Approximation Algorithms and an Integer Program for Multi-level Graph Spanners

Abstract

Given a weighted graph G(V, E) and $t \ge 1$, a subgraph H is a t–spanner of G if the lengths of shortest paths in G are preserved in H up to a multiplicative factor of t. The subsetwise spanner problem aims to preserve distances in G for only a subset of the vertices. We generalize the minimum-cost subsetwise spanner problem to one where vertices appear on multiple levels, which we call the multi-level graph spanner (MLGS) problem, and describe two simple heuristics. Applications of this problem include road/network building and multi-level graph visualization, especially where vertices may require different grades of service.

We formulate a 0–1 integer linear program (ILP) of size $O(|E||V|^2)$ for the more general minimum pairwise spanner problem, which resolves an open question by Sigurd and Zachariasen on whether this problem admits a useful polynomial-size ILP. We extend this ILP formulation to the MLGS problem, and evaluate the heuristic and ILP performance on random graphs of up to 100 vertices and 500 edges.

This work was supported in part by NSF grants CCF-1740858, CCF-1712119, and DMS-1839274.

Appendices

A Proof of Theorem 4

Proof

We use the simple algebraic fact that $\min \{x,y\} \le \alpha x + (1-\alpha )y$ for all $x,y \in \mathbb {R}$ and $\alpha \in [0,1]$. Here, we can also use the fact that $\text {MIN}_1 \le \text {OPT}_1 + \text {OPT}_2 + \ldots + \text {OPT}_{\ell }$, as the RHS equals the cost of $G_1^*$, which is some subsetwise $(T_1 \times T_1)$-spanner. Combining, we have

$$\begin{aligned} \min (\text {TOP}, \text {BOT})&\le \alpha \sum _{i=1}^{\ell } \frac{i(i+1)}{2} \text {OPT}_i + (1-\alpha )\ell \sum _{i=1}^{\ell } \text {OPT}_i \\&= \sum _{i=1}^{\ell } \left[ \left( \frac{i(i+1)}{2} - \ell \right) \alpha + \ell \right] \rho \text {OPT}_i \\ \end{aligned}$$

Since we are comparing $\min \{\text {TOP}, \text {BOT}\}$ to $r \cdot \text {OPT}$ for some approximation ratio $r > 1$, we can compare coefficients and find the smallest $r \ge 1$ such that the system of inequalities

$$\begin{aligned} \left( \frac{\ell (\ell +1)}{2} - \ell \right) \alpha + \ell \rho&\le \ell r \\ \left( \frac{(\ell -1)\ell }{2} - \ell \right) \alpha + \ell \rho&\le (\ell -1)r \\&\vdots \\ \left( \frac{2 \cdot 1}{2} - \ell \right) \alpha + \ell \rho&\le r \end{aligned}$$

has a solution $\alpha \in [0,1]$. Adding the first inequality to $\ell /2$ times the last inequality yields $\frac{\ell ^2 + 2\ell }{2} \le \frac{3\ell r}{2}$, or $r \ge \frac{\ell +2}{3}$. Also, it can be shown algebraically that $(r, \alpha ) = (\frac{\ell +2}{3}, \frac{2}{3})$ simultaneously satisfies the above inequalities. This implies that $\min \{\text {TOP}, \text {BOT}\} \le \frac{\ell +2}{3}\rho \cdot \text {OPT}$.$\square$

B Proof of Theorem 6

Proof

Let $H^*$ denote an optimal pairwise spanner of G with stretch factor t, and let $\text {OPT}$ denote the cost of $H^*$. Let $\text {OPT}_{ILP}$ denote the minimum cost of the objective in the ILP (6). First, given a minimum cost t–spanner $H^*(V,E^*)$, a solution to the ILP can be constructed as follows: for each edge $e \in E^*$, set $x_e = 1$. Then for each unordered pair $(u,v) \in K$ with $u < v$, compute a shortest path $p_{uv}$ from u to v in $H^*$, and set $x_{(i,j)}^{uv} = 1$ for each edge along this path, and $x_{(i,j)}^{uv} = 0$ if (i, j) is not on $p_{uv}$.

As each shortest path $p_{uv}$ necessarily has cost $\le t \cdot d_G(u,v)$, constraint (7) is satisfied. Constraints (8)–(9) are satisfied as $p_{uv}$ is a simple u-v path. Constraint (10) also holds, as $p_{uv}$ should not traverse the same edge twice in opposite directions. In particular, every edge in $H^*$ appears on some shortest path; otherwise, removing such an edge yields a pairwise spanner of lower cost. Hence $\text {OPT}_{ILP} \le \text {OPT}$.

Conversely, an optimal solution to the ILP induces a feasible t–spanner H. Consider an unordered pair $(u,v) \in K$ with $u < v$, and the set of decision variables satisfying $x_{(i,j)}^{uv} = 1$. By (8) and (9), these chosen edges form a simple path from u to v. The sum of the weights of these edges is at most $t \cdot d_G(u,v)$ by (7). Then by constraint (10), the chosen edges corresponding to (u, v) appear in the spanner, which is induced by the set of edges e with $x_e = 1$. Hence $\text {OPT}\le \text {OPT}_{ILP}$.

Combining the above observations, we see that $\text {OPT}=\text {OPT}_{ILP}$.$\square$

C Proof of Theorem 7

Proof

Given an optimal solution to the ILP with cost $\text {OPT}_{ILP}$, construct an MLGS by letting $G_i = (V, E_i)$ where $E_i = \{e \in E \mid y_e \ge i\}$. This clearly gives a nested sequence of subgraphs. Let u and v be terminals in $T_i$ (not necessarily of required grade $R(\cdot ) = i$), with $u < v$, and consider the set of all variables of the form $x_{(i,j)}^{uv}$ equal to 1. By (13)–(15), these selected edges form a path from u to v of length at most $t \cdot d_G(u,v)$, while constraints (16)–(17) imply that these selected edges have grade at least $m_{uv} \ge i$, so the selected path is contained in $E_i$. Hence $G_i$ is a subsetwise $(T_i \times T_i)$–spanner for G with stretch factor t, and the optimal ILP solution gives a feasible MLGS.

Given an optimal MLGS with cost $\text {OPT}$, we can construct a feasible ILP solution with the same cost in a way similar to the proof of Theorem 6. For each $u,v \in T_1$ with $u < v$, set $m_{uv} = \min (R(u), R(v))$. Compute a shortest path in $G_{m_{uv}}$ from u to v, and set $x_{(i,j)}^{uv} = 1$ for all edges along this path. Then for each $e \in E$, consider all pairs $(u_1, v_1), \ldots , (u_k,v_k)$ that use either (i, j) or (j, i), and set $y_e = \max (m_{u_1v_1}, m_{u_2v_2}, \ldots , m_{u_kv_k})$. In particular, $y_e$ is not larger than the grade of e in the MLGS, otherwise this would imply e is on some u-v path at grade greater than its grade of service in the actual solution.$\square$

D Experimental Results on Graphs Generated Using Watts-Strogatz

The results for graphs generated from the Watts–Strogatz model are shown in Figs. 15, 16, 17, 18, 19, 20, 21, 22 and 23, which are organized in the same way as for Erdős–Rényi.

Fig. 15.

Performance with oracle on Watts–Strogatz graphs w.r.t. the number of vertices

Fig. 16.

Performance with oracle on Watts–Strogatz graphs w.r.t. the number of levels

Fig. 17.

Performance with oracle on Watts–Strogatz graphs w.r.t. the stretch factors

Fig. 18.

Performance with oracle on Watts–Strogatz graphs w.r.t. the number of vertices, the number of levels, and the stretch factors

Fig. 19.

Experimental running times for computing exact solutions on Watts–Strogatz graphs w.r.t. the number of vertices, the number of levels, and the stretch factors

Fig. 20.

Performance without oracle on Watts–Strogatz graphs w.r.t. the number of vertices

Fig. 21.

Performance without oracle on Watts–Strogatz graphs w.r.t. the number of levels

Fig. 22.

Performance without oracle on Watts–Strogatz graphs w.r.t. the stretch factors

Fig. 23.

Performance without oracle on Watts–Strogatz graphs w.r.t. the number of vertices, the number of levels, and the stretch factors

E Experimental Results on Large Graphs Using Erdős-Rényi

Figure 24 shows a rough measure of performance for the bottom-up and top-down heuristics on large graphs using the Erdős-Rényi model, where the ratio is defined as the BU or TD cost divided by min(BU, TD). Figure 25 shows the aggregated running times per instance, which significantly worsen as |V| is large.

Fig. 24.

Performance of heuristic bottom-up and top-down on large Erdős–Rényi graphs w.r.t. the number of vertices, the number of levels, and the stretch factors. The ratio is determined by dividing the objective value of the combined (min(BU, TD)) heuristic.

Fig. 25.

Experimental running times for computing heuristic bottom-up, top-down and combined solutions on large Erdős–Rényi graphs w.r.t. the number of vertices, the number of levels, and the stretch factors.