An official website of the United States government
For graphs of average degree $$d$$, positive integer weights bounded by $$W$$, and accuracy parameter $$\epsilon>0$$, [Chazelle, Rubinfeld, Trevisan; SICOMP'05] have shown that the weight of the minimum spanning tree can be $$(1+\epsilon)$$-approximated in $$\tilde{O}(Wd/\epsilon^2)$$ expected time. This algorithm is frequently taught in courses on sublinear time algorithms. However, the $$\tilde{O}(Wd/\epsilon^2)$$-time variant requires an involved analysis, leading to simpler but much slower variations being taught instead. Here we present an alternative that is not only simpler to analyze, but also improves the number of queries, getting closer to the nearly-matching information theoretic lower bound. In addition to estimating the weight of the MST, our algorithm is also a perfect sampler for sampling uniformly at random an edge of the MST. At the core of our result is the insight that halting Prim's algorithm after an expected $$\tilde{O}(d)$$ number of steps, then returning the highest weighted edge of the tree, results in sampling an edge of the MST uniformly at random. Via repeated trials and averaging the results, this immediately implies an algorithm for estimating the weight of the MST. Since our algorithm is based on Prim's, it naturally works for non-integer weighted graphs as well.
more »
« less
Sublinear Algorithms and Lower Bounds for Metric TSP Cost Estimation
We consider the problem of designing sublinear time algorithms for estimating the cost of minimum] metric traveling salesman (TSP) tour. Specifically, given access to a n × n distance matrix D that specifies pairwise distances between n points, the goal is to estimate the TSP cost by performing only sublinear (in the size of D) queries. For the closely related problem of estimating the weight of a metric minimum spanning tree (MST), it is known that for any epsilon > 0, there exists an O^~(n/epsilon^O(1))-time algorithm that returns a (1+epsilon)-approximate estimate of the MST cost. This result immediately implies an O^~(n/epsilon^O(1)) time algorithm to estimate the TSP cost to within a (2 + epsilon) factor for any epsilon > 0. However, no o(n^2)-time algorithms are known to approximate metric TSP to a factor that is strictly better than 2. On the other hand, there were also no known barriers that rule out existence of (1 + epsilon)-approximate estimation algorithms for metric TSP with O^~ (n) time for any fixed epsilon > 0. In this paper, we make progress on both algorithms and lower bounds for estimating metric TSP cost. On the algorithmic side, we first consider the graphic TSP problem where the metric D corresponds to shortest path distances in a connected unweighted undirected graph. We show that there exists an O^~(n) time algorithm that estimates the cost of graphic TSP to within a factor of (2 − epsilon_0) for some epsilon_0 > 0. This is the first sublinear cost estimation algorithm for graphic TSP that achieves an approximation factor less than 2. We also consider another well-studied special case of metric TSP, namely, (1, 2)-TSP where all distances are either 1 or 2, and give an O^~(n ^ 1.5) time algorithm to estimate optimal cost to within a factor of 1.625. Our estimation algorithms for graphic TSP as well as for (1, 2)-TSP naturally lend themselves to O^~(n) space streaming algorithms that give an 11/6-approximation for graphic TSP and a 1.625-approximation for (1, 2)-TSP. These results motivate the natural question if analogously to metric MST, for any epsilon > 0, (1 + epsilon)-approximate estimates can be obtained for graphic TSP and (1, 2)-TSP using O^~ (n) queries. We answer this question in the negative – there exists an epsilon_0 > 0, such that any algorithm that estimates the cost of graphic TSP ((1, 2)-TSP) to within a (1 + epsilon_0)-factor, necessarily requires (n^2) queries. This lower bound result highlights a sharp separation between the metric MST and metric TSP problems. Similarly to many classical approximation algorithms for TSP, our sublinear time estimation algorithms utilize subroutines for estimating the size of a maximum matching in the underlying graph. We show that this is not merely an artifact of our approach, and that for any epsilon > 0, any algorithm that estimates the cost of graphic TSP or (1, 2)-TSP to within a (1 + epsilon)-factor, can also be used to estimate the size of a maximum matching in a bipartite graph to within an epsilon n additive error. This connection allows us to translate known lower bounds for matching size estimation in various models to similar lower bounds for metric TSP cost estimation.
more »
« less
- Award ID(s):
- 1733794
- PAR ID:
- 10185976
- Date Published:
- Journal Name:
- Leibniz international proceedings in informatics
- Volume:
- 168
- ISSN:
- 1868-8969
- Page Range / eLocation ID:
- 30:0 - 30:18
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We show new applications of the nearest-neighbor chain algorithm, a technique that originated in agglomerative hierarchical clustering. We use it to construct the greedy multi-fragment tour for Euclidean TSP in O(n log n) time in any fixed dimension and for Steiner TSP in planar graphs in O(n sqrt(n)log n) time; we compute motorcycle graphs, a central step in straight skeleton algorithms, in O(n^(4/3+epsilon)) time for any epsilon>0.more » « less
-
We consider the problem of estimating the spectral density of the normalized adjacency matrix of an $$n$$-node undirected graph. We provide a randomized algorithm that, with $$O(n\epsilon^{-2})$$ queries to a degree and neighbor oracle and in $$O(n\epsilon^{-3})$$ time, estimates the spectrum up to $$\epsilon$$ accuracy in the Wasserstein-1 metric. This improves on previous state-of-the-art methods, including an $$O(n\epsilon^{-7})$$ time algorithm from [Braverman et al., STOC 2022] and, for sufficiently small $$\epsilon$$, a $$2^{O(\epsilon^{-1})}$$ time method from [Cohen-Steiner et al., KDD 2018]. To achieve this result, we introduce a new notion of graph sparsification, which we call \emph{nuclear sparsification}. We provide an $$O(n\epsilon^{-2})$$-query and $$O(n\epsilon^{-2})$$-time algorithm for computing $$O(n\epsilon^{-2})$$-sparse nuclear sparsifiers. We show that this bound is optimal in both its sparsity and query complexity, and we separate our results from the related notion of additive spectral sparsification. Of independent interest, we show that our sparsification method also yields the first \emph{deterministic} algorithm for spectral density estimation that scales linearly with $$n$$ (sublinear in the representation size of the graph).more » « less
-
null (Ed.)We present a general framework of designing efficient dynamic approximate algorithms for optimization on undirected graphs. In particular, we develop a technique that, given any problem that admits a certain notion of vertex sparsifiers, gives data structures that maintain approximate solutions in sub-linear update and query time. We illustrate the applicability of our paradigm to the following problems. (1) A fully-dynamic algorithm that approximates all-pair maximum-flows/minimum-cuts up to a nearly logarithmic factor in $$\tilde{O}(n^{2/3})$$ amortized time against an oblivious adversary, and $$\tilde{O}(m^{3/4})$$ time against an adaptive adversary. (2) An incremental data structure that maintains $O(1)$-approximate shortest path in $$n^{o(1)}$$ time per operation, as well as fully dynamic approximate all-pair shortest path and transshipment in $$\tilde{O}(n^{2/3+o(1)})$$ amortized time per operation. (3) A fully-dynamic algorithm that approximates all-pair effective resistance up to an $$(1+\eps)$$ factor in $$\tilde{O}(n^{2/3+o(1)} \epsilon^{-O(1)})$$ amortized update time per operation. The key tool behind result (1) is the dynamic maintenance of an algorithmic construction due to Madry [FOCS' 10], which partitions a graph into a collection of simpler graph structures (known as j-trees) and approximately captures the cut-flow and metric structure of the graph. The $O(1)$-approximation guarantee of (2) is by adapting the distance oracles by [Thorup-Zwick JACM `05]. Result (3) is obtained by invoking the random-walk based spectral vertex sparsifier by [Durfee et al. STOC `19] in a hierarchical manner, while carefully keeping track of the recourse among levels in the hierarchy.more » « less
-
We present a weighted approach to compute a maximum cardinality matching in an arbitrary bipartite graph. Our main result is a new algorithm that takes as input a weighted bipartite graph G(A cup B,E) with edge weights of 0 or 1. Let w <= n be an upper bound on the weight of any matching in G. Consider the subgraph induced by all the edges of G with a weight 0. Suppose every connected component in this subgraph has O(r) vertices and O(mr/n) edges. We present an algorithm to compute a maximum cardinality matching in G in O~(m(sqrt{w} + sqrt{r} + wr/n)) time. When all the edge weights are 1 (symmetrically when all weights are 0), our algorithm will be identical to the well-known Hopcroft-Karp (HK) algorithm, which runs in O(m sqrt{n}) time. However, if we can carefully assign weights of 0 and 1 on its edges such that both w and r are sub-linear in n and wr=O(n^{gamma}) for gamma < 3/2, then we can compute maximum cardinality matching in G in o(m sqrt{n}) time. Using our algorithm, we obtain a new O~(n^{4/3}/epsilon^4) time algorithm to compute an epsilon-approximate bottleneck matching of A,B subsetR^2 and an 1/(epsilon^{O(d)}}n^{1+(d-1)/(2d-1)}) poly log n time algorithm for computing epsilon-approximate bottleneck matching in d-dimensions. All previous algorithms take Omega(n^{3/2}) time. Given any graph G(A cup B,E) that has an easily computable balanced vertex separator for every subgraph G'(V',E') of size |V'|^{delta}, for delta in [1/2,1), we can apply our algorithm to compute a maximum matching in O~(mn^{delta/1+delta}) time improving upon the O(m sqrt{n}) time taken by the HK-Algorithm.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.4230/LIPIcs.ICALP.2020.30
