Modular Circulation and Applications to Traffic Management

Abstract

We introduce a variant of the well-known minimum-cost circulation problem in directed networks, where vertex demand values are taken from the integers modulo $\lambda$, for some integer $\lambda \ge 2$. More formally, given a directed network $G = (V,E)$, each of whose edges is associated with a weight and each of whose vertices is associated with a demand taken over the integers modulo $\lambda$, the objective is to compute a flow of minimum weight that satisfies all the vertex demands modulo $\lambda$. This problem is motivated by a problem of computing a periodic schedule for traffic lights in an urban transportation network that minimizes the total delay time of vehicles. We show that this modular circulation problem is solvable in polynomial time when $\lambda = 2$ and that the problem is NP-hard when $\lambda = 3$. We also present a polynomial time algorithm that achieves a $4(\lambda - 1)$-approximation.

Appendix A: NP-hardness Generalization

Fundamental blocks are constructed in the same manner as described in the $\lambda = 3$ example in Sect. 5.1, but for completeness, we describe them here in their general form. A fundamental block begins with $\lambda$ 1-vertices, connected as a star graph (i.e., a central vertex connected to each of the remaining vertices) with edges directed outward. Each 1-vertex is also connected to a $(\lambda -1)$-vertex via an incoming edge. Finally, these $(\lambda -1)$-vertices each have an outgoing interface edge (Fig. 8).

Fig. 8

Fundamental blocks with a$\lambda = 4$ and b$\lambda = 6$

The expansion module, too, remains largely unchanged. It consists of a central fundamental block with $\lambda$ interface edges. A fundamental block is then connected to all but one of these edges, with the remaining edge reserved for connecting to the existing variable gadget. For each expansion module added to a variable gadget, $(\lambda -1)^2-1$ interface edges are added (one interface edge is consumed when connecting to the variable gadget, hence the $-1$). See Fig. 9. Just as before, we must adjust the weights of a small subset of edges so that True and False assignments incur the same cost. To do so, arbitrarily select a fundamental block that is not the hub (i.e., not the fundamental block which connects to the existing variable gadget) and set the outgoing edges from its central 1-vertex to a weight of $\frac{2\lambda -3}{\lambda -1}$.

Fig. 9

Expansion modules built from fundamental blocks (FB) with a$\lambda = 4$ and b$\lambda = 6$. Edges between fundamental blocks are represented with a disc and no direction as a reminder that the blocks are connected via a shared $(\lambda -1)$-vertex

Recall that the variable gadget described in Sect. 5.1 creates $c(v_i)\lambda$ interface edges. In truth, the fundamental block creates $\lambda$ interface edges while each of the $c(v_i)-1$ expansion modules add $(\lambda -1)^2-1$ interface edges. When $\lambda = 3$, these formulae are equivalent, each variable has $c(v_i)\lambda$ interface edges, and each of the $c(v_i)$ clause gadgets will connect to $\lambda$ interface edges.

When $\lambda > 3$, the variable gadgets will have $\lambda + \left[ c(v_i)-1\right] \left[ (\lambda -1)^2-1\right]$ interface edges. As this is not evenly divisible by $c(v_i)$, connecting to the clause gadgets becomes problematic. To correct for this, a modified expansion module, called a seed module, is used to start each variable gadget rather than a fundamental block. Rather than connecting to an existing variable gadget, the edge that was previously reserved for this connection is instead attached to one of the existing interface edges (recall that in actuality a $(\lambda -1)$-vertex is being shared, not that two edges are being connected directly). By doing so, we now have a seed module that has $(\lambda -1)^2-1$ interface edges (Fig. 10) and, when used in conjunction with unaltered expansion modules, creates a variable gadget with $c(v_i)\left[ (\lambda -1)^2-1\right]$ interface edges.

Clause gadgets are constructed as described in Sect. 5.2, with the caveat that they require $(\lambda -1)^2-1$ vertices rather than $\lambda$. Each vertex will still only have three incoming edges, as each clause will only ever have three literals, and a demand of 1.

Fig. 10

Seed modules for creating variable gadgets with a$\lambda = 4$ and b$\lambda = 6$

Despite a small increase in the size of each gadget, the reduction can still be done in polynomial time.

Finally, the cost threshold must be generalized in Lemma 5.4. In the general form, if F is satisfiable there will be a satisfying flow with

$$\begin{aligned} \mathrm {cost}(f) ~ \le ~ \sum _{v_i \in V} [(\lambda ^2-1)c(v_i) + \lambda |C|\gamma ] \end{aligned}$$