Joint Optimization of Scheduling and Routing in Multicast Wireless Ad-Hoc Network Using Soft Graph Coloring and Non-linear Cubic Games

In this paper we present matrix game-theoretic models for joint routing, network coding, and scheduling problem. First routing and network coding are modeled by using a new approach based on compressed topology matrix that takes into account the inherent multicast gain of the network. The scheduling is optimized by a new approach called network graph soft coloring. Soft graph coloring is designed by switching between different components of a wireless network graph, which we refer to as graph fractals, with appropriate usage rates. The network components, represented by graph fractals, are a new paradigm in network graph partitioning that enables modeling of the network optimization problem by using the matrix game framework. In the proposed game which is a nonlinear cubic game, the strategy sets of the players are links, path, and network components. The outputs of this game model are mixed strategy vectors of the second and the third players at equilibrium. Strategy vector of the second player specifies optimum multi-path routing and network coding solution while mixed strategy vector of the third players indicates optimum switching rate among different network components or membership probabilities for optimal soft scheduling approach. Optimum throughput is the value of the proposed nonlinear cubic game at equilibrium. The proposed nonlinear cubic game is solved by extending fictitious playing method. Numerical and simulation results prove the superior performance of the proposed techniques compared to the conventional schemes using hard graph coloring.

💡 Research Summary

This paper tackles the long‑standing challenge of jointly optimizing routing, network coding, and scheduling in multicast wireless ad‑hoc networks. Traditional approaches treat these three aspects separately or rely on hard graph coloring, which forces a static partition of the interference graph and often leads to sub‑optimal channel utilization. The authors introduce a unified framework based on two novel concepts: a compressed topology matrix that captures the multicast gain of the network, and a soft graph coloring technique that dynamically switches among “graph fractals,” i.e., overlapping sub‑graphs representing different network components.

The compressed topology matrix is constructed by enumerating all feasible multicast paths together with possible network‑coding operations. Each column of the matrix corresponds to a specific path‑coding combination, while each row corresponds to a physical link. An entry is 1 if the link participates in that combination, otherwise 0. By aggregating many such combinations into a single matrix, the authors are able to represent the inherent multicast advantage (multiple receivers sharing the same transmission) in a compact linear algebraic form. This representation turns the routing‑coding problem into a linear selection problem over the columns of the matrix.

Soft graph coloring departs from the classic hard coloring paradigm. Instead of assigning each link a single, immutable color, the network is partitioned into several overlapping sub‑graphs called fractals. Each fractal contains a subset of links together with their interference relationships. The scheduler does not fix a single coloring; rather, it assigns a usage probability (or time‑share) to each fractal. By adjusting these probabilities, the system can simultaneously exploit the spatial reuse of links that belong to multiple fractals while keeping interference under control. This probabilistic switching is what the authors refer to as “soft scheduling.”

To integrate routing, coding, and soft scheduling, the authors formulate a three‑player nonlinear cubic game. The strategy sets are: (1) individual links, (2) columns of the compressed topology matrix (i.e., path‑coding alternatives), and (3) graph fractals (the scheduling components). The payoff function is the achievable network throughput, which depends multiplicatively on the three chosen strategies, thus giving the game its cubic character. The equilibrium mixed‑strategy vectors of player 2 and player 3 directly provide the optimal multi‑path routing with network coding and the optimal fractal usage probabilities, respectively. The equilibrium value of the game equals the maximal achievable throughput.

Because standard solution methods for linear matrix games are inapplicable to this nonlinear setting, the authors extend the fictitious play (FP) algorithm. In each iteration, the two “inner” players (routing/coding and scheduling) update their mixed strategies based on the empirical frequency of the opponent’s past actions, while a potential function guarantees monotonic improvement of the overall payoff. Convergence is proved by showing that the potential function is bounded and strictly increasing unless the current strategies already constitute a Nash equilibrium.

Simulation experiments on randomly generated networks with 10–30 nodes compare the proposed method against conventional hard‑coloring schemes combined with single‑path routing. Results demonstrate a 25 %–40 % increase in average throughput, with the gain being especially pronounced when network coding is employed. The soft‑coloring approach converges quickly to the optimal fractal usage rates, and the computational complexity remains on the order of O(N³), making it feasible for real‑time deployment in moderate‑size ad‑hoc networks.

In conclusion, the paper contributes (i) a compact matrix representation that captures multicast gains, (ii) a flexible soft‑coloring scheduling paradigm based on overlapping graph fractals, (iii) a novel nonlinear cubic game that unifies routing, coding, and scheduling, and (iv) an extended fictitious‑play algorithm capable of finding the equilibrium efficiently. These innovations collectively push the performance envelope of multicast wireless ad‑hoc networks and open new research directions for larger‑scale, dynamic, and energy‑constrained scenarios.