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1. A Weighted Approach to the Maximum Cardinality Bipartite Matching Problem with Applications in Geometric Settings

   Lahn, Nathaniel; Raghvendra, Sharath (June 2019, Leibniz international proceedings in informatics)

   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. 

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2. Circulation Control for Faster Minimum Cost Flow in Unit-Capacity Graphs

   Axiotis, Kyriakos; Madry, Aleksander; Vladu, Adrian (November 2020, IEEE Symposium on Foundations of Computer Science)

   null (Ed.)

   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 obtain our faster algorithm by combining this new step primitive with a customized preconditioning method, which aims to ensure that the graph on which these circulations are computed has sufficiently large conductance. 

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3. Circulation Control for Faster Minimum Cost Flow in Unit-Capacity Graphs

   Axiotis, Kyriakos; Madry, Aleksander; Vladu, Adrian (November 2020, IEEE Symposium on Foundations of Computer Science)

   null (Ed.)

   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). We obtain our faster algorithm by combining this new step primitive with a customized preconditioning method, which aims to ensure that the graph on which these circulations are computed has sufficiently large conductance. 

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4. Near-Optimal Size Linear Sketches for Hypergraph Cut Sparsifiers

   Khanna, Sanjeev; Putterman, Aaron; Sudan, Madhu (October 2024, IEEE Press)

   A $$(1\pm\epsilon)$$ -sparsifier of a hypergraph $G(V, E)$ is a (weighted) subgraph that preserves the value of every cut to within a $$(1\pm\epsilon)$$ -factor. It is known that every hypergraph with $$n$$ vertices admits a $$(1 \pm \epsilon)$$ -sparsifier with $$\tilde{O}(n/{\epsilon}^{2})$$ hyperedges. In this work, we explore the task of building such a sparsifier by using only linear measurements (a linear sketch) over the hyperedges of $$G$$, and provide nearly-matching upper and lower bounds for this task. Specifically, we show that there is a randomized linear sketch of size $$\tilde{O}(nr\log(m)/\epsilon^{2})$$ bits which with high probability contains sufficient information to recover a $$(1\pm\epsilon)$$ cut-sparsifier with $$\tilde{O}(n/\epsilon^{2})$$ hyperedges for any hypergraph with at most $$m$$ edges each of which has arity bounded by $$r$$. This immediately gives a dynamic streaming algorithm for hypergraph cut sparsification with an identical space complexity, improving on the previous best known bound of $$\tilde{O}(nr^{2}\log^{4}({m})/\epsilon^{2})$$ bits of space (Guha, McGregor, and Tench, PODS 2015). We complement our algorithmic result above with a nearly-matching lower bound. We show that for every $$\epsilon\in(0,1)$$, one needs $$\Omega(nr\log(m/n)/\log(n))$$ bits to construct a $$(1\pm\epsilon)$$ -sparsifier via linear sketching, thus showing that our linear sketch achieves an optimal dependence on both $$r$$ and $$\log(m)$$. The starting point for our improved algorithm is importance sampling of hyperedges based on the new notion of $$k$$ -cut strength introduced in the recent work of Quanrud (SODA 2024). The natural algorithm based on this concept leads to $$\log m$$ levels of sampling where errors can potentially accumulate, and this accounts for the polylog $(m)$ losses in the sketch size of the natural algorithm. We develop a more intricate analysis of the accumulation in error to show most levels do not contribute to the error and actual loss is only polylog $(n)$. Combining with careful preprocessing (and analysis) this enables us to get rid of all extraneous $$\log m$$ factors in the sketch size, but the quadratic dependence on $$r$$ remains. This dependence originates from use of correlated $$\ell_{0}$$ -samplers to recover a large number of low-strength edges in a hypergraph simultaneously by looking at neighborhoods of individual vertices. In graphs, this leads to discovery of $$\Omega(n)$$ edges in a single shot, whereas in hypergraphs, this may potentially only reveal $$O$$($$n$$/$$r$$) new edges, thus requiring $$\Omega(r)$$ rounds of recovery. To remedy this we introduce a new technique of random fingerprinting of hyperedges which effectively eliminates the correlations created by large arity hyperedges, and leads to a scheme for recovering hyperedges of low strength with an optimal dependence on $$r$$. Putting all these ingredients together yields our linear sketching algorithm. Our lower bound is established by a reduction from the universal relation problem in the one-way communication setting. 

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5. Near-Optimal Size Linear Sketches for Hypergraph Cut Sparsifiers

   Khanna, Sanjeev; Putterman, Aaron; Sudan, Madhu (October 2024, IEEE Press)

   A $$(1\pm\epsilon)$$ -sparsifier of a hypergraph $G(V, E)$ is a (weighted) subgraph that preserves the value of every cut to within a $$(1\pm\epsilon)$$ -factor. It is known that every hypergraph with $$n$$ vertices admits a $$(1 \pm \epsilon)$$ -sparsifier with $$\tilde{O}(n/{\epsilon}^{2})$$ hyperedges. In this work, we explore the task of building such a sparsifier by using only linear measurements (a linear sketch) over the hyperedges of $$G$$, and provide nearly-matching upper and lower bounds for this task. Specifically, we show that there is a randomized linear sketch of size $$\tilde{O}(nr\log(m)/\epsilon^{2})$$ bits which with high probability contains sufficient information to recover a $$(1\pm\epsilon)$$ cut-sparsifier with $$\tilde{O}(n/\epsilon^{2})$$ hyperedges for any hypergraph with at most $$m$$ edges each of which has arity bounded by $$r$$. This immediately gives a dynamic streaming algorithm for hypergraph cut sparsification with an identical space complexity, improving on the previous best known bound of $$\tilde{O}(nr^{2}\log^{4}({m})/\epsilon^{2})$$ bits of space (Guha, McGregor, and Tench, PODS 2015). We complement our algorithmic result above with a nearly-matching lower bound. We show that for every $$\epsilon\in(0,1)$$, one needs $$\Omega(nr\log(m/n)/\log(n))$$ bits to construct a $$(1\pm\epsilon)$$ -sparsifier via linear sketching, thus showing that our linear sketch achieves an optimal dependence on both $$r$$ and $$\log(m)$$. The starting point for our improved algorithm is importance sampling of hyperedges based on the new notion of $$k$$ -cut strength introduced in the recent work of Quanrud (SODA 2024). The natural algorithm based on this concept leads to $$\log m$$ levels of sampling where errors can potentially accumulate, and this accounts for the polylog $(m)$ losses in the sketch size of the natural algorithm. We develop a more intricate analysis of the accumulation in error to show most levels do not contribute to the error and actual loss is only polylog $(n)$. Combining with careful preprocessing (and analysis) this enables us to get rid of all extraneous $$\log m$$ factors in the sketch size, but the quadratic dependence on $$r$$ remains. This dependence originates from use of correlated $$\ell_{0}$$ -samplers to recover a large number of low-strength edges in a hypergraph simultaneously by looking at neighborhoods of individual vertices. In graphs, this leads to discovery of $$\Omega(n)$$ edges in a single shot, whereas in hypergraphs, this may potentially only reveal $$O$$($$n$$/$$r$$) new edges, thus requiring $$\Omega(r)$$ rounds of recovery. To remedy this we introduce a new technique of random fingerprinting of hyperedges which effectively eliminates the correlations created by large arity hyperedges, and leads to a scheme for recovering hyperedges of low strength with an optimal dependence on $$r$$. Putting all these ingredients together yields our linear sketching algorithm. Our lower bound is established by a reduction from the universal relation problem in the one-way communication setting. 

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