Strongly polynomial bounds for multiobjective and parametric global minimum cuts in graphs and hypergraphs

Abstract

We consider multiobjective and parametric versions of the global minimum cut problem in undirected graphs and bounded-rank hypergraphs with multiple edge cost functions. For a fixed number of edge cost functions, we show that the total number of supported non-dominated (SND) cuts is bounded by a polynomial in the numbers of nodes and edges, i.e., is strongly polynomial. This bound also applies to the combinatorial facet complexity of the problem, i.e., the maximum number of facets (linear pieces) of the parametric curve for the parametrized (linear combination) objective, over the set of all parameter vectors such that the parametrized edge costs are nonnegative and the parametrized cut costs are positive. We sharpen this bound in the case of two objectives (the bicriteria problem), for which we also derive a strongly polynomial upper bound on the total number of non-dominated (Pareto optimal) cuts. In particular, the bicriteria global minimum cut problem in an n-node graph admits $O(n^3 \log n)$ SND cuts and $O(n^5 \log n)$ Pareto optimal cuts. These results significantly improve on earlier graph cut results by Mulmuley (SIAM J Comput 28(4):1460–1509, 1999) and Armon and Zwick (Algorithmica 46(1):15–26, 2006). They also imply that the parametric curve and all SND cuts, and, for the bicriteria problems, all Pareto optimal cuts, can be computed in strongly polynomial time when the number of objectives is fixed.

Appendix

In this Appendix we present a proof of Theorem 1 for arbitrary $\alpha \geqslant 1$. We adapt and extend Kogan and Krauthgamer’s [21] approach and some of their notations.

We start with a simple upper bound on the minimum cut cost in a rank-$\rho$ hypergraph. For every $i=2,\dots ,\rho$ let $W_i = \sum \{c(e) : e\in E\text { and }|e|=i\}$ denote the total cost of all size-i edges, and $c(E) = \sum _{i=2}^{\rho } W_i = \sum _{e\in E} c(e)$ the total weight of all edges.

Lemma 13

([21]) The minimum cost of a cut in a rank-$\rho$ hypergraph $G = (V,E)$ with positive edge costs c, is at most $\frac{1}{|V|} \sum _{i=2}^{\rho } i\,W_i \leqslant \frac{\rho }{|V|}\, c(E)$.

Proof

An edge $e\in E$ crosses a singleton cut $C = (\{v\},V{\setminus }\{v\})$ (i.e., $e\in \delta (\{v\})$) if and only if $v\in e$. Thus an edge e crosses exactly |e| singleton cuts. If we choose a node $\mathbf{v}$ uniformly at random in V and consider the resulting (random) singleton cut $\{\mathbf{v}\}$, then

$$\begin{aligned} \min _{X\in \mathcal {C}} c(\delta (X))\le & {} \min _{v\in V} c(\delta (\{v\})) \ \le \ {\mathbb E}\left[ c(\{\mathbf{v}\})\right] \ =\ \sum _{v\in V}\frac{1}{|V|}\sum _{e \,:\, v\in e\in E} c(e)\\= & {} \frac{1}{|V|} \sum _{e\in E} |e|\, c(e) \ =\ \frac{1}{|V|}\sum _{i=2}^{\rho } i\,W_i \ \leqslant \ \frac{\rho }{|V|} \sum _{i=2}^{\rho } W_i \ =\ \frac{\rho }{|V|}\, c(E) \end{aligned}$$

and the Lemma follows. $\square$

Kogan and Krauthgamer use a probabilistic argument which extends to hypergraphs by an approach introduced by Karger [16, 19] for graph cuts. The first step is to define a generalization to hypergraphs of Karger’s randomized edge contraction algorithm. Similar generalizations for finding and counting minimum hypergraph cuts were outlined by Chekuri and Korula [5] and by Queyranne and Guiñez [38]. The algorithm below, adapted from [21], is similar to these, except that (as in [19]), it stops early so as to ensure a sufficiently high probability of generating any fixed $\alpha$-approximate cut.

Theorem 1 will be an easy consequence of the following result, which is adapted from and extends (with minor changes) Theorem 3.4 in [21]. Since we allow $2\alpha$ to be noninteger, we use, as in Karger and Stein [19, proof of Corollary 8.3], generalized binomial coefficients

$$\begin{aligned} \left( {\begin{array}{c}x\\ y\end{array}}\right) = \frac{\varGamma (x+1)}{\varGamma (y+1)\varGamma (x-y+1)} \end{aligned}$$

where, for simplicity, we assume $x > -1$ and $-1 < y < x+1$, and where the Gamma function is the Euler integral $\varGamma (x) = \int _0^{+\infty } t^{x-1} e^{-t} dt$. Since $\varGamma (t+1) = t\,\varGamma (t)$ for all $t>0$, these (generalized) binomial coefficients satisfy the recurrence

$$\begin{aligned} \left( {\begin{array}{c}x\\ y\end{array}}\right) = \frac{x}{x-y}\left( {\begin{array}{c}x-1\\ y\end{array}}\right) = \left( 1+\frac{y}{x-y}\right) \left( {\begin{array}{c}x-1\\ y\end{array}}\right) \end{aligned}$$

(11)

when $x > 0$ and $-1 < y < x$.

Theorem 14

For every real $\alpha \geqslant 1$ and every integer $\rho \geqslant 2$ there exists a constant $K(\alpha ,\rho ) > 0$ such that, for every rank-$\rho$ hypergraph $G = (V,E)$ with $|V| > \alpha r$ vertices, nonnegative edge costs c and positive minimum cut cost, and for every particular $\alpha$-approximate cut X in G, the probability that Algorithm 1 outputs X or $V{\setminus } X$ is at least $K(\alpha ,\rho ) \left( {\begin{array}{c}|V|-\alpha (\rho -2)\\ 2\alpha \end{array}}\right) ^{-1}$.

We prove Theorem 14 below, after first establishing another lemma. In the rest of this Appendix we assume that $\alpha \geqslant 1$ is a fixed real number, and $\rho \geqslant 2$ a fixed integer.

For every integer $t\geqslant 2$ let ${\mathcal G}_t$ denote the set of all edge-weighted rank-$\rho$ hypergraphs (G, c) with t vertices, nonnegative edge costs and positive minimum cut cost. Let ${\mathcal G} = \bigcup _{t\geqslant 2} {\mathcal G}_t$ denote the set of all such edge-weighted hypergraphs with any number of vertices. For $(G,c)\in {\mathcal G}_t$ and cut X in G let $p_t(X \,|\, G,c)$ denote the probability that Algorithm 1, when applied to (G, c), outputs cut X or $V{\setminus } X$. Let

$$\begin{aligned} p_t = \inf \left\{ p_t(X \,|\, G,c) \,:\, (G,c)\in {\mathcal G}_t \text { and }X\text { is an } \alpha \text {-approximate cut in }(G,c)\right\} \end{aligned}$$

Lemma 15

If $t\geqslant \left\lfloor \alpha \rho \right\rfloor +1$ and we have a positive lower bound $\widetilde{p}_u \leqslant p_u$ for every $u = t-\rho +1,\dots ,t-1$, then $p_t \geqslant (t-\alpha \rho )\min _{i=2,\dots ,\rho } \widetilde{p}_{t-i+1}/(t - \alpha (\rho -i))$.

Proof

Assume that, as stated, $t\geqslant \left\lfloor \alpha \rho \right\rfloor +1$ and $0 < \widetilde{p}_u \leqslant p_u$ for every $u = t-\rho +1,\dots ,t-1$. For any $(G,c)\in {\mathcal G}_t$ let $\varepsilon$ denote the (random) edge selected for contraction in step 3 of Algorithm 1. Let (G, c) / e denote the node-weighted hypergraph resulting from the contraction of an edge $e\in E$. Thus $(G,c)/e \in {\mathcal G}_{t-|e|+1}$. Let X be any $\alpha$-approximate cut in (G, c) and let $A(X,\, (G,c))$ represent the event that Algorithm 1, when applied to (G, c), outputs cut X or $V{\setminus } X$. Cut X will survive the current contraction if $\varepsilon \not \in \delta (X)$, otherwise it certainly cannot be output by Algorithm 1. Letting X / e denote the node subset X after contraction of edge $e\not \in \delta (X)$, conditioning over the size |e| of the contracted edge, and using the assumed lower bounds, we have

$$\begin{aligned} {{\mathrm{Prob}}}\left\{ A(X ,\, (G,c)) \right\}= & {} \sum _{e\in E{\setminus }\delta (X)} {{\mathrm{Prob}}}\left\{ A(X/e ,\, (G,c))/e\right\} {{\mathrm{Prob}}}\{\varepsilon = e\}\\= & {} \sum _{i=2}^\rho \ \sum _{e\in E{\setminus }\delta (X)\,:\,|e|=i} {{\mathrm{Prob}}}\left\{ A(X/e ,\, (G,c))/e\right\} {{\mathrm{Prob}}}\{\varepsilon = e\}\\\geqslant & {} \sum _{i=2}^\rho \widetilde{p}_{t-i+1}{{\mathrm{Prob}}}\{\varepsilon \not \in \delta (X)\text { and }|\varepsilon |=i\} \!=\! \sum _{i=2}^\rho \widetilde{p}_{t-i+1}(x_i \!-\! y_i). \end{aligned}$$

where $x_i = {{\mathrm{Prob}}}\{|\varepsilon |=i\}$ and $y_i = {{\mathrm{Prob}}}\{\varepsilon \in \delta (X)\text { and }|\varepsilon |=i\}$. Using the notations in Lemma 13 above, we have $x_i = W_i / c(E)$. Let $\widehat{w}(G,c) = \min _{X\in \mathcal {C}} c(\delta (X))$ denote the minimum cost of a cut in (G, c). By Lemma 13 we have (as in [21]):

$$\begin{aligned} \sum _{i=2}^\rho y_i= & {} {{\mathrm{Prob}}}\{\varepsilon \in \delta (X)\} = \frac{c(\delta (X))}{c(E)}\nonumber \\\leqslant & {} \frac{\alpha \,\widehat{w}(G,c)}{c(E)} \leqslant \frac{\alpha }{t} \sum _{i=2}^{\rho } \frac{i\,W_i}{c(E)} = \frac{\alpha }{t} \sum _{i=2}^{\rho } i\,x_i \end{aligned}$$

(12)

Let $\eta = t\,\min _{i=2,\dots ,\rho } \widetilde{p}_{t-i+1}/(t - \alpha (\rho -i))$, so $\eta > 0$ and for every $i=2,\dots ,\rho$

$$\begin{aligned} \widetilde{p}_{t-i+1} \geqslant \frac{t-\alpha (\rho -i)}{t}\eta \end{aligned}$$

Therefore,⁵

$$\begin{aligned} {{\mathrm{Prob}}}\left\{ A(X ,\, (G,c)) \right\}\geqslant & {} \sum _{i=2}^\rho \frac{t-\alpha (\rho -i)}{t}\eta (x_i-y_i) \end{aligned}$$

(13)

$$\begin{aligned}= & {} \frac{t-\alpha \rho }{t}\eta \sum _{i=2}^\rho (x_i-y_i) + \eta \sum _{i=2}^\rho \frac{\alpha i}{t}(x_i-y_i)\nonumber \\\geqslant & {} \frac{t-\alpha \rho }{t}\eta \sum _{i=2}^\rho x_i + \eta \sum _{i=2}^\rho \left( \frac{\alpha i}{t}x_i-y_i\right) \end{aligned}$$

(14)

$$\begin{aligned}\geqslant & {} \frac{t-\alpha \rho }{t}\eta = (t - \alpha \rho )\,\min _{i=2,\dots ,\rho } \frac{\widetilde{p}_{t-i+1}}{t - \alpha (\rho -i)} \end{aligned}$$

(15)

where in (14) we use $y\geqslant 0$ and, since $i \leqslant r, \alpha i \leqslant \alpha r \leqslant t$; and in (15) we use $\sum _{i=2}^r x_i = 1$, (12), and then the definition of $\eta$.

Since (13)–(15) holds for every $(G,c)\in {\mathcal G}_t$ and every $\alpha$-approximate cut X in (G, c), the proof of Lemma 15 is complete. $\square$

Proof of Theorem 14

For $u = |V| = \left\lfloor \alpha \rho \right\rfloor - \rho + 2,\dots ,\left\lfloor \alpha \rho \right\rfloor$, Algorithm 1 directly chooses in step 6, uniformly at random, one of the $2^u-2$ cuts in $\mathcal {C}$, so the probability it outputs cut X or $V{\setminus } X$ is exactly $\widetilde{p}_u = (2^{u-1}-1)^{-1}$.

For $t = \left\lfloor \alpha \rho \right\rfloor + 1,\dots ,\left\lfloor \alpha \rho \right\rfloor + \rho -1$ recursively define

$$\begin{aligned} \widetilde{p}_t = (t-\alpha \rho )\min _{i=2,\dots ,\rho } \widetilde{p}_{t-i+1}/(t - \alpha (\rho -i)). \end{aligned}$$

By Lemma 15 we have $p_t \geqslant \widetilde{p}_t$ for all such t. Define

$$\begin{aligned} K(\alpha ,\rho ) = \min _{t = \left\lfloor \alpha \rho \right\rfloor + 1,\dots ,\left\lfloor \alpha \rho \right\rfloor + \rho -1} \widetilde{p}_t \left( {\begin{array}{c}t-\alpha (\rho -2)\\ 2\alpha \end{array}}\right) \end{aligned}$$

and $\widehat{p}_t = K(\alpha ,\rho ) \left( {\begin{array}{c}t-\alpha (\rho -2)\\ 2\alpha \end{array}}\right) ^{-1}$, so $p_t \geqslant \widehat{p}_t > 0$ for all such t.

For $t\geqslant \left\lfloor \alpha \rho \right\rfloor + \rho$, applying Lemma 15 using $\widehat{p}_u$ in lieu of $\widetilde{p}_u$ we get

$$\begin{aligned} p_t \geqslant (t-\alpha \rho )\min _{i=2,\dots ,\rho } \frac{K(\alpha ,\rho ) \left( {\begin{array}{c}t-i+1-\alpha (\rho - 2)\\ 2\alpha \end{array}}\right) ^{-1}}{t - \alpha (\rho -i)} \end{aligned}$$

(16)

By induction on $i=2,\dots ,\rho$, equation (11) implies (as in Claim 3.6 in [21]):

$$\begin{aligned} \frac{\left( {\begin{array}{c}t-\alpha (\rho - 2)\\ 2\alpha \end{array}}\right) }{ \left( {\begin{array}{c}t-i+1-\alpha (\rho - 2)\\ 2\alpha \end{array}}\right) }= & {} \prod _{j=2}^{i} \left( 1 +\frac{2\alpha }{t-j+2-\alpha \rho }\right) \\\geqslant & {} \left( 1 +\frac{2\alpha }{t-\alpha \rho }\right) ^{i-1} \geqslant \ 1 +(i-1)\frac{2\alpha }{t-\alpha \rho }\\\geqslant & {} \frac{t - \alpha (\rho -i)}{t-\alpha \rho }\;. \end{aligned}$$

Then (16) implies

$$\begin{aligned} p_t \geqslant K(\alpha ,\rho ) \left( {\begin{array}{c}t-\alpha (\rho - 2)\\ 2\alpha \end{array}}\right) ^{-1}. \end{aligned}$$

completing the proof of Theorem 14. $\square$

Proof of Theorem 1

Theorem 14 implies that the number of $\alpha$-approximate cuts in any $(G,c) \in \mathcal {G}$, where $G=(V,E)$ is a rank-$\rho$ hypergraph, is at most $K(\alpha ,\rho )^{-1} \left( {\begin{array}{c}|V|-\alpha (\rho - 2)\\ 2\alpha \end{array}}\right)$, which is $O\left( |V|^{2\alpha }\right)$ when $\alpha \geqslant 1$ is a fixed real number and $\rho \geqslant 2$ is fixed. This proves Theorem 1. $\square$