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.
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Notes
A reader only interested in graph cuts may skip the rest of this paragraph and the whole of Sect. 2; and in the sequel replace all occurrences of “(bounded-rank) hypergraph(s)” with “graph(s)”, all \(O_{\rho ,\cdot }(\cdot )\) with \(O(\cdot )\), and ignore all mentions of rank \(\rho \geqslant 3\).
This is also called parametric complexity by some authors, such as Mulmuley [28] (see also [24, 27]) for the bicriterion case. We use a different terminology to avoid conflict with a different concept of parametric (or “parametrized”, or “parameterized”) complexity, e.g., [7], which deals with finding efficient algorithms for problems in which certain input or output “parameters” (or properties) are fixed. On the other hand, Fernández-Baca and Venkatachalam [10] use the term combinatorial complexity to refer to the total number of faces of all dimensions (here, 0 to \(k-1\)) of the graph of a parametric function (such as \(c^*\) here), whereas Schrijver [39] uses facet complexity to refer to the maximum input size of a rational linear inequality in a system that defines a polyhedron (such as \(\mathcal {D}\) here).
Their approach also seems to require that \(\alpha \rho \) be integer, but this requirement is not mentioned in [21].
This is also a slight (and parametrized) extension of the more familiar \(\widetilde{O}\) notation. Indeed, recall that \(f(n) = \widetilde{O}(g(n))\) if and only if \(f(n) = O(g(n)\, \log ^p g(n) )\) for some \(p\geqslant 1\). Then \(\widetilde{O}(g(n))\) is \(O_{\rho ,-\infty }(g(n))\), but the converse need not hold, e.g., when \(f(n) = g(n)\,h(n)\) with h(n) “super-polylog” [i.e., \(h(n) = \Omega (\log ^p n)\) for all fixed \(p \geqslant 1\)] and \(h(n) = O_{\rho ,-\infty }(1)\).
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Acknowledgments
We thank Volker Kaibel and Martin Skutella for helpful conversations around the Upper Bound Theorem, and anonymous referees for detailed and perceptive comments, particularly for pointing out reference [21]. The work of the third author was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada. The work of the last author was supported by a Discovery Grant and a Discovery Accelerator Supplement Grant from NSERC, and by the Center for Operations Research and Econometrics (CORE) of the Université Catholique de Louvain, Belgium.
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This paper derives from the conference paper [1], substantially extended, re-written, and corrected.
Appendix
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
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.
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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
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
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
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
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]):
Let \(\eta = t\,\min _{i=2,\dots ,\rho } \widetilde{p}_{t-i+1}/(t - \alpha (\rho -i))\), so \(\eta > 0\) and for every \(i=2,\dots ,\rho \)
Therefore,Footnote 5
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
By Lemma 15 we have \(p_t \geqslant \widetilde{p}_t\) for all such t. Define
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
By induction on \(i=2,\dots ,\rho \), equation (11) implies (as in Claim 3.6 in [21]):
Then (16) implies
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. \(\square \)
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Aissi, H., Mahjoub, A.R., McCormick, S.T. et al. Strongly polynomial bounds for multiobjective and parametric global minimum cuts in graphs and hypergraphs. Math. Program. 154, 3–28 (2015). https://doi.org/10.1007/s10107-015-0944-8
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DOI: https://doi.org/10.1007/s10107-015-0944-8