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On the (near) optimality of extended formulations for multi-way cut in social networks

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Abstract

Given an edge-weighted graph and a subset of the vertices called terminals, the multiway cut problem aims to find a minimum weight set of edges that separates each terminal from all the others. This problem is known to be NP-hard even for \(k=3\). Computational experiments on social networks obtained from the Indian e-commerce company Flipkart show that an integer programming formulation (referred to as EF2) of the problem introduced by Chopra and Owen (1996) provides a very strong LP-relaxation that allows us to solve large problems in a reasonable amount of time. We show that the cardinality EF2 (where all edge weights are 1) on a wheel graph has a primal integer solution and a dual integer solution of the same value. We consider a hub-spoke network of wheel graphs constructed by adding edges connecting the hub vertices of a collection of wheel graphs. We assume that every wheel has a terminal hub or a terminal vertex with three non-terminal neighbors, and show that if the graph connecting the hub vertices is planar, the cardinality EF2 on the hub-spoke network of the wheel graphs has a primal integer solution and a dual integer solution of the same value. Given the prevalence of such structures in our social networks, our results provide some theoretical justification for the strong empirical performance of the EF2 formulation.

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Notes

  1. The second author identified wheel graphs as one of the classes of frequent patterns of regional community graphs in his industrial project supported by an e-commerce firm.

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Acknowledgements

The authors are grateful to the anonymous referees for fruitful comments which significantly improved this paper.

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Correspondence to Sangho Shim.

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Appendices

Appendix A: Example: PDI qualifications on Freund–Karloff graphs

A general multiway cut problem may be reduced to a cardinality multiway cut problem on a multi-graph. Since there is a rational number arbitrarily close to any irrational number, the objective function may be assumed to be integral by multiplying the rational numbers by a common multiple of the denominators. We construct a multi-graph by replacing each edge by the same number of multi-edges as the integer capacity of the edge. The cardinality-multiway cut problem on the multi-graph is equivalent to the general multiway cut problem on the original graph. This reduction is not polynomial, as the input size of the capacity w(e) of an edge is \(\log w(e)\). Given Freund-Karloff simple graph \(\mathrm {FK}(k,w)\) and two integers f and g with \(w=f/g\), we define a Freund–Karloff multi-graph \(\mathrm {FK}(k,f,g)\) replacing every inner edge by f multi-edges and every outer edge by g multi-edges.

In this section, we use two kinds of cuts, referred to as collapsed configuration and home configuration. The collapsed configuration is a cut \((V_1,\ldots ,V_k)\) such that \(V_i=\{ v_i\}\) for \(i<k\) and \(V_k\) contains \(v_k\) and all the inner vertices. The home configuration is a cut \((V_1,\ldots ,V_k)\) defined by

$$\begin{aligned} V_j = \{ v_j\} \cup \{ v(i,j) : i < j \} \text{ for } j\in [k]. \end{aligned}$$

Freund and Karloff (2000) proved that the collapsed configuration is optimal on \(\mathrm {FK}(k,w)\) with cut capacity \((k-1)^2\) if \(w = f/g \ge 3/2k\), and that the home configuration is optimal with cut capacity \(\left( {\begin{array}{c}k\\ 2\end{array}}\right) + w \cdot 2 \left( {\begin{array}{c}k\\ 3\end{array}}\right)\) if \(w = f/g \le 3/2k\). They identified a feasible fractional solution of value \((k-1)(7k-6)/8\) and a fact that the choice of \(w = 3/2k\) maximizes the integrality gap

$$\begin{aligned} \frac{(k-1)^2}{\frac{1}{8}(k-1)(7k-6)} = \frac{8}{7+\frac{1}{k-1}}. \end{aligned}$$

It is thus a lower bound of the integrality gap of the CKR relaxation (on a general graph, not only \(\mathrm {FK}(k,w)\)).

Freund and Karloff (2000) commented that the integrality ratio is 1 if \(w\ge 2/k\). In fact, we can show that the cardinality EF2 is PDI on a Freund-Karloff multi-graph \(\mathrm {FK}(k,f,g)\). Using EF2, we also see that the feasible fractional solution is optimal if \(w \le 2/k\). Note that the number of the outer (simple) edges is \(2\left( {\begin{array}{c}k\\ 2\end{array}}\right)\) and the number of the inner edges is \(3\left( {\begin{array}{c}k\\ 3\end{array}}\right)\).

1.1 If \(w=f/g\ge 2/k\), the integrality ratio is 1

In this section, we assume that \(w=f/g\ge 2/k\). We may assume than f is an even number (by doubling g and f if f is odd). We consider a collapsed configuration \((V_1,\ldots ,V_k)\) with \(V_i=\{v_i\},i<k\) and \(V_k=\{ v_k\}\cup \{ v(i,j):i, j\in [k]\}\). The cut capacity is \((k-1)^2\). The subset graph is the star graph with center subset vertex \(V_k\).

We identify a dual integer solution which satisfy the sufficient condition of Subset Graph Theorem. We only need to take edge-disjoint paths compatible into \(v_k\). Assume that \(v_k\) is active. The f multi-edges connecting two inner vertices v(ij) and v(ik) are all directed from v(ij) to v(ik). For each of the other inner edges \(( v(i,j),v(i',j))\), f/2 multi-edges are directed from v(ij) to \(v(i',j)\), and the other half of the multi-edges are directed the other way around.

From an inactive vertex \(v_j\), we define the following disjoint paths \(P^q [v_j,v_k], q=1,\ldots ,(k-1)g\). Since g multi-edges link \(v_j\) to v(jk) and g multi-edges link v(jk) to \(v_k\), the first g edge-disjoint paths \(P^q [v_j,v_k] = v_j\Rightarrow v(j,k)\Rightarrow v_k\) are well-defined for \(q=1,\ldots ,g\). Consider another set of g edge-disjoint paths beginning with g outer multi-edges \((v_j,v(i,j))\) which are cut edges. The first f of them use the f inner multi-edges (v(ij), v(ik)). The remaining \(g-f\) paths use f/2 paths \(v_j\Rightarrow v(i,j)\Rightarrow v(j,k)\Rightarrow v(i,k)\Rightarrow v_k\), and f/2 paths \(v_j\Rightarrow v(i,j)\Rightarrow v(i',k)\Rightarrow v(i,k)\Rightarrow v_k\) for \(i'\ne [k]{\setminus }\{ i,j,k\}\). The g edge-disjoint paths are well defined by assumption

$$\begin{aligned} g\le f + f/2 + (k-3) f/2 = \frac{k}{2}f. \end{aligned}$$

The edge disjoint paths over all \(j\in [k-1]\) are compatible into \(v_k\) and satisfy the sufficient condition of Subset Graph Theorem.

1.2 If \(w \le 2/k\), the feasible fractional solution is optimal

Consider a primal feasible solution \(y^*\) satisfying \(y^* (v(i,j),i) = y^* (v(i,j),j)=1/2\) for all inner vertices v(ij). We show by strong duality of EF2 that, if \(w \le 1/(k - 1.5)\), the feasible fractional solution is optimal. Since \(\sum _{j}t^* (e,j) = 1.5\) for any multi-edge e, the primal value is the sum (over the outer and the inner edges)

$$\begin{aligned} 3\left( {\begin{array}{c}k\\ 2\end{array}}\right) g + 4.5\left( {\begin{array}{c}k\\ 3\end{array}}\right) f = 1.5\cdot 2\left( {\begin{array}{c}k\\ 2\end{array}}\right) g + 1.5\cdot 3\left( {\begin{array}{c}k\\ 3\end{array}}\right) f. \end{aligned}$$

Over the terminals, the sum of degrees is

$$\begin{aligned} \sum _{i=1}^k \alpha (v_i) = \sum _{i=1}^k \mathrm {degree} (v_i) = k(k-1)g=2\left( {\begin{array}{c}k\\ 2\end{array}}\right) g. \end{aligned}$$

Thus, the sum of \(\alpha (v(i,j))\) over the inner vertices is desired to be

$$\begin{aligned} 3\left( {\begin{array}{c}k\\ 2\end{array}}\right) g + 4.5\left( {\begin{array}{c}k\\ 3\end{array}}\right) f - 2\left( {\begin{array}{c}k\\ 2\end{array}}\right) g = \left( {\begin{array}{c}k\\ 2\end{array}}\right) \left( g + 4.5\frac{\left( {\begin{array}{c}k\\ 3\end{array}}\right) }{\left( {\begin{array}{c}k\\ 2\end{array}}\right) } f \right) . \end{aligned}$$

Note that \(\left( {\begin{array}{c}k\\ 2\end{array}}\right)\) is the number of the inner vertices. Assuming \(w \le 1/(k - 1.5)\), we show

$$\begin{aligned} \alpha (v(i,j)) = g + 4.5\frac{\left( {\begin{array}{c}k\\ 3\end{array}}\right) }{\left( {\begin{array}{c}k\\ 2\end{array}}\right) } f = g + 1.5 (k-2) f. \end{aligned}$$

Let \(v_j\) be active. Consider an inner vertex v(ij). It is easy to see that v(ij) is adjacent to total \(2k-4\) inner vertices; \(k-2\) inner vertices \(v(i',j),i'\ne i\) and the other \(k-2\) inner vertices \(v(i,i'),i'\ne j\). The g arrowheads from the inactive \(v_i\), the f/2 arrowheads from each of the \(k-2\) inner vertices \(v(i',j),i'\ne i\) adjacent to v(ij) and the f arrowheads from each of the other \(k-2\) inner vertices \(v(i',i),i'\ne j\) adjacent to v(ij) sum up to \(g + (k-2)\frac{f}{2} + (k-2) f = g + 1.5 (k-2) f.\) We then consider another inner vertex \(v(i,i')\) where none of \(v_i,v_{i'}\) is the active \(v_j\). Except two vertices \(v(i,j),v(i',j)\), each of the other \(2k-6\) inner vertices adjacent to \(v(i,i')\) provide f/2 arrowheads. Along with 2g arrowheads from its adjacent inactive terminals \(v_i,v_{i'}\), they sum up to \(2g + (k-3)f.\) By assumption \(w\le 2/k\), we see that

$$\begin{aligned} \alpha (v(i,j)) = g + (k-2)\frac{f}{2} + (k-2) f = g + 1.5 (k-2) f\le 2g + (k-3)f, \end{aligned}$$

as desired.

Appendix B: Network motifs found among social networks

See Fig. 11, 12, 13, 14, 15 and 16.

Fig. 11
figure 11

Tadpole \(M_{15}^{(4)}\)

Fig. 12
figure 12

4-Circle \(M_{30}^{(4)}\)

Fig. 13
figure 13

Diamond \(M_{31}^{(4)}\)

Fig. 14
figure 14

4-Complete \(M_{63}^{(4)}\)

Fig. 15
figure 15

m_75_5

Fig. 16
figure 16

m_79_5

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Shim, S., Bandi, C. & Chopra, S. On the (near) optimality of extended formulations for multi-way cut in social networks. Optim Eng 22, 1557–1588 (2021). https://doi.org/10.1007/s11081-021-09648-6

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