Given a weighted planar bipartite graph G ( A ∪ B , E ) where each edge has an integer edge cost, we give an Õ( n 4/3 log nC ) time algorithm to compute minimum-cost perfect matching; here C is the maximum edge cost in the graph. The previous best-known planarity exploiting algorithm has a running time of O ( n 3/2 log n ) and is achieved by using planar separators (Lipton and Tarjan ’80). Our algorithm is based on the bit-scaling paradigm (Gabow and Tarjan ’89). For each scale, our algorithm first executes O ( n 1/3 ) iterations of Gabow and Tarjan’s algorithm in O ( n 4/3 ) time leaving only O ( n 2/3 ) vertices unmatched. Next, it constructs a compressed residual graph H with O ( n 2/3 ) vertices and O ( n ) edges. This is achieved by using an r -division of the planar graph G with r = n 2/3 . For each partition of the r -division, there is an edge between two vertices of H if and only if they are connected by a directed path inside the partition. Using existing efficient shortest-path data structures, themore »
An O(n5/4) Time ∊-Approximation Algorithm for RMS Matching in a Plane
The 2-Wasserstein distance (or RMS distance) is a useful measure of similarity between probability distributions with exciting applications in machine learning. For discrete distributions, the problem of computing this distance can be expressed in terms of finding a minimum-cost perfect matching on a complete bipartite graph given by two multisets of points A, B ⊂ ℝ2, with |A| = |B| = n, where the ground distance between any two points is the squared Euclidean distance between them. Although there is a near-linear time relative ∊-approximation algorithm for the case where the ground distance is Euclidean (Sharathkumar and Agarwal, JACM 2020), all existing relative ∊-approximation algorithms for the RMS distance take Ω(n3/2) time. This is primarily because, unlike Euclidean distance, squared Euclidean distance is not a metric. In this paper, for the RMS distance, we present a new ∊-approximation algorithm that runs in O(n^5/4 poly{log n, 1/∊}) time. Our algorithm is inspired by a recent approach for finding a minimum-cost perfect matching in bipartite planar graphs (Asathulla et al, TALG 2020). Their algorithm depends heavily on the existence of sublinear sized vertex separators as well as shortest path data structures that require planarity. Surprisingly, we are able to design a similar more »
- Award ID(s):
- 1909171
- Publication Date:
- NSF-PAR ID:
- 10232551
- Journal Name:
- Proceedings of the Annual ACMSIAM Symposium on Discrete Algorithms
- Page Range or eLocation-ID:
- 869 - 888
- ISSN:
- 1557-9468
- Sponsoring Org:
- National Science Foundation
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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 problemmore »
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We present an $m^{4/3}+o(1) \log W$ -time algorithm for solving the minimum cost flow problem in graphs with unit capacity, where W is the maximum absolute value of any edge weight. For sparse graphs, this improves over the best known running time for this problem and, by well-known reductions, also implies improved running times for the shortest path problem with negative weights, minimum cost bipartite $b$-matching when $\|b\|_1 = O(m)$, and recovers the running time of the currently fastest algorithm for maximum flow in graphs with unit capacities (Liu-Sidford, 2020). Our algorithm relies on developing an interior point method–based framework which acts on the space of circulations in the underlying graph. From the combinatorial point of view, this framework can be viewed as iteratively improving the cost of a suboptimal solution by pushing flow around circulations. These circulations are derived by computing a regularized version of the standard Newton step, which is partially inspired by previous work on the unit-capacity maximum flow problem (Liu-Sidford, 2019), and subsequently refined based on the very re- cent progress on this problem (Liu-Sidford, 2020). The resulting step problem can then be computed efficiently using the recent work on $l_p$-norm minimizing flows (Kyng-Peng-Sachdeva- Wang, 2019).more »
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We present an m^{4/3+o(1)} log W -time algorithm for solving the minimum cost flow problem in graphs with unit capacity, where W is the maximum absolute value of any edge weight. For sparse graphs, this improves over the best known running time for this problem and, by well-known reductions, also implies improved running times for the shortest path problem with negative weights, minimum cost bipartite b-matching when |b|_1 = O(m), and recovers the running time of the currently fastest algorithm for maximum flow in graphs with unit capacities (Liu-Sidford, 2020). Our algorithm relies on developing an interior point method–based framework which acts on the space of circulations in the underlying graph. From the combinatorial point of view, this framework can be viewed as iteratively improving the cost of a suboptimal solution by pushing flow around circulations. These circulations are derived by computing a regularized version of the standard Newton step, which is partially inspired by previous work on the unit-capacity maximum flow problem (Liu-Sidford, 2019), and subsequently refined based on the very recent progress on this problem (Liu-Sidford, 2020). The resulting step problem can then be computed efficiently using the recent work on l_p-norm minimizing flows (Kyng-Peng-Sachdeva-Wang, 2019). We obtainmore »
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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 logmore »
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- Publisher's Version of Record
- https://doi.org/10.1137/1.9781611976465.55
