This dissertation focuses on computing high-quality solutions for the NP-hard balanced hypergraph partitioning problem: Given a hypergraph and an integer $k$, partition its vertex set into $k$ disjoint blocks of bounded size, while minimizing an objective function over the hyperedges. Here, we consider the two most commonly used objectives: the cut-net metric and the connectivity metric.

Since the problem is computationally intractable, heuristics are used in practice - the most prominent being the three-phase multi-level paradigm: During coarsening, the hypergraph is successively contracted to obtain a hierarchy of smaller instances. After applying an initial partitioning algorithm to the smallest hypergraph, contraction is undone and, at each level, refinement algorithms try to improve the current solution.

With this work, we give a brief overview of the field and present several algorithmic improvements to the multi-level paradigm. Instead of using a logarithmic number of levels like traditional algorithms, we present two coarsening algorithms that create a hierarchy of (nearly) $n$ levels, where $n$ is the number of vertices. This makes consecutive levels as similar as possible and provides many opportunities for refinement algorithms to improve the partition. This approach is made feasible in practice by tailoring all algorithms and data structures to the $n$-level paradigm, and developing lazy-evaluation techniques, caching mechanisms and early stopping criteria to speed up the partitioning process. Furthermore, we propose a sparsification algorithm based on locality-sensitive hashing that improves the running time for hypergraphs with large hyperedges, and show that incorporating global information about the community structure into the coarsening process improves quality. Moreover, we present a portfolio-based initial partitioning approach, and propose three refinement algorithms. Two are based on the Fiduccia-Mattheyses (FM) heuristic, but perform a highly localized search at each level. While one is designed for two-way partitioning, the other is the first FM-style algorithm that can be efficiently employed in the multi-level setting to directly improve $k$-way partitions. The third algorithm uses max-flow computations on pairs of blocks to refine $k$-way partitions. Finally, we present the first memetic multi-level hypergraph partitioning algorithm for an extensive exploration of the global solution space.

All contributions are made available through our open-source framework KaHyPar. In a comprehensive experimental study, we compare KaHyPar with hMETIS, PaToH, Mondriaan, Zoltan-AlgD, and HYPE on a wide range of hypergraphs from several application areas. Our results indicate that KaHyPar, already without the memetic component, computes better solutions than all competing algorithms for both the cut-net and the connectivity metric, while being faster than Zoltan-AlgD and equally fast as hMETIS. Moreover, KaHyPar compares favorably with the current best graph partitioning system KaFFPa - both in terms of solution quality and running time.

High-Quality Hypergraph Partitioning

The minimum $s$-$t$ cut problem in graphs is one of the most fundamental problems in combinatorial optimization, and graph cuts underlie algorithms throughout discrete mathematics, theoretical computer science, operations research, and data science. While graphs are a standard model for pairwise relationships, hypergraphs provide the flexibility to model multi-way relationships, and are now a standard model for complex data and systems. However, when generalizing from graphs to hypergraphs, the notion of a "cut hyperedge" is less clear, as a hyperedge's nodes can be split in several ways. Here, we develop a framework for hypergraph cuts by considering the problem of separating two terminal nodes in a hypergraph in a way that minimizes a sum of penalties at split hyperedges. In our setup, different ways of splitting the same hyperedge have different penalties, and the penalty is encoded by what we call a splitting function. 
Our framework opens a rich space on the foundations of hypergraph cuts. We first identify a natural class of cardinality-based hyperedge splitting functions that depend only on the number of nodes on each side of the split. In this case, we show that the general hypergraph $s$-$t$ cut problem can be reduced to a tractable graph $s$-$t$ cut problem if and only if the splitting functions are submodular. We also identify a wide regime of non-submodular splitting functions for which the problem is NP-hard. We also analyze extensions to multiway cuts with at least three terminal nodes and identify a natural class of splitting functions for which the problem can be reduced in an approximation-preserving way to the node-weighted multiway cut problem in graphs, again subject to a submodularity property. Finally, we outline several open questions on general hypergraph cut problems.

Hypergraph Cuts with General Splitting Functions.

We study algorithmic and structural aspects of connectivity in hypergraphs. Given a hypergraph $H=(V,E)$ with $n = |V|$, $m = |E|$, and $p = \sum_{e \in E} |e|$ the fastest known algorithm to compu...

Minimum Cuts and Sparsification in Hypergraphs

Applications of Polynomial Systems

Karger used spanning tree packings [D R Karger, J ACM, 47 (2000), pp 46--76] to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the numbe

Lp relaxation and tree packing for minimum k-CUT

In the hypergraph k-cut problem, the input is a hypergraph, and the goal is to find a smallest subset of hyperedges whose removal ensures that the remaining hypergraph has at least k connected components. This problem is known to be at least as hard as the densest k-subgraph problem when k is part of the input (Chekuri-Li, 2015). We present a randomized polynomial time algorithm to solve the hypergraph k-cut problem for constant k. Our algorithm solves the more general hedge k-cut problem when the subgraph induced by every hedge has a constant number of connected components. In the hedge k-cut problem, the input is a hedgegraph specified by a vertex set and a disjoint set of hedges, where each hedge is a subset of edges defined over the vertices. The goal is to find a smallest subset of hedges whose removal ensures that the number of connected components in the remaining underlying (multi-)graph is at least k. Our algorithm is based on random contractions akin to Karger's min cut algorithm. Our main technical contribution is a distribution over the hedges (hyperedges) so that random contraction of hedges (hyperedges) chosen from the distribution succeeds in returning an optimum solution with large probability.

/pdf/hypergraph-k-cut-in-randomized-polynomial-time-4wquc3txhx.pdf

Hypergraph k-cut in randomized polynomial time

For a fixed integer $$k\ge 2$$
 , the hypergraph k-cut problem asks for a smallest subset of hyperedges whose removal leads to at least k connected components in the remaining hypergraph. While graph k-cut is solvable efficiently (Goldschmidt and Hochbaum in Math. Oper. Res. 19(1):24–37, 1994), the complexity of hypergraph k-cut has been open. In this work, we present a randomized polynomial time algorithm to solve the hypergraph k-cut problem. Our algorithmic technique extends to solve the more general hedge k-cut problem when the subgraph induced by every hedge has a constant number of connected components. Our algorithm is based on random contractions akin to Karger’s min cut algorithm. Our main technical contribution is a non-uniform distribution over the hedges (hyperedges) so that random contraction of hedges (hyperedges) chosen from the distribution succeeds in returning an optimum solution with large probability. In addition, we present an alternative contraction based randomized polynomial time approximation scheme for hedge k-cut in arbitrary hedgegraphs (i.e., hedgegraphs whose hedges could have a large number of connected components). Our algorithm and analysis also lead to bounds on the number of optimal solutions to the respective problems.

Hypergraph k -cut in randomized polynomial time

Algorithms for detecting dependencies and rigid subsystems for CAD

One of the Grand Challenges in Science is the construction of the Tree of Life, an evolutionary tree containing several million species, spanning all life on earth. However, the construction of the Tree of Life is enormously computationally challenging, as all the current most accurate methods are either heuristics for NP-hard optimization problems or Bayesian MCMC methods that sample from tree space. One of the most promising approaches for improving scalability and accuracy for phylogeny estimation uses divide-and-conquer: a set of species is divided into overlapping subsets, trees are constructed on the subsets, and then merged together using a "supertree method". Here, we present Exact-RFS-2, the first polynomial-time algorithm to find an optimal supertree of two trees, using the Robinson-Foulds Supertree (RFS) criterion (a major approach in supertree estimation that is related to maximum likelihood supertrees), and we prove that finding the RFS of three input trees is NP-hard. We also present GreedyRFS (a greedy heuristic that operates by repeatedly using Exact-RFS-2 on pairs of trees, until all the trees are merged into a single supertree). We evaluate Exact-RFS-2 and GreedyRFS, and show that they have better accuracy than the current leading heuristic for RFS. Exact-RFS-2 and GreedyRFS are available in open source form on Github at https://github.com/yuxilin51/GreedyRFS.

/pdf/advancing-divide-and-conquer-phylogeny-estimation-amgtqmqty6.pdf

Advancing Divide-and-Conquer Phylogeny Estimation

One of the Grand Challenges in Science is the construction of the Tree of Life, an evolutionary tree containing several million species, spanning all life on earth. However, the construction of the Tree of Life is enormously computationally challenging, as all the current most accurate methods are either heuristics for NP-hard optimization problems or Bayesian MCMC methods that sample from tree space. One of the most promising approaches for improving scalability and accuracy for phylogeny estimation uses divide-and-conquer: a set of species is divided into overlapping subsets, trees are constructed on the subsets, and then merged together using a "supertree method". Here, we present Exact-RFS-2, the first polynomial-time algorithm to find an optimal supertree of two trees, using the Robinson-Foulds Supertree (RFS) criterion (a major approach in supertree estimation that is related to maximum likelihood supertrees), and we prove that finding the RFS of three input trees is NP-hard. We also present GreedyRFS (a greedy heuristic that operates by repeatedly using Exact-RFS-2 on pairs of trees, until all the trees are merged into a single supertree). We evaluate Exact-RFS-2 and GreedyRFS, and show that they have better accuracy than the current leading heuristic for RFS.

https://drops.dagstuhl.de/opus/volltexte/2020/12804/pdf/LIPIcs-WABI-2020-15.pdf/

Xilin Yu

Papers

Hypergraph k-cut in randomized polynomial time

Hypergraph k -cut in randomized polynomial time

Algorithms for detecting dependencies and rigid subsystems for CAD

Advancing Divide-and-Conquer Phylogeny Estimation

Advancing divide-and-conquer phylogeny estimation using robinson-foulds supertrees