We initiate the study of matroid problems in a new oracle model called dynamic oracle. Our algorithms in this model lead to new bounds for some classic problems, and a "unified" algorithm whose performance matches previous results developed in various papers. We also show a lower bound that answers some open problems from a few decades ago. Concretely, our results are as follows.
* We show an algorithm with $\tilde{O}_k(n+r\sqrt{r})$ dynamic-rank-query and time complexities for the matroid union problem over $k$ matroids. This implies the following consequences. (i) An improvement over the $\tilde{O}_k(n\sqrt{r})$ bound implied by [Chakrabarty-Lee-Sidford-Singla-Wong FOCS'19] for matroid union in the traditional rank-query model. (ii) An $\tilde{O}_k(|E|+|V|\sqrt{|V|})$-time algorithm for the $k$-disjoint spanning tree problem. This improves the $\tilde{O}_k(|V|\sqrt{|E|})$ bounds of Gabow-Westermann [STOC'88] and Gabow [STOC'91].
* We show a matroid intersection algorithm with $\tilde{O}(n\sqrt{r})$ dynamic-rank-query and time complexities. This implies new bounds for some problems and bounds that match the classic ones obtained in various papers, e.g. colorful spanning tree [Gabow-Stallmann ICALP'85], graphic matroid intersection [Gabow-Xu FOCS'89], simple scheduling matroid intersection [Xu-Gabow ISAAC'94], and Hopcroft-Karp combinatorial bipartite matching. More importantly, this is done via a "unified" algorithm in the sense that an improvement over our dynamic-rank-query algorithm would imply improved bounds for all the above problems simultaneously.
* We show simple super-linear ($Ω(n\log n)$) query lower bounds for matroid intersection in our dynamic-rank-oracle and the traditional independence-query models; the latter improves the previous $\log_2(3)n - o(n)$ bound by Harvey [SODA'08] and answers an open problem raised by, e.g., Welsh [1976] and CLSSW [FOCS'19].
Matroid intersection is one of the most powerful frameworks of matroid theory that generalizes various problems in combinatorial optimization. Edmonds' fundamental theorem provides a min-max characterization for the unweighted setting, while Frank's weight-splitting theorem provides one for the weighted case. Several efficient algorithms were developed for these problems, all relying on the usage of one of the conventional oracles for both matroids. In the present paper, we consider the tractability of the matroid intersection problem under restricted oracles. In particular, we focus on the ra…
Accelerating Matroid Optimization through Fast Imprecise Oracles
Institute of Mathematics, Technical University of Berlin（柏林工业大学数学研究所）；Faculty of Mathematics and Computer Science, University of Bremen（不来梅大学数学与计算机科学学院）；Zhejiang University（浙江大学）；Centrum Wiskunde & Informatica（荷兰国家数学和计算机科学研究院）
Given two matroids $\mathcal{M}_1 = (V, \mathcal{I}_1)$ and $\mathcal{M}_2 = (V, \mathcal{I}_2)$ over an $n$-element integer-weighted ground set $V$, the weighted matroid intersection problem aims to find a common independent set $S^{*} \in \mathcal{I}_1 \cap \mathcal{I}_2$ maximizing the weight of $S^{*}$. In this paper, we present a simple deterministic algorithm for weighted matroid intersection using $\tilde{O}(nr^{3/4}\log{W})$ rank queries, where $r$ is the size of the largest intersection of $\mathcal{M}_1$ and $\mathcal{M}_2$ and $W$ is the maximum weight. This improves upon the best p…
Eötvös Loránd University（厄特沃什·罗兰大学）；HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利研究网络阿尔弗雷德·雷尼数学研究所）；Hokkaido University（北海道大学）；RIKEN（理化学研究所）；Osaka University（大阪大学）；Institute of Science Tokyo（东京科学大学）
Submodular function minimization (SFM) and matroid intersection are fundamental discrete optimization problems with applications in many fields. It is well known that both of these can be solved making $\mathrm{poly}(N)$ queries to a relevant oracle (evaluation oracle for SFM and rank oracle for matroid intersection), where $N$ denotes the universe size. However, all known polynomial query algorithms are highly adaptive, requiring at least $N$ rounds of querying the oracle. A natural question is whether these can be efficiently solved in a highly parallel manner, namely, with $\mathrm{poly}(N)…
In this paper we consider the classic matroid intersection problem: given two matroids $\M_{1}=(V,\I_{1})$ and $\M_{2}=(V,\I_{2})$ defined over a common ground set $V$, compute a set $S\in\I_{1}\cap\I_{2}$ of largest possible cardinality, denoted by $r$. We consider this problem both in the setting where each $\M_{i}$ is accessed through an independence oracle, i.e. a routine which returns whether or not a set $S\in\I_{i}$ in $\indep$ time, and the setting where each $\M_{i}$ is accessed through a rank oracle, i.e. a routine which returns the size of the largest independent subset of $S$ in $\…
In the matroid intersection problem, we are given two matroids of rank $r$ on a common ground set $E$ of $n$ elements and the goal is to find the maximum set that is independent in both matroids. In this note, we show that Cunningham's algorithm for matroid intersection can be implemented to use $O(nr\log^2(r))$ independent oracle calls.
Let $M$ be a matroid defined on a finite set $E$ and $L\subset E$. $L$ is locked in $M$ if $M|L$ and $M^*|(E\backslash L)$ are 2-connected, and $min\{r(L), r^*(E\backslash L)\} \geq 2$. Locked subsets characterize nontrivial facets of the bases polytope. In this paper, we give a new axiom system for matroids based on locked subsets. We deduce an algorithm for recognizing matroids improving the running time complexity of the best known till today. This algorithm induces a polynomial time algorithm for recognizing uniform matroids. This latter problem is intractable if we use an independence ora…
A Deterministic $0.401$-Approximation for Non-monotone Submodular Maximization under Matroid Constraints
State Key Lab of Processors, Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所处理器重点实验室）；School of Computer Science and Technology, University of Chinese Academy of Sciences（中国科学院大学计算机科学与技术学院）
The matroid parity (or matroid matching) problem, introduced as a common generalization of matching and matroid intersection problems, is so general that it requires an exponential number of oracle calls. Nevertheless, Lovász (1980) showed that this problem admits a min-max formula and a polynomial algorithm for linearly represented matroids. Since then efficient algorithms have been developed for the linear matroid parity problem. In this paper, we present a combinatorial, deterministic, polynomial-time algorithm for the weighted linear matroid parity problem. The algorithm builds on a polyno…
Fast algorithms for submodular maximization problems have a vast potential use in applicative settings, such as machine learning, social networks, and economics. Though fast algorithms were known for some special cases, only recently Badanidiyuru and Vondrák (2014) were the first to explicitly look for such algorithms in the general case of maximizing a monotone submodular function subject to a matroid independence constraint. The algorithm of Badanidiyuru and Vondrák matches the best possible approximation guarantee, while trying to reduce the number of value oracle queries the algorithm perf…
In this work we present the first practical $\left(\frac{1}{e}-ε\right)$-approximation algorithm to maximise a general non-negative submodular function subject to a matroid constraint. Our algorithm is based on combining the decreasing-threshold procedure of Badanidiyuru and Vondrak (SODA 2014) with a smoother version of the measured continuous greedy algorithm of Feldman et al. (FOCS 2011). This enables us to obtain an algorithm that requires $O(\frac{nr^2}{ε^4} \big(\frac{a+b}{a}\big)^2 \log^2({\frac{n}ε}))$ value oracle calls, where $n$ is the cardinality of the ground set, $r$ is the matro…
CNRS, DI ENS, PSL（法国国家科学研究中心、巴黎高等师范学院计算机科学系、巴黎文理研究大学）；Keio University（庆应义塾大学）；Université de Paris, IRIF, CNRS（巴黎大学、法国国家科学研究中心信息基础与交互研究所）；National Institute of Informatics（国立信息学研究所）
Maximizing a monotone submodular function under various constraints is a classical and intensively studied problem. However, in the single-pass streaming model, where the elements arrive one by one and an algorithm can store only a small fraction of input elements, there is much gap in our knowledge, even though several approximation algorithms have been proposed in the literature. In this work, we present the first lower bound on the approximation ratios for cardinality and matroid constraints that beat $1-\frac{1}{e}$ in the single-pass streaming model. Let $n$ be the number of elements in t…
Matroids are a fundamental object of study in combinatorial optimization. Three closely related and important problems involving matroids are maximizing the size of the union of $k$ independent sets (that is, $k$-fold matroid union), computing $k$ disjoint bases (a.k.a. matroid base packing), and covering the elements by $k$ bases (a.k.a. matroid base covering). These problems generalize naturally to integral and real-valued capacities on the elements. This work develops faster exact and/or approximation problems for these and some other closely related problems such as optimal reinforcement a…
A $k$-submodular function is an extension of a submodular function in that its input is given by $k$ disjoint subsets instead of a single subset. For unconstrained nonnegative $k$-submodular maximization, Ward and Živný proposed a constant-factor approximation algorithm, which was improved by the recent work of Iwata, Tanigawa and Yoshida presenting a $1/2$-approximation algorithm. Iwata et al. also provided a $k/(2k-1)$-approximation algorithm for monotone $k$-submodular maximization and proved that its approximation ratio is asymptotically tight. More recently, Ohsaka and Yoshida proposed co…
We consider fast algorithms for monotone submodular maximization subject to a matroid constraint. We assume that the matroid is given as input in an explicit form, and the goal is to obtain the best possible running times for important matroids. We develop a new algorithm for a \emph{general matroid constraint} with a $1 - 1/e - ε$ approximation that achieves a fast running time provided we have a fast data structure for maintaining a maximum weight base in the matroid through a sequence of decrease weight operations. We construct such data structures for graphic matroids and partition matroid…
The matroid intersection problem is a fundamental problem that has been extensively studied for half a century. In the classic version of this problem, we are given two matroids $\mathcal{M}_1 = (V, \mathcal{I}_1)$ and $\mathcal{M}_2 = (V, \mathcal{I}_2)$ on a comment ground set $V$ of $n$ elements, and then we have to find the largest common independent set $S \in \mathcal{I}_1 \cap \mathcal{I}_2$ by making independence oracle queries of the form "Is $S \in \mathcal{I}_1$?" or "Is $S \in \mathcal{I}_2$?" for $S \subseteq V$. The goal is to minimize the number of queries. Beating the existing …
In the \textit{Matroid Secretary Problem} (MSP), the elements of the ground set of a Matroid are revealed on-line one by one, each together with its value. An algorithm for the MSP is \textit{Matroid-Unknown} if, at every stage of its execution: (i) it only knows the elements that have been revealed so far and their values, and (ii) it has access to an oracle for testing whether or not any subset of the elements that have been revealed so far is an independent set. An algorithm is \textit{Known-Cardinality} if, in addition to (i) and (ii), it also initially knows the cardinality of the ground …
Combinatorial Optimization using Comparison Oracles
Google Research（谷歌研究）；Bocconi University（博科尼大学）；New York University（纽约大学）；University of Michigan（密歇根大学）；Duke University（杜克大学）；Carnegie Mellon University（卡内基梅隆大学）
We give a deterministic polynomial time $2^{O(r)}$-approximation algorithm for the number of bases of a given matroid of rank $r$ and the number of common bases of any two matroids of rank $r$. To the best of our knowledge, this is the first nontrivial deterministic approximation algorithm that works for arbitrary matroids. Based on a lower bound of Azar, Broder, and Frieze [ABF94] this is almost the best possible result assuming oracle access to independent sets of the matroid. There are two main ingredients in our result: For the first, we build upon recent results of Adiprasito, Huh, and Ka…
We study the online submodular maximization problem with free disposal under a matroid constraint. Elements from some ground set arrive one by one in rounds, and the algorithm maintains a feasible set that is independent in the underlying matroid. In each round when a new element arrives, the algorithm may accept the new element into its feasible set and possibly remove elements from it, provided that the resulting set is still independent. The goal is to maximize the value of the final feasible set under some monotone submodular function, to which the algorithm has oracle access. For $k$-unif…
Simon Fraser University（西蒙菲莎大学）；National Institute of Informatics（信息科学研究所）
The problem of maximizing nonnegative monotone submodular functions under a certain constraint has been intensively studied in the last decade, and a wide range of efficient approximation algorithms have been developed for this problem. Many machine learning problems, including data summarization and influence maximization, can be naturally modeled as the problem of maximizing monotone submodular functions. However, when such applications involve sensitive data about individuals, their privacy concerns should be addressed. In this paper, we study the problem of maximizing monotone submodular f…
Recent progress in (semi-)streaming algorithms for monotone submodular function maximization has led to tight results for a simple cardinality constraint. However, current techniques fail to give a similar understanding for natural generalizations, including matroid constraints. This paper aims at closing this gap. For a single matroid of rank $k$ (i.e., any solution has cardinality at most $k$), our main results are: 1) a single-pass streaming algorithm that uses $\widetilde{O}(k)$ memory and achieves an approximation guarantee of $0.3178$, and 2) a multi-pass streaming algorithm that uses $\…
For a given a graph, a distance oracle is a data structure that answers distance queries between pairs of vertices. We introduce an $O(n^{5/3})$-space distance oracle which answers exact distance queries in $O(\log n)$ time for $n$-vertex planar edge-weighted digraphs. All previous distance oracles for planar graphs with truly subquadratic space i.e., space $O(n^{2 - ε})$ for some constant $ε> 0$) either required query time polynomial in $n$ or could only answer approximate distance queries. Furthermore, we show how to trade-off time and space: for any $S \ge n^{3/2}$, we show how to obtain an…
We consider the problem of optimizing a coverage function under a $\ell$-matchoid of rank $k$. We design fixed-parameter algorithms as well as streaming algorithms to compute an exact solution. Unlike previous work that presumes linear representativity of matroids, we consider the general oracle model. For the special case where the coverage function is linear, we give a deterministic fixed-parameter algorithm parameterized by $\ell$ and $k$. This result, combined with the lower bounds of Lovasz, and Jensen and Korte demonstrates a separation between the $\ell$-matchoid and the matroid $\ell$-…
When a network is prone to failures, it is very expensive to compute the shortest paths every time from the scratch. Distance sensitivity oracle provides this privilege to find the new shortest paths faster and with lower cost by once pre-computing an oracle in advance. Although several efficient solutions are proposed in the literature to support the single failure, few efforts are done to devise an efficient method regarding the case of multiple failures. In this paper, we present a novel distance sensitivity oracle based on Markov Tensor Theory \cite{golnari2017markov} to support replacemen…
