 Award ID(s):
 1909111
 NSFPAR ID:
 10462948
 Date Published:
 Journal Name:
 Leibniz international proceedings in informatics
 Volume:
 251
 ISSN:
 18688969
 Page Range / eLocation ID:
 20:1  20:22
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Given a metric space ℳ = (X,δ), a weighted graph G over X is a metric tspanner of ℳ if for every u,v ∈ X, δ(u,v) ≤ δ_G(u,v) ≤ t⋅ δ(u,v), where δ_G is the shortest path metric in G. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points (s₁, …, s_n), where the points are presented one at a time (i.e., after i steps, we have seen S_i = {s₁, … , s_i}). The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to remove any edge from the spanner. The goal is to maintain a tspanner G_i for S_i for all i, while minimizing the number of edges, and their total weight. Under the L₂norm in ℝ^d for arbitrary constant d ∈ ℕ, we present an online (1+ε)spanner algorithm with competitive ratio O_d(ε^{d} log n), improving the previous bound of O_d(ε^{(d+1)}log n). Moreover, the spanner maintained by the algorithm has O_d(ε^{1d}log ε^{1})⋅ n edges, almost matching the (offline) optimal bound of O_d(ε^{1d})⋅ n. In the plane, a tighter analysis of the same algorithm provides an almost quadratic improvement of the competitive ratio to O(ε^{3/2}logε^{1}log n), by comparing the online spanner with an instanceoptimal spanner directly, bypassing the comparison to an MST (i.e., lightness). As a counterpart, we design a sequence of points that yields a Ω_d(ε^{d}) lower bound for the competitive ratio for online (1+ε)spanner algorithms in ℝ^d under the L₁norm. Then we turn our attention to online spanners in general metrics. Note that, it is not possible to obtain a spanner with stretch less than 3 with a subquadratic number of edges, even in the offline setting, for general metrics. We analyze an online version of the celebrated greedy spanner algorithm, dubbed ordered greedy. With stretch factor t = (2k1)(1+ε) for k ≥ 2 and ε ∈ (0,1), we show that it maintains a spanner with O(ε^{1}logε^{1})⋅ n^{1+1/k} edges and O(ε^{1}n^{1/k}log² n) lightness for a sequence of n points in a metric space. We show that these bounds cannot be significantly improved, by introducing an instance that achieves an Ω(1/k⋅ n^{1/k}) competitive ratio on both sparsity and lightness. Furthermore, we establish the tradeoff among stretch, number of edges and lightness for points in ultrametrics, showing that one can maintain a (2+ε)spanner for ultrametrics with O(ε^{1}logε^{1})⋅ n edges and O(ε^{2}) lightness.more » « less

Given a metric space ℳ = (X,δ), a weighted graph G over X is a metric tspanner of ℳ if for every u,v ∈ X, δ(u,v) ≤ δ_G(u,v) ≤ t⋅ δ(u,v), where δ_G is the shortest path metric in G. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points (s₁, …, s_n), where the points are presented one at a time (i.e., after i steps, we have seen S_i = {s₁, … , s_i}). The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to remove any edge from the spanner. The goal is to maintain a tspanner G_i for S_i for all i, while minimizing the number of edges, and their total weight. Under the L₂norm in ℝ^d for arbitrary constant d ∈ ℕ, we present an online (1+ε)spanner algorithm with competitive ratio O_d(ε^{d} log n), improving the previous bound of O_d(ε^{(d+1)}log n). Moreover, the spanner maintained by the algorithm has O_d(ε^{1d}log ε^{1})⋅ n edges, almost matching the (offline) optimal bound of O_d(ε^{1d})⋅ n. In the plane, a tighter analysis of the same algorithm provides an almost quadratic improvement of the competitive ratio to O(ε^{3/2}logε^{1}log n), by comparing the online spanner with an instanceoptimal spanner directly, bypassing the comparison to an MST (i.e., lightness). As a counterpart, we design a sequence of points that yields a Ω_d(ε^{d}) lower bound for the competitive ratio for online (1+ε)spanner algorithms in ℝ^d under the L₁norm. Then we turn our attention to online spanners in general metrics. Note that, it is not possible to obtain a spanner with stretch less than 3 with a subquadratic number of edges, even in the offline setting, for general metrics. We analyze an online version of the celebrated greedy spanner algorithm, dubbed ordered greedy. With stretch factor t = (2k1)(1+ε) for k ≥ 2 and ε ∈ (0,1), we show that it maintains a spanner with O(ε^{1}logε^{1})⋅ n^{1+1/k} edges and O(ε^{1}n^{1/k}log² n) lightness for a sequence of n points in a metric space. We show that these bounds cannot be significantly improved, by introducing an instance that achieves an Ω(1/k⋅ n^{1/k}) competitive ratio on both sparsity and lightness. Furthermore, we establish the tradeoff among stretch, number of edges and lightness for points in ultrametrics, showing that one can maintain a (2+ε)spanner for ultrametrics with O(ε^{1}logε^{1})⋅ n edges and O(ε^{2}) lightness.more » « less

One of the most important and wellstudied settings for network design is edgeconnectivity requirements. This encompasses uniform demands such as the Minimum kEdgeConnected Spanning Subgraph problem (kECSS), as well as nonuniform demands such as the Survivable Network Design problem. A weakness of these formulations, though, is that we are not able to ask for faulttolerance larger than the connectivity. Taking inspiration from recent definitions and progress in graph spanners, we introduce and study new variants of these problems under a notion of relative faulttolerance. Informally, we require not that two nodes are connected if there are a bounded number of faults (as in the classical setting), but that two nodes are connected if there are a bounded number of faults and the two nodes are connected in the underlying graph postfaults. That is, the subgraph we build must "behave" identically to the underlying graph with respect to connectivity after bounded faults. We define and introduce these problems, and provide the first approximation algorithms: a (1+4/k)approximation for the unweighted relative version of kECSS, a 2approximation for the weighted relative version of kECSS, and a 27/4approximation for the special case of Relative Survivable Network Design with only a single demand with a connectivity requirement of 3. To obtain these results, we introduce a number of technical ideas that may of independent interest. First, we give a generalization of Jain’s iterative rounding analysis that works even when the cutrequirement function is not weakly supermodular, but instead satisfies a weaker definition we introduce and term local weak supermodularity. Second, we prove a structure theorem and design an approximation algorithm utilizing a new decomposition based on important separators, which are structures commonly used in fixedparameter algorithms that have not commonly been used in approximation algorithms.more » « less

Abstract In a Merlin–Arthur proof system, the proof verifier (Arthur) accepts valid proofs (from Merlin) with probability 1, and rejects invalid proofs with probability arbitrarily close to 1. The running time of such a system is defined to be the length of Merlin’s proof plus the running time of Arthur. We provide new Merlin–Arthur proof systems for some key problems in finegrained complexity. In several cases our proof systems have optimal running time. Our main results include:
Certifying that a list of
n integers has no 3SUM solution can be done in Merlin–Arthur time . Previously, Carmosino et al. [ITCS 2016] showed that the problem has a nondeterministic algorithm running in$$\tilde{O}(n)$$ $\stackrel{~}{O}\left(n\right)$ time (that is, there is a proof system with proofs of length$$\tilde{O}(n^{1.5})$$ $\stackrel{~}{O}\left({n}^{1.5}\right)$ and a deterministic verifier running in$$\tilde{O}(n^{1.5})$$ $\stackrel{~}{O}\left({n}^{1.5}\right)$ time).$$\tilde{O}(n^{1.5})$$ $\stackrel{~}{O}\left({n}^{1.5}\right)$Counting the number of
k cliques with total edge weight equal to zero in ann node graph can be done in Merlin–Arthur time (where$${\tilde{O}}(n^{\lceil k/2\rceil })$$ $\stackrel{~}{O}\left({n}^{\lceil k/2\rceil}\right)$ ). For odd$$k\ge 3$$ $k\ge 3$k , this bound can be further improved for sparse graphs: for example, counting the number of zeroweight triangles in anm edge graph can be done in Merlin–Arthur time . Previous Merlin–Arthur protocols by Williams [CCC’16] and Björklund and Kaski [PODC’16] could only count$${\tilde{O}}(m)$$ $\stackrel{~}{O}\left(m\right)$k cliques in unweighted graphs, and had worse running times for smallk .Computing the AllPairs Shortest Distances matrix for an
n node graph can be done in Merlin–Arthur time . Note this is optimal, as the matrix can have$$\tilde{O}(n^2)$$ $\stackrel{~}{O}\left({n}^{2}\right)$ nonzero entries in general. Previously, Carmosino et al. [ITCS 2016] showed that this problem has an$$\Omega (n^2)$$ $\Omega \left({n}^{2}\right)$ nondeterministic time algorithm.$$\tilde{O}(n^{2.94})$$ $\stackrel{~}{O}\left({n}^{2.94}\right)$Certifying that an
n variablek CNF is unsatisfiable can be done in Merlin–Arthur time . We also observe an algebrization barrier for the previous$$2^{n/2  n/O(k)}$$ ${2}^{n/2n/O\left(k\right)}$ time Merlin–Arthur protocol of R. Williams [CCC’16] for$$2^{n/2}\cdot \textrm{poly}(n)$$ ${2}^{n/2}\xb7\text{poly}\left(n\right)$ SAT: in particular, his protocol algebrizes, and we observe there is no algebrizing protocol for$$\#$$ $\#$k UNSAT running in time. Therefore we have to exploit nonalgebrizing properties to obtain our new protocol.$$2^{n/2}/n^{\omega (1)}$$ ${2}^{n/2}/{n}^{\omega \left(1\right)}$ Due to the centrality of these problems in finegrained complexity, our results have consequences for many other problems of interest. For example, our work implies that certifying there is no Subset Sum solution toCertifying a Quantified Boolean Formula is true can be done in Merlin–Arthur time
. Previously, the only nontrivial result known along these lines was an Arthur–Merlin–Arthur protocol (where Merlin’s proof depends on some of Arthur’s coins) running in$$2^{4n/5}\cdot \textrm{poly}(n)$$ ${2}^{4n/5}\xb7\text{poly}\left(n\right)$ time.$$2^{2n/3}\cdot \textrm{poly}(n)$$ ${2}^{2n/3}\xb7\text{poly}\left(n\right)$n integers can be done in Merlin–Arthur time , improving on the previous best protocol by Nederlof [IPL 2017] which took$$2^{n/3}\cdot \textrm{poly}(n)$$ ${2}^{n/3}\xb7\text{poly}\left(n\right)$ time.$$2^{0.49991n}\cdot \textrm{poly}(n)$$ ${2}^{0.49991n}\xb7\text{poly}\left(n\right)$ 
null (Ed.)A longstanding conjecture by Kotzig, Ringel, and Rosa states that every tree admits a graceful labeling. That is, for any tree $T$ with $n$~edges, it is conjectured that there exists a labeling $f\colon V(T) \to \{0,1,\ldots,n\}$ such that the set of induced edge labels $\bigl\{ f(u)f(v) : \{u,v\}\in E(T) \bigr\}$ is exactly $\{1,2,\ldots,n\}$. We extend this concept to allow for multigraphs with edge multiplicity at most~$2$. A \emph{2fold graceful labeling} of a graph (or multigraph) $G$ with $n$~edges is a onetoone function $f\colon V(G) \to \{0,1,\ldots,n\}$ such that the multiset of induced edge labels is comprised of two copies of each element in $\bigl\{ 1,2,\ldots, \lfloor n/2 \rfloor \bigr\}$ and, if $n$ is odd, one copy of $\bigl\{ \lceil n/2 \rceil \bigr\}$. When $n$ is even, this concept is similar to that of 2equitable labelings which were introduced by Bloom and have been studied for several classes of graphs. We show that caterpillars, cycles of length $n \not\equiv 1 \pmod{4}$, and complete bipartite graphs admit 2fold graceful labelings. We also show that under certain conditions, the join of a tree and an empty graph (i.e., a graph with vertices but no edges) is $2$fold graceful.more » « less