{"title":"A Polynomial Time Algorithm for Finding a Minimum 4-Partition of a Submodular Function","authors":"Tsuyoshi Hirayama, Yuhao Liu, K. Makino, Ke Shi, Chao Xu","doi":"10.1137/1.9781611977554.ch64","DOIUrl":"https://doi.org/10.1137/1.9781611977554.ch64","url":null,"abstract":"","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"1 1","pages":"1680-1691"},"PeriodicalIF":0.0,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76170463","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}
The problem of matching markets has been studied for a long time in the literature due to its wide range of applications. Finding a stable matching is a common equilibrium objective in this problem. Since market participants are usually uncertain of their preferences, a rich line of recent works study the online setting where one-side participants (players) learn their unknown preferences from iterative interactions with the other side (arms). Most previous works in this line are only able to derive theoretical guarantees for player-pessimal stable regret, which is defined compared with the players' least-preferred stable matching. However, under the pessimal stable matching, players only obtain the least reward among all stable matchings. To maximize players' profits, player-optimal stable matching would be the most desirable. Though citet{basu21beyond} successfully bring an upper bound for player-optimal stable regret, their result can be exponentially large if players' preference gap is small. Whether a polynomial guarantee for this regret exists is a significant but still open problem. In this work, we provide a new algorithm named explore-then-Gale-Shapley (ETGS) and show that the optimal stable regret of each player can be upper bounded by $O(Klog T/Delta^2)$ where $K$ is the number of arms, $T$ is the horizon and $Delta$ is the players' minimum preference gap among the first $N+1$-ranked arms. This result significantly improves previous works which either have a weaker player-pessimal stable matching objective or apply only to markets with special assumptions. When the preferences of participants satisfy some special conditions, our regret upper bound also matches the previously derived lower bound.
匹配市场问题由于其广泛的应用范围，在文献中已经被研究了很长时间。寻找一个稳定的匹配是这一问题中常见的平衡目标。由于市场参与者通常不确定自己的偏好，最近有大量研究在线环境的工作，其中一方参与者(玩家)从与另一方(手臂)的反复互动中了解他们未知的偏好。这方面以往的研究大多只能推导出玩家-悲观稳定遗憾的理论保证，而玩家-悲观稳定遗憾是通过玩家最不喜欢的稳定匹配来定义的。而在悲观稳定匹配下，参与者在所有稳定匹配中获得的奖励最少。为了最大化玩家的利益，玩家最优的稳定匹配是最理想的。虽然citet{basu21beyond}成功地给出了玩家最优稳定后悔的上限，但如果玩家的偏好差距很小，他们的结果可能会呈指数级增长。这种遗憾是否存在多项式保证是一个重要但仍未解决的问题。在这项工作中，我们提供了一种名为探索-然后- gale - shapley (ETGS)的新算法，并表明每个参与者的最优稳定后悔可以由$O(Klog T/Delta^2)$上界，其中$K$是手臂的数量，$T$是视界，$Delta$是参与者在前$N+1$ -排名手臂之间的最小偏好差距。这一结果显著改善了以前的工作，这些工作要么具有较弱的参与者悲观稳定匹配目标，要么仅适用于具有特殊假设的市场。当参与者的偏好满足某些特殊条件时，我们的遗憾上界也与之前导出的下界匹配。
{"title":"Player-optimal Stable Regret for Bandit Learning in Matching Markets","authors":"Fang-yuan Kong, Shuai Li","doi":"10.1137/1.9781611977554.ch55","DOIUrl":"https://doi.org/10.1137/1.9781611977554.ch55","url":null,"abstract":"The problem of matching markets has been studied for a long time in the literature due to its wide range of applications. Finding a stable matching is a common equilibrium objective in this problem. Since market participants are usually uncertain of their preferences, a rich line of recent works study the online setting where one-side participants (players) learn their unknown preferences from iterative interactions with the other side (arms). Most previous works in this line are only able to derive theoretical guarantees for player-pessimal stable regret, which is defined compared with the players' least-preferred stable matching. However, under the pessimal stable matching, players only obtain the least reward among all stable matchings. To maximize players' profits, player-optimal stable matching would be the most desirable. Though citet{basu21beyond} successfully bring an upper bound for player-optimal stable regret, their result can be exponentially large if players' preference gap is small. Whether a polynomial guarantee for this regret exists is a significant but still open problem. In this work, we provide a new algorithm named explore-then-Gale-Shapley (ETGS) and show that the optimal stable regret of each player can be upper bounded by $O(Klog T/Delta^2)$ where $K$ is the number of arms, $T$ is the horizon and $Delta$ is the players' minimum preference gap among the first $N+1$-ranked arms. This result significantly improves previous works which either have a weaker player-pessimal stable matching objective or apply only to markets with special assumptions. When the preferences of participants satisfy some special conditions, our regret upper bound also matches the previously derived lower bound.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"16 1","pages":"1512-1522"},"PeriodicalIF":0.0,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84219989","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}
Squares (fragments of the form $xx$, for some string $x$) are arguably the most natural type of repetition in strings. The basic algorithmic question concerning squares is to check if a given string of length $n$ is square-free, that is, does not contain a fragment of such form. Main and Lorentz [J. Algorithms 1984] designed an $mathcal{O}(nlog n)$ time algorithm for this problem, and proved a matching lower bound assuming the so-called general alphabet, meaning that the algorithm is only allowed to check if two characters are equal. However, their lower bound also assumes that there are $Omega(n)$ distinct symbols in the string. As an open question, they asked if there is a faster algorithm if one restricts the size of the alphabet. Crochemore [Theor. Comput. Sci. 1986] designed a linear-time algorithm for constant-size alphabets, and combined with more recent results his approach in fact implies such an algorithm for linearly-sortable alphabets. Very recently, Ellert and Fischer [ICALP 2021] significantly relaxed this assumption by designing a linear-time algorithm for general ordered alphabets, that is, assuming a linear order on the characters that permits constant time order comparisons. However, the open question of Main and Lorentz from 1984 remained unresolved for general (unordered) alphabets. In this paper, we show that testing square-freeness of a length-$n$ string over general alphabet of size $sigma$ can be done with $mathcal{O}(nlog sigma)$ comparisons, and cannot be done with $o(nlog sigma)$ comparisons. We complement this result with an $mathcal{O}(nlog sigma)$ time algorithm in the Word RAM model. Finally, we extend the algorithm to reporting all the runs (maximal repetitions) in the same complexity.
{"title":"Optimal Square Detection Over General Alphabets","authors":"J. Ellert, Paweł Gawrychowski, Garance Gourdel","doi":"10.1137/1.9781611977554.ch189","DOIUrl":"https://doi.org/10.1137/1.9781611977554.ch189","url":null,"abstract":"Squares (fragments of the form $xx$, for some string $x$) are arguably the most natural type of repetition in strings. The basic algorithmic question concerning squares is to check if a given string of length $n$ is square-free, that is, does not contain a fragment of such form. Main and Lorentz [J. Algorithms 1984] designed an $mathcal{O}(nlog n)$ time algorithm for this problem, and proved a matching lower bound assuming the so-called general alphabet, meaning that the algorithm is only allowed to check if two characters are equal. However, their lower bound also assumes that there are $Omega(n)$ distinct symbols in the string. As an open question, they asked if there is a faster algorithm if one restricts the size of the alphabet. Crochemore [Theor. Comput. Sci. 1986] designed a linear-time algorithm for constant-size alphabets, and combined with more recent results his approach in fact implies such an algorithm for linearly-sortable alphabets. Very recently, Ellert and Fischer [ICALP 2021] significantly relaxed this assumption by designing a linear-time algorithm for general ordered alphabets, that is, assuming a linear order on the characters that permits constant time order comparisons. However, the open question of Main and Lorentz from 1984 remained unresolved for general (unordered) alphabets. In this paper, we show that testing square-freeness of a length-$n$ string over general alphabet of size $sigma$ can be done with $mathcal{O}(nlog sigma)$ comparisons, and cannot be done with $o(nlog sigma)$ comparisons. We complement this result with an $mathcal{O}(nlog sigma)$ time algorithm in the Word RAM model. Finally, we extend the algorithm to reporting all the runs (maximal repetitions) in the same complexity.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"110 1","pages":"5220-5242"},"PeriodicalIF":0.0,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81611563","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}
Given a simple $n$-vertex, $m$-edge graph $G$ undergoing edge insertions and deletions, we give two new fully dynamic algorithms for exactly maintaining the edge connectivity of $G$ in $tilde{O}(n)$ worst-case update time and $tilde{O}(m^{1-1/16})$ amortized update time, respectively. Prior to our work, all dynamic edge connectivity algorithms assumed bounded edge connectivity, guaranteed approximate solutions, or were restricted to edge insertions only. Our results answer in the affirmative an open question posed by Thorup [Combinatorica'07].
{"title":"Fully Dynamic Exact Edge Connectivity in Sublinear Time","authors":"Gramoz Goranci, M. Henzinger, Danupon Nanongkai, Thatchaphol Saranurak, M. Thorup, Christian Wulff-Nilsen","doi":"10.1137/1.9781611977554.ch3","DOIUrl":"https://doi.org/10.1137/1.9781611977554.ch3","url":null,"abstract":"Given a simple $n$-vertex, $m$-edge graph $G$ undergoing edge insertions and deletions, we give two new fully dynamic algorithms for exactly maintaining the edge connectivity of $G$ in $tilde{O}(n)$ worst-case update time and $tilde{O}(m^{1-1/16})$ amortized update time, respectively. Prior to our work, all dynamic edge connectivity algorithms assumed bounded edge connectivity, guaranteed approximate solutions, or were restricted to edge insertions only. Our results answer in the affirmative an open question posed by Thorup [Combinatorica'07].","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"547 1","pages":"70-86"},"PeriodicalIF":0.0,"publicationDate":"2023-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77070501","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}
The emph{maximal $k$-edge-connected subgraphs} problem is a classical graph clustering problem studied since the 70's. Surprisingly, no non-trivial technique for this problem in weighted graphs is known: a very straightforward recursive-mincut algorithm with $Omega(mn)$ time has remained the fastest algorithm until now. All previous progress gives a speed-up only when the graph is unweighted, and $k$ is small enough (e.g.~Henzinger~et~al.~(ICALP'15), Chechik~et~al.~(SODA'17), and Forster~et~al.~(SODA'20)). We give the first algorithm that breaks through the long-standing $tilde{O}(mn)$-time barrier in emph{weighted undirected} graphs. More specifically, we show a maximal $k$-edge-connected subgraphs algorithm that takes only $tilde{O}(mcdotmin{m^{3/4},n^{4/5}})$ time. As an immediate application, we can $(1+epsilon)$-approximate the emph{strength} of all edges in undirected graphs in the same running time. Our key technique is the first local cut algorithm with emph{exact} cut-value guarantees whose running time depends only on the output size. All previous local cut algorithms either have running time depending on the cut value of the output, which can be arbitrarily slow in weighted graphs or have approximate cut guarantees.
emph{极大$k$边连通子图}问题是70年代以来研究的一个经典的图聚类问题。令人惊讶的是，对于加权图中的这个问题，还没有已知的非平凡技术:到目前为止，一个非常简单的递归最小切算法($Omega(mn)$时间)仍然是最快的算法。所有之前的进展只有在图未加权且$k$足够小时才会加速(例如Henzinger et al. (ICALP'15)， Chechik et al. (SODA'17)和Forster et al. (SODA'20))。我们给出了第一个突破emph{加权无向}图中存在已久的$tilde{O}(mn)$时间障碍的算法。更具体地说，我们展示了一个极大的$k$ -边连接子图算法，它只需要$tilde{O}(mcdotmin{m^{3/4},n^{4/5}})$时间。作为一个直接的应用，我们可以$(1+epsilon)$ -在相同的运行时间内近似无向图中所有边的emph{强度}。我们的关键技术是第一个具有精emph{确切}值保证的局部切算法，其运行时间仅取决于输出大小。所有以前的局部切算法的运行时间取决于输出的切值，这在加权图中可能会任意慢，或者有近似切保证。
{"title":"Maximal k-Edge-Connected Subgraphs in Weighted Graphs via Local Random Contraction","authors":"Chaitanya Nalam, Thatchaphol Saranurak","doi":"10.48550/arXiv.2302.02290","DOIUrl":"https://doi.org/10.48550/arXiv.2302.02290","url":null,"abstract":"The emph{maximal $k$-edge-connected subgraphs} problem is a classical graph clustering problem studied since the 70's. Surprisingly, no non-trivial technique for this problem in weighted graphs is known: a very straightforward recursive-mincut algorithm with $Omega(mn)$ time has remained the fastest algorithm until now. All previous progress gives a speed-up only when the graph is unweighted, and $k$ is small enough (e.g.~Henzinger~et~al.~(ICALP'15), Chechik~et~al.~(SODA'17), and Forster~et~al.~(SODA'20)). We give the first algorithm that breaks through the long-standing $tilde{O}(mn)$-time barrier in emph{weighted undirected} graphs. More specifically, we show a maximal $k$-edge-connected subgraphs algorithm that takes only $tilde{O}(mcdotmin{m^{3/4},n^{4/5}})$ time. As an immediate application, we can $(1+epsilon)$-approximate the emph{strength} of all edges in undirected graphs in the same running time. Our key technique is the first local cut algorithm with emph{exact} cut-value guarantees whose running time depends only on the output size. All previous local cut algorithms either have running time depending on the cut value of the output, which can be arbitrarily slow in weighted graphs or have approximate cut guarantees.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"11 1","pages":"183-211"},"PeriodicalIF":0.0,"publicationDate":"2023-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90168105","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 consider the allocation of $m$ balls (jobs) into $n$ bins (servers). In the standard Two-Choice process, at each step $t=1,2,ldots,m$ we first sample two bins uniformly at random and place a ball in the least loaded bin. It is well-known that for any $m geq n$, this results in a gap (difference between the maximum and average load) of $log_2 log n + Theta(1)$ (with high probability). In this work, we consider the Memory process [Mitzenmacher, Prabhakar and Shah 2002] where instead of two choices, we only sample one bin per step but we have access to a cache which can store the location of one bin. Mitzenmacher, Prabhakar and Shah showed that in the lightly loaded case ($m = n$), the Memory process achieves a gap of $mathcal{O}(log log n)$. Extending the setting of Mitzenmacher et al. in two ways, we first allow the number of balls $m$ to be arbitrary, which includes the challenging heavily loaded case where $m geq n$. Secondly, we follow the heterogeneous bins model of Wieder [Wieder 2007], where the sampling distribution of bins can be biased up to some arbitrary multiplicative constant. Somewhat surprisingly, we prove that even in this setting, the Memory process still achieves an $mathcal{O}(log log n)$ gap bound. This is in stark contrast with the Two-Choice (or any $d$-Choice with $d=mathcal{O}(1)$) process, where it is known that the gap diverges as $m rightarrow infty$ [Wieder 2007]. Further, we show that for any sampling distribution independent of $m$ (but possibly dependent on $n$) the Memory process has a gap that can be bounded independently of $m$. Finally, we prove a tight gap bound of $mathcal{O}(log n)$ for Memory in another relaxed setting with heterogeneous (weighted) balls and a cache which can only be maintained for two steps.
我们考虑将$m$球(作业)分配到$n$箱(服务器)中。在标准的两选过程中，在每一步$t=1,2,ldots,m$，我们首先均匀随机地对两个箱子进行抽样，并将一个球放入装载最少的箱子中。众所周知，对于任何$m geq n$，这会导致(高概率)$log_2 log n + Theta(1)$的差距(最大和平均负载之间的差异)。在这项工作中，我们考虑内存过程[Mitzenmacher, Prabhakar and Shah 2002]，其中我们每一步只采样一个bin，而不是两个选择，但我们可以访问可以存储一个bin位置的缓存。Mitzenmacher, Prabhakar和Shah表明，在轻负载情况下($m = n$)， Memory进程实现了$mathcal{O}(log log n)$的间隙。以两种方式扩展Mitzenmacher等人的设置，我们首先允许球的数量$m$是任意的，其中包括具有挑战性的重载情况$m geq n$。其次，我们遵循Wieder [Wieder 2007]的异构箱模型，其中箱的抽样分布可以偏置到一些任意的乘法常数。有些令人惊讶的是，我们证明即使在这种设置下，Memory进程仍然达到$mathcal{O}(log log n)$间隙界限。这与两种选择(或任何$d$ -选择与$d=mathcal{O}(1)$)过程形成鲜明对比，其中已知差距发散为$m rightarrow infty$ [Wieder 2007]。此外，我们表明，对于任何独立于$m$(但可能依赖于$n$)的抽样分布，Memory进程都有一个可以独立于$m$限定的间隙。最后，我们证明了在另一种具有异构(加权)球和只能维持两步的缓存的宽松设置下内存的紧密间隙界$mathcal{O}(log n)$。
{"title":"Balanced Allocations with Heterogeneous Bins: The Power of Memory","authors":"Dimitrios Los, Thomas Sauerwald, John Sylvester","doi":"10.1137/1.9781611977554.ch169","DOIUrl":"https://doi.org/10.1137/1.9781611977554.ch169","url":null,"abstract":"We consider the allocation of $m$ balls (jobs) into $n$ bins (servers). In the standard Two-Choice process, at each step $t=1,2,ldots,m$ we first sample two bins uniformly at random and place a ball in the least loaded bin. It is well-known that for any $m geq n$, this results in a gap (difference between the maximum and average load) of $log_2 log n + Theta(1)$ (with high probability). In this work, we consider the Memory process [Mitzenmacher, Prabhakar and Shah 2002] where instead of two choices, we only sample one bin per step but we have access to a cache which can store the location of one bin. Mitzenmacher, Prabhakar and Shah showed that in the lightly loaded case ($m = n$), the Memory process achieves a gap of $mathcal{O}(log log n)$. Extending the setting of Mitzenmacher et al. in two ways, we first allow the number of balls $m$ to be arbitrary, which includes the challenging heavily loaded case where $m geq n$. Secondly, we follow the heterogeneous bins model of Wieder [Wieder 2007], where the sampling distribution of bins can be biased up to some arbitrary multiplicative constant. Somewhat surprisingly, we prove that even in this setting, the Memory process still achieves an $mathcal{O}(log log n)$ gap bound. This is in stark contrast with the Two-Choice (or any $d$-Choice with $d=mathcal{O}(1)$) process, where it is known that the gap diverges as $m rightarrow infty$ [Wieder 2007]. Further, we show that for any sampling distribution independent of $m$ (but possibly dependent on $n$) the Memory process has a gap that can be bounded independently of $m$. Finally, we prove a tight gap bound of $mathcal{O}(log n)$ for Memory in another relaxed setting with heterogeneous (weighted) balls and a cache which can only be maintained for two steps.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"45 1","pages":"4448-4477"},"PeriodicalIF":0.0,"publicationDate":"2023-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83649691","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}
Knapsack and Partition are two important additive problems whose fine-grained complexities in the $(1-varepsilon)$-approximation setting are not yet settled. In this work, we make progress on both problems by giving improved algorithms. - Knapsack can be $(1 - varepsilon)$-approximated in $tilde O(n + (1/varepsilon) ^ {2.2} )$ time, improving the previous $tilde O(n + (1/varepsilon) ^ {2.25} )$ by Jin (ICALP'19). There is a known conditional lower bound of $(n+varepsilon)^{2-o(1)}$ based on $(min,+)$-convolution hypothesis. - Partition can be $(1 - varepsilon)$-approximated in $tilde O(n + (1/varepsilon) ^ {1.25} )$ time, improving the previous $tilde O(n + (1/varepsilon) ^ {1.5} )$ by Bringmann and Nakos (SODA'21). There is a known conditional lower bound of $(1/varepsilon)^{1-o(1)}$ based on Strong Exponential Time Hypothesis. Both of our new algorithms apply the additive combinatorial results on dense subset sums by Galil and Margalit (SICOMP'91), Bringmann and Wellnitz (SODA'21). Such techniques have not been explored in the context of Knapsack prior to our work. In addition, we design several new methods to speed up the divide-and-conquer steps which naturally arise in solving additive problems.
{"title":"Approximating Knapsack and Partition via Dense Subset Sums","authors":"Mingyang Deng, Ce Jin, Xiao Mao","doi":"10.1137/1.9781611977554.ch113","DOIUrl":"https://doi.org/10.1137/1.9781611977554.ch113","url":null,"abstract":"Knapsack and Partition are two important additive problems whose fine-grained complexities in the $(1-varepsilon)$-approximation setting are not yet settled. In this work, we make progress on both problems by giving improved algorithms. - Knapsack can be $(1 - varepsilon)$-approximated in $tilde O(n + (1/varepsilon) ^ {2.2} )$ time, improving the previous $tilde O(n + (1/varepsilon) ^ {2.25} )$ by Jin (ICALP'19). There is a known conditional lower bound of $(n+varepsilon)^{2-o(1)}$ based on $(min,+)$-convolution hypothesis. - Partition can be $(1 - varepsilon)$-approximated in $tilde O(n + (1/varepsilon) ^ {1.25} )$ time, improving the previous $tilde O(n + (1/varepsilon) ^ {1.5} )$ by Bringmann and Nakos (SODA'21). There is a known conditional lower bound of $(1/varepsilon)^{1-o(1)}$ based on Strong Exponential Time Hypothesis. Both of our new algorithms apply the additive combinatorial results on dense subset sums by Galil and Margalit (SICOMP'91), Bringmann and Wellnitz (SODA'21). Such techniques have not been explored in the context of Knapsack prior to our work. In addition, we design several new methods to speed up the divide-and-conquer steps which naturally arise in solving additive problems.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"72 1","pages":"2961-2979"},"PeriodicalIF":0.0,"publicationDate":"2023-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86909724","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 consider the weighted $k$-set packing problem, in which we are given a collection of weighted sets, each with at most $k$ elements and must return a collection of pairwise disjoint sets with maximum total weight. For $k = 3$, this problem generalizes the classical 3-dimensional matching problem listed as one of the Karp's original 21 NP-complete problems. We give an algorithm attaining an approximation factor of $1.786$ for weighted 3-set packing, improving on the recent best result of $2-frac{1}{63,700,992}$ due to Neuwohner. Our algorithm is based on the local search procedure of Berman that attempts to improve the sum of squared weights rather than the problem's objective. When using exchanges of size at most $k$, this algorithm attains an approximation factor of $frac{k+1}{2}$. Using exchanges of size $k^2(k-1) + k$, we provide a relatively simple analysis to obtain an approximation factor of 1.811 when $k = 3$. We then show that the tools we develop can be adapted to larger exchanges of size $2k^2(k-1) + k$ to give an approximation factor of 1.786. Although our primary focus is on the case $k = 3$, our approach in fact gives slightly stronger improvements on the factor $frac{k+1}{2}$ for all $k>3$. As in previous works, our guarantees hold also for the more general problem of finding a maximum weight independent set in a $(k+1)$-claw free graph.
{"title":"An Improved Approximation for Maximum Weighted k-Set Packing","authors":"Theophile Thiery, J. Ward","doi":"10.48550/arXiv.2301.07537","DOIUrl":"https://doi.org/10.48550/arXiv.2301.07537","url":null,"abstract":"We consider the weighted $k$-set packing problem, in which we are given a collection of weighted sets, each with at most $k$ elements and must return a collection of pairwise disjoint sets with maximum total weight. For $k = 3$, this problem generalizes the classical 3-dimensional matching problem listed as one of the Karp's original 21 NP-complete problems. We give an algorithm attaining an approximation factor of $1.786$ for weighted 3-set packing, improving on the recent best result of $2-frac{1}{63,700,992}$ due to Neuwohner. Our algorithm is based on the local search procedure of Berman that attempts to improve the sum of squared weights rather than the problem's objective. When using exchanges of size at most $k$, this algorithm attains an approximation factor of $frac{k+1}{2}$. Using exchanges of size $k^2(k-1) + k$, we provide a relatively simple analysis to obtain an approximation factor of 1.811 when $k = 3$. We then show that the tools we develop can be adapted to larger exchanges of size $2k^2(k-1) + k$ to give an approximation factor of 1.786. Although our primary focus is on the case $k = 3$, our approach in fact gives slightly stronger improvements on the factor $frac{k+1}{2}$ for all $k>3$. As in previous works, our guarantees hold also for the more general problem of finding a maximum weight independent set in a $(k+1)$-claw free graph.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"31 1","pages":"1138-1162"},"PeriodicalIF":0.0,"publicationDate":"2023-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81996149","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}
Estimating the empirical distribution of a scalar-valued data set is a basic and fundamental task. In this paper, we tackle the problem of estimating an empirical distribution in a setting with two challenging features. First, the algorithm does not directly observe the data; instead, it only asks a limited number of threshold queries about each sample. Second, the data are not assumed to be independent and identically distributed; instead, we allow for an arbitrary process generating the samples, including an adaptive adversary. These considerations are relevant, for example, when modeling a seller experimenting with posted prices to estimate the distribution of consumers' willingness to pay for a product: offering a price and observing a consumer's purchase decision is equivalent to asking a single threshold query about their value, and the distribution of consumers' values may be non-stationary over time, as early adopters may differ markedly from late adopters. Our main result quantifies, to within a constant factor, the sample complexity of estimating the empirical CDF of a sequence of elements of $[n]$, up to $varepsilon$ additive error, using one threshold query per sample. The complexity depends only logarithmically on $n$, and our result can be interpreted as extending the existing logarithmic-complexity results for noisy binary search to the more challenging setting where noise is non-stochastic. Along the way to designing our algorithm, we consider a more general model in which the algorithm is allowed to make a limited number of simultaneous threshold queries on each sample. We solve this problem using Blackwell's Approachability Theorem and the exponential weights method. As a side result of independent interest, we characterize the minimum number of simultaneous threshold queries required by deterministic CDF estimation algorithms.
{"title":"Non-Stochastic CDF Estimation Using Threshold Queries","authors":"Princewill Okoroafor, Vaishnavi Gupta, Robert D. Kleinberg, Eleanor Goh","doi":"10.48550/arXiv.2301.05682","DOIUrl":"https://doi.org/10.48550/arXiv.2301.05682","url":null,"abstract":"Estimating the empirical distribution of a scalar-valued data set is a basic and fundamental task. In this paper, we tackle the problem of estimating an empirical distribution in a setting with two challenging features. First, the algorithm does not directly observe the data; instead, it only asks a limited number of threshold queries about each sample. Second, the data are not assumed to be independent and identically distributed; instead, we allow for an arbitrary process generating the samples, including an adaptive adversary. These considerations are relevant, for example, when modeling a seller experimenting with posted prices to estimate the distribution of consumers' willingness to pay for a product: offering a price and observing a consumer's purchase decision is equivalent to asking a single threshold query about their value, and the distribution of consumers' values may be non-stationary over time, as early adopters may differ markedly from late adopters. Our main result quantifies, to within a constant factor, the sample complexity of estimating the empirical CDF of a sequence of elements of $[n]$, up to $varepsilon$ additive error, using one threshold query per sample. The complexity depends only logarithmically on $n$, and our result can be interpreted as extending the existing logarithmic-complexity results for noisy binary search to the more challenging setting where noise is non-stochastic. Along the way to designing our algorithm, we consider a more general model in which the algorithm is allowed to make a limited number of simultaneous threshold queries on each sample. We solve this problem using Blackwell's Approachability Theorem and the exponential weights method. As a side result of independent interest, we characterize the minimum number of simultaneous threshold queries required by deterministic CDF estimation algorithms.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"24 1","pages":"3551-3572"},"PeriodicalIF":0.0,"publicationDate":"2023-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83522318","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}
{"title":"Improved girth approximation in weighted undirected graphs","authors":"Avi Kadria, L. Roditty, Aaron Sidford, V. V. Williams, Uri Zwick","doi":"10.1137/1.9781611977554.ch85","DOIUrl":"https://doi.org/10.1137/1.9781611977554.ch85","url":null,"abstract":"","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"16 1","pages":"2242-2255"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76971335","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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