We prove that a polynomial fraction of the set of $k$-component forests in the $m \times n$ grid graph have equal numbers of vertices in each component, for any constant $k$. This resolves a conjecture of Charikar, Liu, Liu, and Vuong, and establishes the first provably polynomial-time algorithm for (exactly or approximately) sampling balanced grid graph partitions according to the spanning tree distribution, which weights each $k$-partition according to the product, across its $k$ pieces, of the number of spanning trees of each piece. Our result follows from a careful analysis of the probability a uniformly random spanning tree of the grid can be cut into balanced pieces.
Beyond grids, we show that for a broad family of lattice-like graphs, we achieve balance up to any multiplicative $(1 \pm \varepsilon)$ constant with constant probability. More generally, we show that, with constant probability, components derived from uniform spanning trees can approximate any given partition of a planar region specified by Jordan curves. This implies polynomial-time algorithms for sampling approximately balanced tree-weighted partitions for lattice-like graphs.
Our results have applications to understanding political districtings, where there is an underlying graph of indivisible geographic units that must be partitioned into $k$ population-balanced connected subgraphs. In this setting, tree-weighted partitions have interesting geometric properties, and this has stimulated significant effort to develop methods to sample them.
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Computing Balanced Convex Partitions of Lines
Dujmovic and Langerman (2013) proved a ham-sandwich cut theorem for an arrangement of lines in the plane. Recently, Xue and Soberon (2019) generalized it to balanced convex partitions of lines in the plane. In this paper, we study the computational problems of computing a ham-sandwich cut balanced convex partitions for an arrangement of lines in the plane. We show that both problems can be solved in polynomial time.
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- Award ID(s):
- 1718994
- NSF-PAR ID:
- 10196682
- Date Published:
- Journal Name:
- Latin American Theoretical Informatics
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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null (Ed.)We consider the classical Minimum Balanced Cut problem: given a graph $G$, compute a partition of its vertices into two subsets of roughly equal volume, while minimizing the number of edges connecting the subsets. We present the first {\em deterministic, almost-linear time} approximation algorithm for this problem. Specifically, our algorithm, given an $n$-vertex $m$-edge graph $G$ and any parameter $1\leq r\leq O(\log n)$, computes a $(\log m)^{r^2}$-approximation for Minimum Balanced Cut on $G$, in time $O\left ( m^{1+O(1/r)+o(1)}\cdot (\log m)^{O(r^2)}\right )$. In particular, we obtain a $(\log m)^{1/\epsilon}$-approximation in time $m^{1+O(1/\sqrt{\epsilon})}$ for any constant $\epsilon$, and a $(\log m)^{f(m)}$-approximation in time $m^{1+o(1)}$, for any slowly growing function $m$. We obtain deterministic algorithms with similar guarantees for the Sparsest Cut and the Lowest-Conductance Cut problems. Our algorithm for the Minimum Balanced Cut problem in fact provides a stronger guarantee: it either returns a balanced cut whose value is close to a given target value, or it certifies that such a cut does not exist by exhibiting a large subgraph of $G$ that has high conductance. We use this algorithm to obtain deterministic algorithms for dynamic connectivity and minimum spanning forest, whose worst-case update time on an $n$-vertex graph is $n^{o(1)}$, thus resolving a major open problem in the area of dynamic graph algorithms. Our work also implies deterministic algorithms for a host of additional problems, whose time complexities match, up to subpolynomial in $n$ factors, those of known randomized algorithms. The implications include almost-linear time deterministic algorithms for solving Laplacian systems and for approximating maximum flows in undirected graphs.more » « less
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In this paper, we consider two fundamental cut approximation problems on large graphs. We prove new lower bounds for both problems that are optimal up to logarithmic factors. The first problem is approximating cuts in balanced directed graphs, where the goal is to build a data structure to provide a $(1 \pm \epsilon)$-estimation of the cut values of a graph on $n$ vertices. For this problem, there are tight bounds for undirected graphs, but for directed graphs, such a data structure requires $\Omega(n^2)$ bits even for constant $\epsilon$. To cope with this, recent works consider $\beta$-balanced graphs, meaning that for every directed cut, the total weight of edges in one direction is at most $\beta$ times the total weight in the other direction. We consider the for-each model, where the goal is to approximate a fixed cut with high probability, and the for-all model, where the data structure must simultaneously preserve all cuts. We improve the previous $\Omega(n \sqrt{\beta/\epsilon})$ lower bound in the for-each model to $\tilde\Omega(n \sqrt{\beta}/\epsilon)$ and we improve the previous $\Omega(n \beta/\epsilon)$ lower bound in the for-all model to $\Omega(n \beta/\epsilon^2)$. This resolves the main open questions of (Cen et al., ICALP, 2021). The second problem is approximating the global minimum cut in the local query model where we can only access the graph through degree, edge, and adjacency queries. We prove an $\Omega(\min\{m, \frac{m}{\epsilon^2 k}\})$ lower bound for this problem, which improves the previous $\Omega(\frac{m}{k})$ lower bound, where $m$ is the number of edges of the graph, $k$ is the minimum cut size, and we seek a $(1+\epsilon)$-approximation. In addition, we observe that existing upper bounds with minor modifications match our lower bound up to logarithmic factors.more » « less
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