4,817 research outputs found

    On the Hardness of Partially Dynamic Graph Problems and Connections to Diameter

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    Conditional lower bounds for dynamic graph problems has received a great deal of attention in recent years. While many results are now known for the fully-dynamic case and such bounds often imply worst-case bounds for the partially dynamic setting, it seems much more difficult to prove amortized bounds for incremental and decremental algorithms. In this paper we consider partially dynamic versions of three classic problems in graph theory. Based on popular conjectures we show that: -- No algorithm with amortized update time O(n1ε)O(n^{1-\varepsilon}) exists for incremental or decremental maximum cardinality bipartite matching. This significantly improves on the O(m1/2ε)O(m^{1/2-\varepsilon}) bound for sparse graphs of Henzinger et al. [STOC'15] and O(n1/3ε)O(n^{1/3-\varepsilon}) bound of Kopelowitz, Pettie and Porat. Our linear bound also appears more natural. In addition, the result we present separates the node-addition model from the edge insertion model, as an algorithm with total update time O(mn)O(m\sqrt{n}) exists for the former by Bosek et al. [FOCS'14]. -- No algorithm with amortized update time O(m1ε)O(m^{1-\varepsilon}) exists for incremental or decremental maximum flow in directed and weighted sparse graphs. No such lower bound was known for partially dynamic maximum flow previously. Furthermore no algorithm with amortized update time O(n1ε)O(n^{1-\varepsilon}) exists for directed and unweighted graphs or undirected and weighted graphs. -- No algorithm with amortized update time O(n1/2ε)O(n^{1/2 - \varepsilon}) exists for incremental or decremental (4/3ε)(4/3-\varepsilon')-approximating the diameter of an unweighted graph. We also show a slightly stronger bound if node additions are allowed. [...]Comment: To appear at ICALP'16. Abstract truncated to fit arXiv limit

    Conditional Lower Bounds for Space/Time Tradeoffs

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    In recent years much effort has been concentrated towards achieving polynomial time lower bounds on algorithms for solving various well-known problems. A useful technique for showing such lower bounds is to prove them conditionally based on well-studied hardness assumptions such as 3SUM, APSP, SETH, etc. This line of research helps to obtain a better understanding of the complexity inside P. A related question asks to prove conditional space lower bounds on data structures that are constructed to solve certain algorithmic tasks after an initial preprocessing stage. This question received little attention in previous research even though it has potential strong impact. In this paper we address this question and show that surprisingly many of the well-studied hard problems that are known to have conditional polynomial time lower bounds are also hard when concerning space. This hardness is shown as a tradeoff between the space consumed by the data structure and the time needed to answer queries. The tradeoff may be either smooth or admit one or more singularity points. We reveal interesting connections between different space hardness conjectures and present matching upper bounds. We also apply these hardness conjectures to both static and dynamic problems and prove their conditional space hardness. We believe that this novel framework of polynomial space conjectures can play an important role in expressing polynomial space lower bounds of many important algorithmic problems. Moreover, it seems that it can also help in achieving a better understanding of the hardness of their corresponding problems in terms of time

    Distributed PCP Theorems for Hardness of Approximation in P

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    We present a new distributed model of probabilistically checkable proofs (PCP). A satisfying assignment x{0,1}nx \in \{0,1\}^n to a CNF formula φ\varphi is shared between two parties, where Alice knows x1,,xn/2x_1, \dots, x_{n/2}, Bob knows xn/2+1,,xnx_{n/2+1},\dots,x_n, and both parties know φ\varphi. The goal is to have Alice and Bob jointly write a PCP that xx satisfies φ\varphi, while exchanging little or no information. Unfortunately, this model as-is does not allow for nontrivial query complexity. Instead, we focus on a non-deterministic variant, where the players are helped by Merlin, a third party who knows all of xx. Using our framework, we obtain, for the first time, PCP-like reductions from the Strong Exponential Time Hypothesis (SETH) to approximation problems in P. In particular, under SETH we show that there are no truly-subquadratic approximation algorithms for Bichromatic Maximum Inner Product over {0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate Regular Expression Matching, and Diameter in Product Metric. All our inapproximability factors are nearly-tight. In particular, for the first two problems we obtain nearly-polynomial factors of 2(logn)1o(1)2^{(\log n)^{1-o(1)}}; only (1+o(1))(1+o(1))-factor lower bounds (under SETH) were known before

    Algorithms and Conditional Lower Bounds for Planning Problems

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    We consider planning problems for graphs, Markov decision processes (MDPs), and games on graphs. While graphs represent the most basic planning model, MDPs represent interaction with nature and games on graphs represent interaction with an adversarial environment. We consider two planning problems where there are k different target sets, and the problems are as follows: (a) the coverage problem asks whether there is a plan for each individual target set, and (b) the sequential target reachability problem asks whether the targets can be reached in sequence. For the coverage problem, we present a linear-time algorithm for graphs and quadratic conditional lower bound for MDPs and games on graphs. For the sequential target problem, we present a linear-time algorithm for graphs, a sub-quadratic algorithm for MDPs, and a quadratic conditional lower bound for games on graphs. Our results with conditional lower bounds establish (i) model-separation results showing that for the coverage problem MDPs and games on graphs are harder than graphs and for the sequential reachability problem games on graphs are harder than MDPs and graphs; (ii) objective-separation results showing that for MDPs the coverage problem is harder than the sequential target problem.Comment: Accepted at ICAPS'1
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