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Added by
Yasamin Nazari
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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A $k$-spanner of a graph $G$ is a sparse subgraph that preserves its shortest path distances up to a multiplicative stretch factor of $k$, and a $k$-emulator is similar but not required to be a subgraph of $G$. A classic theorem by Thorup and Zwick [JACM 05] shows that, despite the extra flexibility available to emulators, the size/stretch tradeoffs for spanners and emulators are equivalent. Our main result is that this equivalence in tradeoffs no longer holds in the commonly-studied setting of graphs with vertex failures. That is: we introduce a natural definition of vertex fault-tolerant emulators, and then we show a three-way tradeoff between size, stretch, and fault-tolerance for these emulators that polynomially surpasses the tradeoff known to be optimal for spanners. We complement our emulator upper bound with a lower bound construction that is essentially tight (within $log n$ factors of the upper bound) when the stretch is $2k-1$ and $k$ is either a fixed odd integer or $2$. We also show constructions of fault-tolerant emulators with additive error, demonstrating that these also enjoy significantly improved tradeoffs over those available for fault-tolerant additive spanners.
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Recent work has established that, for every positive integer $k$, every $n$-node graph has a $(2k-1)$-spanner on $O(f^{1-1/k} n^{1+1/k})$ edges that is resilient to $f$ edge or vertex faults. For vertex faults, this bound is tight. However, the case of edge faults is not as well understood: the best known lower bound for general $k$ is $Omega(f^{frac12 - frac{1}{2k}} n^{1+1/k} +fn)$. Our main result is to nearly close this gap with an improved upper bound, thus separating the cases of edge and vertex faults. For odd $k$, our new upper bound is $O_k(f^{frac12 - frac{1}{2k}} n^{1+1/k} + fn)$, which is tight up to hidden $poly(k)$ factors. For even $k$, our new upper bound is $O_k(f^{1/2} n^{1+1/k} +fn)$, which leaves a gap of $poly(k) f^{1/(2k)}$. Our proof is an analysis of the fault-tolerant greedy algorithm, which requires exponential time, but we also show that there is a polynomial-time algorithm which creates edge fault tolerant spanners that are larger only by factors of $k$.
Recent work has pinned down the existentially optimal size bounds for vertex fault-tolerant spanners: for any positive integer $k$, every $n$-node graph has a $(2k-1)$-spanner on $O(f^{1-1/k} n^{1+1/k})$ edges resilient to $f$ vertex faults, and there are examples of input graphs on which this bound cannot be improved. However, these proofs work by analyzing the output spanner of a certain exponential-time greedy algorithm. In this work, we give the first algorithm that produces vertex fault tolerant spanners of optimal size and which runs in polynomial time. Specifically, we give a randomized algorithm which takes $widetilde{O}left( f^{1-1/k} n^{2+1/k} + mf^2right)$ time. We also derandomize our algorithm to give a deterministic algorithm with similar bounds. This reflects an exponential improvement in runtime over [Bodwin-Patel PODC 19], the only previously known algorithm for constructing optimal vertex fault-tolerant spanners.
In this work, we initiate the study of fault tolerant Max Cut, where given an edge-weighted undirected graph $G=(V,E)$, the goal is to find a cut $Ssubseteq V$ that maximizes the total weight of edges that cross $S$ even after an adversary removes $k$ vertices from $G$. We consider two types of adversaries: an adaptive adversary that sees the outcome of the random coin tosses used by the algorithm, and an oblivious adversary that does not. For any constant number of failures $k$ we present an approximation of $(0.878-epsilon)$ against an adaptive adversary and of $alpha_{GW}approx 0.8786$ against an oblivious adversary (here $alpha_{GW}$ is the approximation achieved by the random hyperplane algorithm of [Goemans-Williamson J. ACM `95]). Additionally, we present a hardness of approximation of $alpha_{GW}$ against both types of adversaries, rendering our results (virtually) tight. The non-linear nature of the fault tolerant objective makes the design and analysis of algorithms harder when compared to the classic Max Cut. Hence, we employ approaches ranging from multi-objective optimization to LP duality and the ellipsoid algorithm to obtain our results.
The paper presents fault-tolerant (FT) labeling schemes for general graphs, as well as, improved FT routing schemes. For a given $n$-vertex graph $G$ and a bound $f$ on the number of faults, an $f$-FT connectivity labeling scheme is a distributed data structure that assigns each of the graph edges and vertices a short label, such that given the labels of the vertices $s$ and $t$, and at most $f$ failing edges $F$, one can determine if $s$ and $t$ are connected in $G setminus F$. The primary complexity measure is the length of the individual labels. Since their introduction by [Courcelle, Twigg, STACS 07], compact FT labeling schemes have been devised only for a limited collection of graph families. In this work, we fill in this gap by proposing two (independent) FT connectivity labeling schemes for general graphs, with a nearly optimal label length. This serves the basis for providing also FT approximate distance labeling schemes, and ultimately also routing schemes. Our main results for an $n$-vertex graph and a fault bound $f$ are: -- There is a randomized FT connectivity labeling scheme with a label length of $O(f+log n)$ bits, hence optimal for $f=O(log n)$. This scheme is based on the notion of cycle space sampling [Pritchard, Thurimella, TALG 11]. -- There is a randomized FT connectivity labeling scheme with a label length of $O(log^3 n)$ bits (independent of the number of faults $f$). This scheme is based on the notion of linear sketches of [Ahn et al., SODA 12]. -- For $kgeq 1$, there is a randomized routing scheme that routes a message from $s$ to $t$ in the presence of a set $F$ of faulty edges, with stretch $O(|F|^2 k)$ and routing tables of size $tilde{O}(f^3 n^{1/k})$. This significantly improves over the state-of-the-art bounds by [Chechik, ICALP 11], providing the first scheme with sub-linear FT labeling and routing schemes for general graphs.
It was recently shown that a version of the greedy algorithm gives a construction of fault-tolerant spanners that is size-optimal, at least for vertex faults. However, the algorithm to construct this spanner is not polynomial-time, and the best-known polynomial time algorithm is significantly suboptimal. Designing a polynomial-time algorithm to construct (near-)optimal fault-tolerant spanners was given as an explicit open problem in the two most recent papers on fault-tolerant spanners ([Bodwin, Dinitz, Parter, Vassilevka Williams SODA 18] and [Bodwin, Patel PODC 19]). We give a surprisingly simple algorithm which runs in polynomial time and constructs fault-tolerant spanners that are extremely close to optimal (off by only a linear factor in the stretch) by modifying the greedy algorithm to run in polynomial time. To complement this result, we also give simple distributed constructions in both the LOCAL and CONGEST models.
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