Network Flow-Based Refinement for Multilevel Hypergraph Partitioning

Driven by applications in VLSI design and scientific computing, HGP has evolved into a broad research area since the 1990s. We refer to [5, 8, 43, 50] for an extensive overview. Well-known multilevel HGP software packages with certain distinguishing characteristics include PaToH [12] (originating from scientific computing), hMetis [32, 33] (originating from VLSI design), KaHyPar [2, 28, 48] (general purpose, $n$-level), Mondriaan [54] (sparse matrix partitioning), MLPart [4] (circuit partitioning), Zoltan [16], Parkway [51] and SHP [31] (distributed), UMPa [52] (directed hypergraph model, multi-objective), and kPaToH (multiple constraints, fixed vertices) [7]. All of these tools either use variations of the Kernighan-Lin (KL) [34, 49] or the Fiduccia-Mattheyses (FM) heuristic [19, 46], or algorithms that greedily move vertices [33] or nets [32] to improve solution quality in the refinement phase.

While flow-based approaches have not yet been considered as refinement algorithms for multilevel HGP, several works deal with flow-based hypergraph min-cut computation. The problem of finding minimum $(\mathpzc{s},\mathpzc{t})$-cuts in hypergraphs was first considered by Lawler [36], who showed that it can be reduced to computing maximum flows in directed graphs. Hu and Moerder [29] present an augmenting path algorithm to compute a minimum-weight vertex separator on the star-expansion of the hypergraph. Their vertex-capacitated network can also be transformed into an edge-capacitated network using a transformation due to Lawler [37]. Yang and Wong [57] use repeated, incremental max-flow min-cut computations on the Lawler network [36] to find $\varepsilon$-balanced hypergraph bipartitions. Solution quality and running time of this algorithm are improved by Lillis and Cheng [39] by introducing advanced heuristics to select source and sink nodes. Furthermore, they present a preflow-based [22] min-cut algorithm that implicitly operates on the star-expanded hypergraph. Pistorius and Minoux [45] generalize the algorithm of Edmonds and Karp [18] to hypergraphs by labeling both vertices and nets. Liu and Wong [40] simplify Lawler’s hypergraph flow network [36] by explicitly distinguishing between graph edges and hyperedges with three or more pins. This approach significantly reduces the size of flow networks derived from VLSI hypergraphs, since most of the nets in a circuit are graph edges. Note that the above-mentioned approaches to model hypergraphs as flow networks for max-flow min-cut computations do not contradict the negative results of Ihler et al. [30], who show that, in general, there does not exist an edge-weighted graph $G=(V,E)$ that correctly represents the min-cut properties of the corresponding hypergraph $H=(V,E)$.

Flow-based refinement algorithms for graph partitioning include Improve [6] and MQI [35], which improve expansion or conductance of bipartitions. MQI also yields as small improvement when used as a post processing technique on hypergraph bipartitions initially computed by hMetis [35]. FlowCutter [24] uses an approach similar to Yang and Wong [57] to compute graph bisections that are Pareto-optimal in regard to cut size and balance. Sanders and Schulz [47] present a flow-based refinement framework for their direct $k$-way graph partitioner KaFFPa. The algorithm works on pairs of adjacent blocks and constructs flow problems such that each min-cut in the flow network is a feasible solution in regard to the original partitioning problem.

Since our algorithm is integrated into the KaHyPar framework, we briefly review its core components. While traditional multilevel HGP algorithms contract matchings or clusterings and therefore work with a coarsening hierarchy of $\mathcal{O}\!\left(\log n\right)$ levels, KaHyPar instantiates the multilevel paradigm in the extreme $n$-level version, removing only a single vertex between two levels. After coarsening, a portfolio of simple algorithms is used to create an initial partition of the coarsest hypergraph. During uncoarsening, strong localized local search heuristics based on the FM algorithm [19, 46] are used to refine the solution. Our work builds on KaHyPar-CA [28], which is a direct $k$-way partitioning algorithm for optimizing the $(\lambda-1)$-metric. It uses an improved coarsening scheme that incorporates global information about the community structure of the hypergraph into the coarsening process.

Given a hypergraph $H=(V,E,c,\omega)$ and two distinct nodes $\mathpzc{s}$ and $\mathpzc{t}$, an $(\mathpzc{s},\mathpzc{t})$-min-cut can be computed by finding a minimum-capacity cut in the following flow network $\mathcal{N}=(\mathcal{V},\mathcal{E})$:

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  $\mathcal{V}$ contains all vertices in $V$.

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  For each multi-pin net $e\in E$ with $|e|\geq 3$, add two bridging nodes $e^{\prime}$ and $e^{\prime\prime}$ to $\mathcal{V}$ and a bridging edge $(e^{\prime},e^{\prime\prime})$ with capacity $\mathpzc{c}(e^{\prime},e^{\prime\prime})=\omega(e)$ to $\mathcal{E}$. For each pin $p\in e$, add two edges $(p,e^{\prime})$ and $(e^{\prime\prime},p)$ with capacity $\infty$ to $\mathcal{E}$.

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  For each two-pin net $e=(u,v)\in E$, add two bridging edges $(u,v)$ and $(v,u)$ with capacity $\omega(e)$ to $\mathcal{E}$.

The flow network of Lawler [36] does not distinguish between two-pin and multi-pin nets. This increases the size of the network by two vertices and three edges per two-pin net. Figure 1 shows an example of the Lawler and Liu-Wong hypergraph flow networks as well as of our network described in the following paragraph.

We further decrease the size of the network by using the observation that the problem of finding an $(\mathpzc{s},\mathpzc{t})$-min-cut of $H$ can be reduced to finding a minimum-weight $(\mathpzc{s},\mathpzc{t})$-vertex-separator in the star-expansion, where the capacity of each star-node is the weight of the corresponding net and all other nodes (corresponding to vertices in $H$) have infinite capacity [29]. Since the separator has to be a subset of the star-nodes, it is possible to replace any infinite-capacity node by adding a clique between all adjacent star-nodes without affecting the separator. The key observation now is that an infinite-capacity node $v$ with degree $d(v)$ induces $2d(v)$ infinite-capacity edges in the Lawler network [36], while a clique between star-nodes induces $d(v)(d(v)-1)$ edges. For hypernodes with $d(v)\leq 3$, it therefore holds that $d(v)(d(v)-1)\leq 2d(v)$. Thus we can reduce the number of nodes and edges of the Liu-Wong network as follows. Before applying the transformation on the star-expansion of $H$, we remove all infinite-capacity nodes $v$ corresponding to hypernodes with $d(v)\leq 3$ that are not incident to any two-pin nets and add a clique between all star-nodes adjacent to $v$. In case $v$ was a source or sink node, we create a multi-source multi-sink problem by adding all adjacent star-nodes to the set of sources resp. sinks [20].

After computing an $(\mathpzc{s},\mathpzc{t})$-max-flow in the Lawler or Liu-Wong network, an $(\mathpzc{s},\mathpzc{t})$-min-cut of $H$ can be computed by a BFS in the residual graph starting from $\mathpzc{s}$. Let $S$ be the set of nodes corresponding to vertices of $H$ reached by the BFS. Then $(S,V\setminus S)$ is an $(\mathpzc{s},\mathpzc{t})$-min-cut. Since our network does not contain low degree hypernodes, we use the following lemma to compute an $(\mathpzc{s},\mathpzc{t})$-min-cut of $H$:

Thus $(A,V\setminus A)$ is an $(\mathpzc{s},\mathpzc{t})$-min-cut of $H$, where $A:=\{v\in e\leavevmode\nobreak\ |\leavevmode\nobreak\ \exists e\in E:\langle\mathpzc{s},\dots,e^{\prime\prime}\rangle\leavevmode\nobreak\ \text{in}\leavevmode\nobreak\ \mathcal{N}_{f}\}$. Furthermore this allows us to search for more balanced min-cuts using the Picard-Queyranne-DAG of $\mathcal{N}_{f}$ as described in Section 2.3. By the definition of closed sets it follows that if a bridging node $e^{\prime\prime}$ is contained in a closed set $C$, then all nodes $v\in\mathrm{\Gamma}(e^{\prime\prime})$ (which correspond to vertices of $H$) are also contained in $C$. Thus we can use the respective bridging nodes $e^{\prime\prime}$ as representatives of removed low degree hypernodes.

In KaFFPa, this is done by directly connecting internal border nodes $\overrightarrow{B}$ to $\mathpzc{s}$ and $\mathpzc{t}$. This approach can also be used for hypergraphs. In the hypergraph flow problem $\mathcal{F}_{G}$, the source $\mathpzc{s}$ is connected to all nodes $\mathcal{S}=\overrightarrow{B}\cap V_{i}$ and all nodes $\mathcal{T}=\overrightarrow{B}\cap V_{j}$ are connected to $\mathpzc{t}$ using directed edges with infinite capacity. While this ensures that $\mathcal{F}_{G}$ has the cut property, applying the graph-based model to hypergraphs unnecessarily restricts the search space. Since all internal border nodes $\overrightarrow{B}$ are connected to either $\mathpzc{s}$ or $\mathpzc{t}$, every min-cut $(S,V_{B}\setminus S)$ will have $\mathcal{S}\subseteq S$ and $\mathcal{T}\subseteq V_{B}\setminus S$. The KaFFPa model therefore prevents all min-cuts in which any non-cut border net (i.e., $e\in\overleftrightarrow{E_{B}}$ with $\lambda(e)=1$) becomes part of the cut-set. This restricts the space of possible solutions, since corridor $B$ was computed such that even a min-cut along either side of the border would result in a feasible cut in $H_{B}$. Thus, ideally, all vertices $v\in B$ should be able to change their block as result of an $(\mathpzc{s},\mathpzc{t})$-max-flow computation on $\mathcal{F}_{G}$ – not only vertices $v\in B\setminus\overrightarrow{B}$. This limitation becomes increasingly relevant for hypergraphs with large nets as well as for partitioning problems with small imbalance $\varepsilon$, since large nets are likely to be only partially contained in $H_{B}$ and tight balance constraints enforce small $B$-corridors. While the former is a problem only for HGP, the latter also applies to GP.

We propose a more general model that allows an $(\mathpzc{s},\mathpzc{t})$-max-flow computation to also cut through border nets by exploiting the structure of hypergraph flow networks. Instead of directly connecting $\mathpzc{s}$ and $\mathpzc{t}$ to internal border nodes $\overrightarrow{B}$ and thus preventing all min-cuts in which these nodes switch blocks, we conceptually extend $H_{B}$ to contain all external border nodes $\overleftarrow{B}$ and all border nets $\overleftrightarrow{E_{B}}$. The resulting hypergraph is $\overleftarrow{H_{B}}=(V_{B}\cup\overleftarrow{B},\{e\in E\leavevmode\nobreak\ |\leavevmode\nobreak\ e\cap V_{B}\neq\emptyset\})$. The key insight now is that by using the flow network of $\overleftarrow{H_{B}}$ and connecting $\mathpzc{s}$ resp. $\mathpzc{t}$ to the external border nodes $\overleftarrow{B}\cap V_{i}$ resp. $\overleftarrow{B}\cap V_{j}$, we get a flow problem that does not lock any node $v\in V_{B}$ in its block, since none of these nodes is directly connected to either $\mathpzc{s}$ or $\mathpzc{t}$. Due to the max-flow min-cut theorem [21], this flow problem furthermore has the cut property, since all border nets of $H_{B}$ are now internal nets and all external border nodes $\overleftarrow{B}$ are locked inside their block. However, it is not necessary to use $\overleftarrow{H_{B}}$ instead of $H_{B}$ to achieve this result. For all vertices $v\in\overleftarrow{B}$ the flow network of $\overleftarrow{H_{B}}$ contains paths $\langle\mathpzc{s},v,e^{\prime}\rangle$ and $\langle e^{\prime\prime},v,\mathpzc{t}\rangle$ that only involve infinite-capacity edges. Therefore we can remove all nodes $v\in\overleftarrow{B}$ by directly connecting $\mathpzc{s}$ and $\mathpzc{t}$ to the corresponding bridging nodes $e^{\prime},e^{\prime\prime}$ via infinite-capacity edges without affecting the maximal flow [20]. More precisely, in the hypergraph flow problem $\mathcal{F}_{H}$, we connect $\mathpzc{s}$ to all bridging nodes $e^{\prime}$ corresponding to border nets $e\in\overleftrightarrow{E_{B}}:e\subset\overleftarrow{B}\cap V_{i}$ and all bridging nodes $e^{\prime\prime}$ corresponding to border nets $e\in\overleftrightarrow{E_{B}}:e\subset\overleftarrow{B}\cap V_{j}$ to $\mathpzc{t}$ using directed, infinite-capacity edges.

Furthermore, we model border nets with $|e\cap\overrightarrow{B}|=1$ more efficiently. For such a net $e$, the flow problem contains paths of the form $\langle\mathpzc{s},e^{\prime},e^{\prime\prime},v\rangle$ or $\langle v,e^{\prime},e^{\prime\prime},\mathpzc{t}\rangle$ which can be replaced by paths of the form $\langle\mathpzc{s},e^{\prime},v\rangle$ or $\langle v,e^{\prime\prime},\mathpzc{t}\rangle$ with $\mathpzc{c}(e^{\prime},v)=\omega(e)$ resp. $\mathpzc{c}(v,e^{\prime\prime})=\omega(e)$. In both cases we can thus remove one bridging node and two infinite-capacity edges. A comparison of $\mathcal{F}_{G}$ and $\mathcal{F}_{H}$ is shown in Figure 2.

To analyze the effects of the different hypergraph flow networks we compute five bipartitions for each hypergraph of set B with KaHyPar-CA using different seeds. Statistics of the hypergraphs are shown in Table 2. The bipartitions are then used to generate hypergraph flow networks for a corridor of size $|B|=\numprint{25000}$ hypernodes around the cut. Figure 3 (top) summarizes the sizes of the respective flow networks in terms of number of nodes $|\mathcal{V}|$ and number of edges $|\mathcal{E}|$ for each instance class. The flow networks of primal and literal SAT instances are the largest in terms of both numbers of nodes and edges. High average vertex degree combined with low average net sizes leads to subhypergraphs $H_{B}$ containing many small nets, which then induce many nodes and (infinite-capacity) edges in $\mathcal{N}_{L}$. Dual instances with low average degree and large average net size on the other hand lead to smaller flow networks. For VLSI instances (DAC, ISPD) both average degree and average net sizes are low, while for SPM hypergraphs the opposite is the case. This explains why SPM flow networks have significantly more edges, despite the number of nodes being comparable in both classes.

As expected, the Lawler-Network $\mathcal{N}_{L}$ induces the biggest flow problems. Looking at the Liu-Wong network $\mathcal{N}_{W}$, we can see that distinguishing between graph edges and nets with $|e|\geq 3$ pins has an effect for all hypergraphs with many small nets (i.e., DAC, ISPD, Primal, Literal). While this technique alone does not improve dual SAT instances, we see that the combination of the Liu-Wong approach and our removal of low degree hypernodes in $\mathcal{N}_{\text{Our}}$ reduces the size of the networks for all instance classes except SPM. Both techniques only have a limited effect on these instances, since both hypernode degrees and net sizes are large on average. Since our flow problems are based on $B$-corridor induced subhypergraphs, $\mathcal{N}_{\text{Our}}^{1}$ additionally models single-pin border nets more efficiently as described in Section 3.2. This further reduces the network sizes significantly. As expected, the reduction in numbers of nodes and edges is most pronounced for hypergraphs with low average net sizes because these instances are likely to contain many single-pin border nets.

To further see how these reductions in network size translate to improved running times of max-flow algorithms, we use these networks to create flow problems using our flow model $\mathcal{F}_{H}$ and compute min-cuts using two highly tuned max-flow algorithms, namely the BK-algorithm³³3Available from: https://github.com/gerddie/maxflow [10] and the incremental breadth-first search (IBFS) algorithm⁴⁴4Available from: http://www.cs.tau.ac.il/~sagihed/ibfs/code.html [23]. These algorithms were chosen because they performed best in preliminary experiments [27]. We then compare the speedups of these algorithms when executed on $\mathcal{N}_{W}$, $\mathcal{N}_{\text{Our}}$, and $\mathcal{N}_{\text{Our}}^{1}$ to the execution on the Lawler network $\mathcal{N}_{L}$. As can be seen in Figure 3 (bottom) both algorithms benefit from improved network models and the speedups directly correlate with the reductions in network size. While $\mathcal{N}_{W}$ significantly reduces the running times for Primal and Literal instances, $\mathcal{N}_{\text{Our}}$ additionally leads to a speedup for Dual instances. By additionally considering single-pin border nets, $\mathcal{N}_{\text{Our}}^{1}$ results in an average speedup between $1.52$ and $2.21$ (except for SPM instances). Since IBFS outperformed the BK algorithm in [27], we use $\mathcal{N}_{\text{Our}}^{1}$ and IBFS in all following experiments.