph $G=(V,E)$, together with a collection $T$ of terminal pairs and the goal is to find a subset $S$ of minimum size that hits a shortest path between every pair of terminals. We show that this problem is W[2]-hard with respect to solution size. On the positive side, we show that TMS is fixed parameter tractable with respect to solution size plus distance to cluster, solution size plus neighborhood diversity, and feedback edge number. For the weighted version of the problem, we obtain a FPT algorithm with respect to vertex cover number, and for a relaxed version of the problem, we show that it is W[1]-hard with respect to solution size plus feedback vertex number.
{"title":"The Parameterized Complexity of Terminal Monitoring Set","authors":"N. R. Aravind, Roopam Saxena","doi":"arxiv-2406.01730","DOIUrl":"https://doi.org/arxiv-2406.01730","url":null,"abstract":"In Terminal Monitoring Set (TMS), the input is an undirected graph $G=(V,E)$,\u0000together with a collection $T$ of terminal pairs and the goal is to find a\u0000subset $S$ of minimum size that hits a shortest path between every pair of\u0000terminals. We show that this problem is W[2]-hard with respect to solution\u0000size. On the positive side, we show that TMS is fixed parameter tractable with\u0000respect to solution size plus distance to cluster, solution size plus\u0000neighborhood diversity, and feedback edge number. For the weighted version of\u0000the problem, we obtain a FPT algorithm with respect to vertex cover number, and\u0000for a relaxed version of the problem, we show that it is W[1]-hard with respect\u0000to solution size plus feedback vertex number.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141258755","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}
Finding a few solutions for a given problem that are diverse, as opposed to finding a single best solution to solve the problem, has recently become a notable topic in theoretical computer science. Recently, Baste, Fellows, Jaffke, Masav{r}'ik, Oliveira, Philip, and Rosamond showed that under a standard structural parameterization by treewidth, one can find a set of diverse solutions for many problems with only a very small additional cost [Artificial Intelligence 2022]. In this paper, we investigate a much stronger graph parameter, the cliquewidth, which can additionally describe some dense graph classes. Broadly speaking, it describes graphs that can be recursively constructed by a few operations defined on graphs whose vertices are divided into a bounded number of groups while each such group behaves uniformly with respect to any operation. We show that for any vertex problem, if we are given a dynamic program solving that problem on cliquewidth decomposition, we can modify it to produce a few solutions that are as diverse as possible with as little overhead as in the above-mentioned treewidth paper. As a consequence, we prove that a diverse version of any MSO$_1$ expressible problem can be solved in FPT time parameterized by cliquewidth, the number of sought solutions, and the number of quantifiers in the formula. That was an important missing piece in the complexity landscape of structural graph parameters and logic. We prove our results allowing for a more general natural collection of diversity functions compared to only two mostly studied diversity functions previously. That might be of independent interest as a larger pool of different diversity functions can highlight various aspects of different solutions to a problem.
{"title":"Finding Diverse Solutions Parameterized by Cliquewidth","authors":"Karolina Drabik, Tomáš Masařík","doi":"arxiv-2405.20931","DOIUrl":"https://doi.org/arxiv-2405.20931","url":null,"abstract":"Finding a few solutions for a given problem that are diverse, as opposed to\u0000finding a single best solution to solve the problem, has recently become a\u0000notable topic in theoretical computer science. Recently, Baste, Fellows,\u0000Jaffke, Masav{r}'ik, Oliveira, Philip, and Rosamond showed that under a\u0000standard structural parameterization by treewidth, one can find a set of\u0000diverse solutions for many problems with only a very small additional cost\u0000[Artificial Intelligence 2022]. In this paper, we investigate a much stronger\u0000graph parameter, the cliquewidth, which can additionally describe some dense\u0000graph classes. Broadly speaking, it describes graphs that can be recursively\u0000constructed by a few operations defined on graphs whose vertices are divided\u0000into a bounded number of groups while each such group behaves uniformly with\u0000respect to any operation. We show that for any vertex problem, if we are given a dynamic program\u0000solving that problem on cliquewidth decomposition, we can modify it to produce\u0000a few solutions that are as diverse as possible with as little overhead as in\u0000the above-mentioned treewidth paper. As a consequence, we prove that a diverse\u0000version of any MSO$_1$ expressible problem can be solved in FPT time\u0000parameterized by cliquewidth, the number of sought solutions, and the number of\u0000quantifiers in the formula. That was an important missing piece in the\u0000complexity landscape of structural graph parameters and logic. We prove our\u0000results allowing for a more general natural collection of diversity functions\u0000compared to only two mostly studied diversity functions previously. That might\u0000be of independent interest as a larger pool of different diversity functions\u0000can highlight various aspects of different solutions to a problem.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141258568","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}
A preference matrix $M$ has an entry for each pair of candidates in an election whose value $p_{ij}$ represents the proportion of voters that prefer candidate $i$ over candidate $j$. The matrix is rationalizable if it is consistent with a set of voters whose preferences are total orders. A celebrated open problem asks for a concise characterization of rationalizable preference matrices. In this paper, we generalize this matrix rationalizability question and study when a preference matrix is consistent with a set of voters whose preferences are partial orders of width $alpha$. The width (the maximum cardinality of an antichain) of the partial order is a natural measure of the rationality of a voter; indeed, a partial order of width $1$ is a total order. Our primary focus concerns the rationality number, the minimum width required to rationalize a preference matrix. We present two main results. The first concerns the class of half-integral preference matrices, where we show the key parameter required in evaluating the rationality number is the chromatic number of the undirected unanimity graph associated with the preference matrix $M$. The second concerns the class of integral preference matrices, where we show the key parameter now is the dichromatic number of the directed voting graph associated with $M$.
{"title":"Matrix Rationalization via Partial Orders","authors":"Agnes Totschnig, Rohit Vasishta, Adrian Vetta","doi":"arxiv-2405.20976","DOIUrl":"https://doi.org/arxiv-2405.20976","url":null,"abstract":"A preference matrix $M$ has an entry for each pair of candidates in an\u0000election whose value $p_{ij}$ represents the proportion of voters that prefer\u0000candidate $i$ over candidate $j$. The matrix is rationalizable if it is\u0000consistent with a set of voters whose preferences are total orders. A\u0000celebrated open problem asks for a concise characterization of rationalizable\u0000preference matrices. In this paper, we generalize this matrix rationalizability\u0000question and study when a preference matrix is consistent with a set of voters\u0000whose preferences are partial orders of width $alpha$. The width (the maximum\u0000cardinality of an antichain) of the partial order is a natural measure of the\u0000rationality of a voter; indeed, a partial order of width $1$ is a total order.\u0000Our primary focus concerns the rationality number, the minimum width required\u0000to rationalize a preference matrix. We present two main results. The first\u0000concerns the class of half-integral preference matrices, where we show the key\u0000parameter required in evaluating the rationality number is the chromatic number\u0000of the undirected unanimity graph associated with the preference matrix $M$.\u0000The second concerns the class of integral preference matrices, where we show\u0000the key parameter now is the dichromatic number of the directed voting graph\u0000associated with $M$.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259043","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}
Frederik Wenkel, Semih Cantürk, Michael Perlmutter, Guy Wolf
Graph neural networks (GNNs) have achieved great success for a variety of tasks such as node classification, graph classification, and link prediction. However, the use of GNNs (and machine learning more generally) to solve combinatorial optimization (CO) problems is much less explored. Here, we introduce a novel GNN architecture which leverages a complex filter bank and localized attention mechanisms designed to solve CO problems on graphs. We show how our method differentiates itself from prior GNN-based CO solvers and how it can be effectively applied to the maximum clique, minimum dominating set, and maximum cut problems in a self-supervised learning setting. In addition to demonstrating competitive overall performance across all tasks, we establish state-of-the-art results for the max cut problem.
图神经网络（GNN）在节点分类、图分类和链接预测等各种任务中取得了巨大成功。然而，利用 GNN（以及更广泛的机器学习）解决组合优化（CO）问题的探索却少得多。在这里，我们介绍了一种新颖的 GNN 架构，它利用复杂的滤波器库和定位注意力机制来解决图上的 CO 问题。我们展示了我们的方法如何区别于之前基于 GNN 的 CO 求解器，以及它如何有效地应用于自我监督学习环境中的最大簇、最小支配集和最大切割问题。除了在所有任务中展示了具有竞争力的整体性能外，我们还在最大切割问题上取得了最先进的结果。
{"title":"Towards a General GNN Framework for Combinatorial Optimization","authors":"Frederik Wenkel, Semih Cantürk, Michael Perlmutter, Guy Wolf","doi":"arxiv-2405.20543","DOIUrl":"https://doi.org/arxiv-2405.20543","url":null,"abstract":"Graph neural networks (GNNs) have achieved great success for a variety of\u0000tasks such as node classification, graph classification, and link prediction.\u0000However, the use of GNNs (and machine learning more generally) to solve\u0000combinatorial optimization (CO) problems is much less explored. Here, we\u0000introduce a novel GNN architecture which leverages a complex filter bank and\u0000localized attention mechanisms designed to solve CO problems on graphs. We show\u0000how our method differentiates itself from prior GNN-based CO solvers and how it\u0000can be effectively applied to the maximum clique, minimum dominating set, and\u0000maximum cut problems in a self-supervised learning setting. In addition to\u0000demonstrating competitive overall performance across all tasks, we establish\u0000state-of-the-art results for the max cut problem.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141258643","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}
Let $M$ be a perfect matching on a set of points in the plane where every edge is a line segment between two points. We say that $M$ is globally maximum if it is a maximum-length matching on all points. We say that $M$ is $k$-local maximum if for any subset $M'={a_1b_1,dots,a_kb_k}$ of $k$ edges of $M$ it holds that $M'$ is a maximum-length matching on points ${a_1,b_1,dots,a_k,b_k}$. We show that local maximum matchings are good approximations of global ones. Let $mu_k$ be the infimum ratio of the length of any $k$-local maximum matching to the length of any global maximum matching, over all finite point sets in the Euclidean plane. It is known that $mu_kgeqslant frac{k-1}{k}$ for any $kgeqslant 2$. We show the following improved bounds for $kin{2,3}$: $sqrt{3/7}leqslantmu_2< 0.93 $ and $sqrt{3}/2leqslantmu_3< 0.98$. We also show that every pairwise crossing matching is unique and it is globally maximum. Towards our proof of the lower bound for $mu_2$ we show the following result which is of independent interest: If we increase the radii of pairwise intersecting disks by factor $2/sqrt{3}$, then the resulting disks have a common intersection.
{"title":"Euclidean Maximum Matchings in the Plane---Local to Global","authors":"Ahmad Biniaz, Anil Maheshwari, Michiel Smid","doi":"arxiv-2405.20424","DOIUrl":"https://doi.org/arxiv-2405.20424","url":null,"abstract":"Let $M$ be a perfect matching on a set of points in the plane where every\u0000edge is a line segment between two points. We say that $M$ is globally maximum\u0000if it is a maximum-length matching on all points. We say that $M$ is $k$-local\u0000maximum if for any subset $M'={a_1b_1,dots,a_kb_k}$ of $k$ edges of $M$ it\u0000holds that $M'$ is a maximum-length matching on points\u0000${a_1,b_1,dots,a_k,b_k}$. We show that local maximum matchings are good\u0000approximations of global ones. Let $mu_k$ be the infimum ratio of the length of any $k$-local maximum\u0000matching to the length of any global maximum matching, over all finite point\u0000sets in the Euclidean plane. It is known that $mu_kgeqslant frac{k-1}{k}$\u0000for any $kgeqslant 2$. We show the following improved bounds for\u0000$kin{2,3}$: $sqrt{3/7}leqslantmu_2< 0.93 $ and $sqrt{3}/2leqslantmu_3<\u00000.98$. We also show that every pairwise crossing matching is unique and it is\u0000globally maximum. Towards our proof of the lower bound for $mu_2$ we show the following result\u0000which is of independent interest: If we increase the radii of pairwise\u0000intersecting disks by factor $2/sqrt{3}$, then the resulting disks have a\u0000common intersection.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141258665","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}
Neural networks with ReLU activation play a key role in modern machine learning. In view of safety-critical applications, the verification of trained networks is of great importance and necessitates a thorough understanding of essential properties of the function computed by a ReLU network, including characteristics like injectivity and surjectivity. Recently, Puthawala et al. [JMLR 2022] came up with a characterization for injectivity of a ReLU layer, which implies an exponential time algorithm. However, the exact computational complexity of deciding injectivity remained open. We answer this question by proving coNP-completeness of deciding injectivity of a ReLU layer. On the positive side, as our main result, we present a parameterized algorithm which yields fixed-parameter tractability of the problem with respect to the input dimension. In addition, we also characterize surjectivity for two-layer ReLU networks with one-dimensional output. Remarkably, the decision problem turns out to be the complement of a basic network verification task. We prove NP-hardness for surjectivity, implying a stronger hardness result than previously known for the network verification problem. Finally, we reveal interesting connections to computational convexity by formulating the surjectivity problem as a zonotope containment problem
{"title":"Complexity of Deciding Injectivity and Surjectivity of ReLU Neural Networks","authors":"Vincent Froese, Moritz Grillo, Martin Skutella","doi":"arxiv-2405.19805","DOIUrl":"https://doi.org/arxiv-2405.19805","url":null,"abstract":"Neural networks with ReLU activation play a key role in modern machine\u0000learning. In view of safety-critical applications, the verification of trained\u0000networks is of great importance and necessitates a thorough understanding of\u0000essential properties of the function computed by a ReLU network, including\u0000characteristics like injectivity and surjectivity. Recently, Puthawala et al.\u0000[JMLR 2022] came up with a characterization for injectivity of a ReLU layer,\u0000which implies an exponential time algorithm. However, the exact computational\u0000complexity of deciding injectivity remained open. We answer this question by\u0000proving coNP-completeness of deciding injectivity of a ReLU layer. On the\u0000positive side, as our main result, we present a parameterized algorithm which\u0000yields fixed-parameter tractability of the problem with respect to the input\u0000dimension. In addition, we also characterize surjectivity for two-layer ReLU\u0000networks with one-dimensional output. Remarkably, the decision problem turns\u0000out to be the complement of a basic network verification task. We prove\u0000NP-hardness for surjectivity, implying a stronger hardness result than\u0000previously known for the network verification problem. Finally, we reveal\u0000interesting connections to computational convexity by formulating the\u0000surjectivity problem as a zonotope containment problem","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"2011 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141194532","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}
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