arXiv subjects

Monika Henzinger

Publications and source records attributed to Monika Henzinger.

At least 127 records · Page 7Linked to original sources

arXiv2017

Improved Set-based Symbolic Algorithms for Parity Games

Graph games with ω-regular winning conditions provide a mathematical framework to analyze a wide range of problems in the analysis of reactive systems and programs (such as the synthesis of reactive systems, program repair, and the verification of branching time properties). Parity conditions are canonical forms to specify ω-regular winning conditions. Graph games with parity conditions are equivalent to μ-calculus model checking, and thus a very important algorithmic problem. Symbolic algorithms are of great significance because they provide scalable algorithms for the analysis of large finite-state systems, as well as algorithms for the analysis of infinite-state systems with finite quotient. A set-based symbolic algorithm uses the basic set operations and the one-step predecessor operators. We consider graph games with $n$ vertices and parity conditions with $c$ priorities. While many explicit algorithms exist for graph games with parity conditions, for set-based symbolic algorithms there are only two algorithms (notice that we use space to refer to the number of sets stored by a symbolic algorithm): (a) the basic algorithm that requires $O(n^c)$ symbolic operations and linear space; and (b) an improved algorithm that requires $O(n^{c/2+1})$ symbolic operations but also $O(n^{c/2+1})$ space (i.e., exponential space). In this work we present two set-based symbolic algorithms for parity games: (a) our first algorithm requires $O(n^{c/2+1})$ symbolic operations and only requires linear space; and (b) developing on our first algorithm, we present an algorithm that requires $O(n^{c/3+1})$ symbolic operations and only linear space. We also present the first linear space set-based symbolic algorithm for parity games that requires at most a sub-exponential number of symbolic operations.

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arXiv2017

Fully Dynamic Approximate Maximum Matching and Minimum Vertex Cover in $O(\log^3 n)$ Worst Case Update Time

We consider the problem of maintaining an approximately maximum (fractional) matching and an approximately minimum vertex cover in a dynamic graph. Starting with the seminal paper by Onak and Rubinfeld [STOC 2010], this problem has received significant attention in recent years. There remains, however, a polynomial gap between the best known worst case update time and the best known amortised update time for this problem, even after allowing for randomisation. Specifically, Bernstein and Stein [ICALP 2015, SODA 2016] have the best known worst case update time. They present a deterministic data structure with approximation ratio $(3/2+ε)$ and worst case update time $O(m^{1/4}/ε^2)$, where $m$ is the number of edges in the graph. In recent past, Gupta and Peng [FOCS 2013] gave a deterministic data structure with approximation ratio $(1+ε)$ and worst case update time $O(\sqrt{m}/ε^2)$. No known randomised data structure beats the worst case update times of these two results. In contrast, the paper by Onak and Rubinfeld [STOC 2010] gave a randomised data structure with approximation ratio $O(1)$ and amortised update time $O(\log^2 n)$, where $n$ is the number of nodes in the graph. This was later improved by Baswana, Gupta and Sen [FOCS 2011] and Solomon [FOCS 2016], leading to a randomised date structure with approximation ratio $2$ and amortised update time $O(1)$. We bridge the polynomial gap between the worst case and amortised update times for this problem, without using any randomisation. We present a deterministic data structure with approximation ratio $(2+ε)$ and worst case update time $O(\log^3 n)$, for all sufficiently small constants $ε$.

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arXiv2017

Conditional Hardness for Sensitivity Problems

In recent years it has become popular to study dynamic problems in a sensitivity setting: Instead of allowing for an arbitrary sequence of updates, the sensitivity model only allows to apply batch updates of small size to the original input data. The sensitivity model is particularly appealing since recent strong conditional lower bounds ruled out fast algorithms for many dynamic problems, such as shortest paths, reachability, or subgraph connectivity. In this paper we prove conditional lower bounds for sensitivity problems. For example, we show that under the Boolean Matrix Multiplication (BMM) conjecture combinatorial algorithms cannot compute the (4/3 - ε)-approximate diameter of an undirected unweighted dense graph with truly subcubic preprocessing time and truly subquadratic update/query time. This result is surprising since in the static setting it is not clear whether a reduction from BMM to diameter is possible. We further show under the BMM conjecture that many problems, such as reachability or approximate shortest paths, cannot be solved faster than by recomputation from scratch even after only one or two edge insertions. We give more lower bounds under the Strong Exponential Time Hypothesis and the All Pairs Shortest Paths Conjecture. Many of our lower bounds also hold for static oracle data structures where no sensitivity is required. Finally, we give the first algorithm for the (1 + ε)-approximate radius, diameter, and eccentricity problems in directed or undirected unweighted graphs in case of single edges failures. The algorithm has a truly subcubic running time for graphs with a truly subquadratic number of edges; it is tight w.r.t. the conditional lower bounds we obtain.

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arXiv2016

Improved Algorithms for Decremental Single-Source Reachability on Directed Graphs

Recently we presented the first algorithm for maintaining the set of nodes reachable from a source node in a directed graph that is modified by edge deletions with $o(mn)$ total update time, where $m$ is the number of edges and $n$ is the number of nodes in the graph [Henzinger et al. STOC 2014]. The algorithm is a combination of several different algorithms, each for a different $m$ vs. $n$ trade-off. For the case of $m = Θ(n^{1.5})$ the running time is $O(n^{2.47})$, just barely below $mn = Θ(n^{2.5})$. In this paper we simplify the previous algorithm using new algorithmic ideas and achieve an improved running time of $\tilde O(\min(m^{7/6} n^{2/3}, m^{3/4} n^{5/4 + o(1)}, m^{2/3} n^{4/3+o(1)} + m^{3/7} n^{12/7+o(1)}))$. This gives, e.g., $O(n^{2.36})$ for the notorious case $m = Θ(n^{1.5})$. We obtain the same upper bounds for the problem of maintaining the strongly connected components of a directed graph undergoing edge deletions. Our algorithms are correct with high probabililty against an oblivious adversary.

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arXiv2016

Incremental Exact Min-Cut in Poly-logarithmic Amortized Update Time

We present a deterministic incremental algorithm for \textit{exactly} maintaining the size of a minimum cut with $\widetilde{O}(1)$ amortized time per edge insertion and $O(1)$ query time. This result partially answers an open question posed by Thorup [Combinatorica 2007]. It also stays in sharp contrast to a polynomial conditional lower-bound for the fully-dynamic weighted minimum cut problem. Our algorithm is obtained by combining a recent sparsification technique of Kawarabayashi and Thorup [STOC 2015] and an exact incremental algorithm of Henzinger [J. of Algorithm 1997]. We also study space-efficient incremental algorithms for the minimum cut problem. Concretely, we show that there exists an ${O}(n\log n/\varepsilon^2)$ space Monte-Carlo algorithm that can process a stream of edge insertions starting from an empty graph, and with high probability, the algorithm maintains a $(1+\varepsilon)$-approximation to the minimum cut. The algorithm has $\widetilde{O}(1)$ amortized update-time and constant query-time.

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arXiv2016

Deterministic Fully Dynamic Approximate Vertex Cover and Fractional Matching in $O(1)$ Amortized Update Time

We consider the problems of maintaining an approximate maximum matching and an approximate minimum vertex cover in a dynamic graph undergoing a sequence of edge insertions/deletions. Starting with the seminal work of Onak and Rubinfeld [STOC 2010], this problem has received significant attention in recent years. Very recently, extending the framework of Baswana, Gupta and Sen [FOCS 2011], Solomon [FOCS 2016] gave a randomized dynamic algorithm for this problem that has an approximation ratio of $2$ and an amortised update time of $O(1)$ with high probability. This algorithm requires the assumption of an {\em oblivious adversary}, meaning that the future sequence of edge insertions/deletions in the graph cannot depend in any way on the algorithm's past output. A natural way to remove the assumption on oblivious adversary is to give a deterministic dynamic algorithm for the same problem in $O(1)$ update time. In this paper, we resolve this question. We present a new {\em deterministic} fully dynamic algorithm that maintains a $O(1)$-approximate minimum vertex cover and maximum fractional matching, with an amortised update time of $O(1)$. Previously, the best deterministic algorithm for this problem was due to Bhattacharya, Henzinger and Italiano [SODA 2015]; it had an approximation ratio of $(2+ε)$ and an amortised update time of $O(\log n/ε^2)$. Our results also extend to a fully dynamic $O(f^3)$-approximate algorithm with $O(f^2)$ amortized update time for the hypergraph vertex cover and fractional hypergraph matching problems, where every hyperedge has at most $f$ vertices.

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arXiv2016

Maximizing a Submodular Function with Viability Constraints

We study the problem of maximizing a monotone submodular function with viability constraints. This problem originates from computational biology, where we are given a phylogenetic tree over a set of species and a directed graph, the so-called food web, encoding viability constraints between these species. These food webs usually have constant {depth}. The goal is to select a subset of $k$ species that satisfies the viability constraints and has maximal phylogenetic diversity. As this problem is known to be NP-hard, we investigate approximation algorithms. We present the first constant factor approximation algorithm if the depth is constant. Its approximation ratio is $(1-\frac{1}{\sqrt{e}})$. This algorithm not only applies to phylogenetic trees with viability constraints but for arbitrary monotone submodular set functions with viability constraints. Second, we show that there is no $(1-1/e+ε)$-approximation algorithm for our problem setting (even for additive functions) and that there is no approximation algorithm for a slight extension of this setting.

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arXiv2016

Incremental and Fully Dynamic Subgraph Connectivity For Emergency Planning

During the last 10 years it has become popular to study dynamic graph problems in a emergency planning or sensitivity setting: Instead of considering the general fully dynamic problem, we only have to process a single batch update of size $d$; after the update we have to answer queries. In this paper, we consider the dynamic subgraph connectivity problem with sensitivity $d$: We are given a graph of which some vertices are activated and some are deactivated. After that we get a single update in which the states of up to d vertices are changed. Then we get a sequence of connectivity queries in the subgraph of activated vertices. We present the first fully dynamic algorithm for this problem which has an update and query time only slightly worse than the best decremental algorithm. In addition, we present the first incremental algorithm which is tight with respect to the best known conditional lower bound; moreover, the algorithm is simple and we believe it is implementable and efficient in practice.

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arXiv2016

Conditionally Optimal Algorithms for Generalized Büchi Games

Games on graphs provide the appropriate framework to study several central problems in computer science, such as the verification and synthesis of reactive systems. One of the most basic objectives for games on graphs is the liveness (or Büchi) objective that given a target set of vertices requires that some vertex in the target set is visited infinitely often. We study generalized Büchi objectives (i.e., conjunction of liveness objectives), and implications between two generalized Büchi objectives (known as GR(1) objectives), that arise in numerous applications in computer-aided verification. We present improved algorithms and conditional super-linear lower bounds based on widely believed assumptions about the complexity of (A1) combinatorial Boolean matrix multiplication and (A2) CNF-SAT. We consider graph games with $n$ vertices, $m$ edges, and generalized Büchi objectives with $k$ conjunctions. First, we present an algorithm with running time $O(k \cdot n^2)$, improving the previously known $O(k \cdot n \cdot m)$ and $O(k^2 \cdot n^2)$ worst-case bounds. Our algorithm is optimal for dense graphs under (A1). Second, we show that the basic algorithm for the problem is optimal for sparse graphs when the target sets have constant size under (A2). Finally, we consider GR(1) objectives, with $k_1$ conjunctions in the antecedent and $k_2$ conjunctions in the consequent, and present an $O(k_1 \cdot k_2 \cdot n^{2.5})$-time algorithm, improving the previously known $O(k_1 \cdot k_2 \cdot n \cdot m)$-time algorithm for $m > n^{1.5}$.

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arXiv2016

Sublinear-Time Decremental Algorithms for Single-Source Reachability and Shortest Paths on Directed Graphs

We consider dynamic algorithms for maintaining Single-Source Reachability (SSR) and approximate Single-Source Shortest Paths (SSSP) on $n$-node $m$-edge directed graphs under edge deletions (decremental algorithms). The previous fastest algorithm for SSR and SSSP goes back three decades to Even and Shiloach [JACM 1981]; it has $ O(1) $ query time and $ O (mn) $ total update time (i.e., linear amortized update time if all edges are deleted). This algorithm serves as a building block for several other dynamic algorithms. The question whether its total update time can be improved is a major, long standing, open problem. In this paper, we answer this question affirmatively. We obtain a randomized algorithm with an expected total update time of $ O(\min (m^{7/6} n^{2/3 + o(1)}, m^{3/4} n^{5/4 + o(1)}) ) = O (m n^{9/10 + o(1)}) $ for SSR and $(1+ε)$-approximate SSSP if the edge weights are integers from $ 1 $ to $ W \leq 2^{\log^c{n}} $ and $ ε\geq 1 / \log^c{n} $ for some constant $ c $. We also extend our algorithm to achieve roughly the same running time for Strongly Connected Components (SCC), improving the algorithm of Roditty and Zwick [FOCS 2002]. Our algorithm is most efficient for sparse and dense graphs. When $ m = Θ(n) $ its running time is $ O (n^{1 + 5/6 + o(1)}) $ and when $ m = Θ(n^2) $ its running time is $ O (n^{2 + 3/4 + o(1)}) $. For SSR we also obtain an algorithm that is faster for dense graphs and has a total update time of $ O ( m^{2/3} n^{4/3 + o(1)} + m^{3/7} n^{12/7 + o(1)}) $ which is $ O (n^{2 + 2/3}) $ when $ m = Θ(n^2) $. All our algorithms have constant query time in the worst case and are correct with high probability against an oblivious adversary.

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arXiv2016

Polynomial-Time Algorithms for Energy Games with Special Weight Structures

Energy games belong to a class of turn-based two-player infinite-duration games}played on a weighted directed graph. It is one of the rare and intriguing combinatorial problems that lie in ${\sf NP} \cap {\sf co\mbox{-}NP}$, but are not known to be in ${\sf P}$. The existence of polynomial-time algorithms has been a major open problem for decades and apart from pseudopolynomial algorithms there is no algorithm that solves any non-trivial subclass in polynomial time. In this paper, we give several results based on the weight structures of the graph. First, we identify a notion of penalty and present a polynomial-time algorithm when the penalty is large. Our algorithm is the first polynomial-time algorithm on a large class of weighted graphs. It includes several worst-case instances on which previous algorithms, such as value iteration and random facet algorithms, require at least sub-exponential time. Our main technique is developing the first non-trivial approximation algorithm and showing how to convert it to an exact algorithm. Moreover, we show that in a practical case in verification where weights are clustered around a constant number of values, the energy game problem can be solved in polynomial time. We also show that the problem is still as hard as in general when the clique-width is bounded or the graph is strongly ergodic, suggesting that restricting the graph structure does not necessarily help.

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arXiv2016

Graph Minors for Preserving Terminal Distances Approximately - Lower and Upper Bounds

Given a graph where vertices are partitioned into $k$ terminals and non-terminals, the goal is to compress the graph (i.e., reduce the number of non-terminals) using minor operations while preserving terminal distances approximately.The distortion of a compressed graph is the maximum multiplicative blow-up of distances between all pairs of terminals. We study the trade-off between the number of non-terminals and the distortion. This problem generalizes the Steiner Point Removal (SPR) problem, in which all non-terminals must be removed. We introduce a novel black-box reduction to convert any lower bound on distortion for the SPR problem into a super-linear lower bound on the number of non-terminals, with the same distortion, for our problem. This allows us to show that there exist graphs such that every minor with distortion less than $2~/~2.5~/~3$ must have $Ω(k^2)~/~Ω(k^{5/4})~/~Ω(k^{6/5})$ non-terminals, plus more trade-offs in between. The black-box reduction has an interesting consequence: if the tight lower bound on distortion for the SPR problem is super-constant, then allowing any $O(k)$ non-terminals will not help improving the lower bound to a constant. We also build on the existing results on spanners, distance oracles and connected 0-extensions to show a number of upper bounds for general graphs, planar graphs, graphs that exclude a fixed minor and bounded treewidth graphs. Among others, we show that any graph admits a minor with $O(\log k)$ distortion and $O(k^{2})$ non-terminals, and any planar graph admits a minor with $1+\varepsilon$ distortion and $\widetilde{O}((k/\varepsilon)^{2})$ non-terminals.

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arXiv2016

Ad Exchange: Envy-Free Auctions with Mediators

Ad exchanges are an emerging platform for trading advertisement slots on the web with billions of dollars revenue per year. Every time a user visits a web page, the publisher of that web page can ask an ad exchange to auction off the ad slots on this page to determine which advertisements are shown at which price. Due to the high volume of traffic, ad networks typically act as mediators for individual advertisers at ad exchanges. If multiple advertisers in an ad network are interested in the ad slots of the same auction, the ad network might use a "local" auction to resell the obtained ad slots among its advertisers. In this work we want to deepen the theoretical understanding of these new markets by analyzing them from the viewpoint of combinatorial auctions. Prior work studied mostly single-item auctions, while we allow the advertisers to express richer preferences over multiple items. We develop a game-theoretic model for the entanglement of the central auction at the ad exchange with the local auctions at the ad networks. We consider the incentives of all three involved parties and suggest a three-party competitive equilibrium, an extension of the Walrasian equilibrium that ensures envy-freeness for all participants. We show the existence of a three-party competitive equilibrium and a polynomial-time algorithm to find one for gross-substitute bidder valuations.

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arXiv2016

Online Ad Assignment with an Ad Exchange

Ad exchanges are becoming an increasingly popular way to sell advertisement slots on the internet. An ad exchange is basically a spot market for ad impressions. A publisher who has already signed contracts reserving advertisement impressions on his pages can choose between assigning a new ad impression for a new page view to a contracted advertiser or to sell it at an ad exchange. This leads to an online revenue maximization problem for the publisher. Given a new impression to sell decide whether (a) to assign it to a contracted advertiser and if so to which one or (b) to sell it at the ad exchange and if so at which reserve price. We make no assumptions about the distribution of the advertiser valuations that participate in the ad exchange and show that there exists a simple primal-dual based online algorithm, whose lower bound for the revenue converges to $R_{ADX} + R_A (1 - 1/e)$, where $R_{ADX}$ is the revenue that the optimum algorithm achieves from the ad exchange and $R_A$ is the revenue that the optimum algorithm achieves from the contracted advertisers.

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arXiv2016

New Deterministic Approximation Algorithms for Fully Dynamic Matching

We present two deterministic dynamic algorithms for the maximum matching problem. (1) An algorithm that maintains a $(2+ε)$-approximate maximum matching in general graphs with $O(\text{poly}(\log n, 1/ε))$ update time. (2) An algorithm that maintains an $α_K$ approximation of the {\em value} of the maximum matching with $O(n^{2/K})$ update time in bipartite graphs, for every sufficiently large constant positive integer $K$. Here, $1\leq α_K < 2$ is a constant determined by the value of $K$. Result (1) is the first deterministic algorithm that can maintain an $o(\log n)$-approximate maximum matching with polylogarithmic update time, improving the seminal result of Onak et al. [STOC 2010]. Its approximation guarantee almost matches the guarantee of the best {\em randomized} polylogarithmic update time algorithm [Baswana et al. FOCS 2011]. Result (2) achieves a better-than-two approximation with {\em arbitrarily small polynomial} update time on bipartite graphs. Previously the best update time for this problem was $O(m^{1/4})$ [Bernstein et al. ICALP 2015], where $m$ is the current number of edges in the graph.

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arXiv2016

Design of Dynamic Algorithms via Primal-Dual Method

We develop a dynamic version of the primal-dual method for optimization problems, and apply it to obtain the following results. (1) For the dynamic set-cover problem, we maintain an $O(f^2)$-approximately optimal solution in $O(f \cdot \log (m+n))$ amortized update time, where $f$ is the maximum "frequency" of an element, $n$ is the number of sets, and $m$ is the maximum number of elements in the universe at any point in time. (2) For the dynamic $b$-matching problem, we maintain an $O(1)$-approximately optimal solution in $O(\log^3 n)$ amortized update time, where $n$ is the number of nodes in the graph.

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arXiv2016

Model and Objective Separation with Conditional Lower Bounds: Disjunction is Harder than Conjunction

Given a model of a system and an objective, the model-checking question asks whether the model satisfies the objective. We study polynomial-time problems in two classical models, graphs and Markov Decision Processes (MDPs), with respect to several fundamental $ω$-regular objectives, e.g., Rabin and Streett objectives. For many of these problems the best-known upper bounds are quadratic or cubic, yet no super-linear lower bounds are known. In this work our contributions are two-fold: First, we present several improved algorithms, and second, we present the first conditional super-linear lower bounds based on widely believed assumptions about the complexity of CNF-SAT and combinatorial Boolean matrix multiplication. A separation result for two models with respect to an objective means a conditional lower bound for one model that is strictly higher than the existing upper bound for the other model, and similarly for two objectives with respect to a model. Our results establish the following separation results: (1) A separation of models (graphs and MDPs) for disjunctive queries of reachability and Büchi objectives. (2) Two kinds of separations of objectives, both for graphs and MDPs, namely, (2a) the separation of dual objectives such as reachability/safety (for disjunctive questions) and Streett/Rabin objectives, and (2b) the separation of conjunction and disjunction of multiple objectives of the same type such as safety, Büchi, and coBüchi. In summary, our results establish the first model and objective separation results for graphs and MDPs for various classical $ω$-regular objectives. Quite strikingly, we establish conditional lower bounds for the disjunction of objectives that are strictly higher than the existing upper bounds for the conjunction of the same objectives.

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arXiv2015

Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture

Consider the following Online Boolean Matrix-Vector Multiplication problem: We are given an $n\times n$ matrix $M$ and will receive $n$ column-vectors of size $n$, denoted by $v_1,\ldots,v_n$, one by one. After seeing each vector $v_i$, we have to output the product $Mv_i$ before we can see the next vector. A naive algorithm can solve this problem using $O(n^3)$ time in total, and its running time can be slightly improved to $O(n^3/\log^2 n)$ [Williams SODA'07]. We show that a conjecture that there is no truly subcubic ($O(n^{3-ε})$) time algorithm for this problem can be used to exhibit the underlying polynomial time hardness shared by many dynamic problems. For a number of problems, such as subgraph connectivity, Pagh's problem, $d$-failure connectivity, decremental single-source shortest paths, and decremental transitive closure, this conjecture implies tight hardness results. Thus, proving or disproving this conjecture will be very interesting as it will either imply several tight unconditional lower bounds or break through a common barrier that blocks progress with these problems. This conjecture might also be considered as strong evidence against any further improvement for these problems since refuting it will imply a major breakthrough for combinatorial Boolean matrix multiplication and other long-standing problems if the term "combinatorial algorithms" is interpreted as "non-Strassen-like algorithms" [Ballard et al. SPAA'11]. The conjecture also leads to hardness results for problems that were previously based on diverse problems and conjectures, such as 3SUM, combinatorial Boolean matrix multiplication, triangle detection, and multiphase, thus providing a uniform way to prove polynomial hardness results for dynamic algorithms; some of the new proofs are also simpler or even become trivial. The conjecture also leads to stronger and new, non-trivial, hardness results.

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