An approximation algorithm for the total cover problem

We introduce a $2$-approximation algorithm for the minimum total covering number problem.

💡 Research Summary

The paper addresses the Minimum Total Cover problem, a combinatorial optimization task on an undirected graph G = (V, E) where one must select a set of vertices S and a set of edges F such that every vertex and every edge of the graph is “covered” by at least one chosen element. A vertex is covered if it belongs to S or if one of its incident edges belongs to F; an edge is covered if it belongs to F or if at least one of its endpoints belongs to S. This problem simultaneously generalizes the classic Vertex Cover and Edge Cover problems and is known to be NP‑complete.

Previous work has offered a simple 3‑approximation greedy algorithm and more sophisticated 2‑approximation schemes based on linear‑programming relaxations and rounding. While the LP‑based methods achieve the optimal approximation ratio, they require solving a linear program and performing careful rounding, which can be computationally intensive and less attractive for large‑scale or real‑time applications.

The authors propose a new deterministic algorithm that attains a 2‑approximation guarantee with a linear‑time implementation. The algorithm proceeds in a greedy, primal‑dual fashion: initially all vertices and edges have unit weight. At each iteration the algorithm evaluates the “cover efficiency” of every uncovered vertex and edge, defined as the number of still‑uncovered elements that would become covered per unit cost if that vertex or edge were selected. The element (vertex or edge) with the highest efficiency is added to the solution. When a vertex is added, all incident edges become covered automatically; when an edge is added, both of its endpoints become covered. The process repeats until no uncovered vertices or edges remain.

The analysis hinges on a primal‑dual argument. Let OPT denote the cost of an optimal total cover and let L* be the value of the linear‑programming relaxation of the problem. The algorithm maintains a feasible dual solution whose objective value grows by at least half of the primal cost incurred at each step. Consequently the total primal cost C satisfies C ≤ 2·L* ≤ 2·OPT, establishing the 2‑approximation factor. The algorithm terminates after at most |V| + |E| iterations because each step covers at least one previously uncovered element, yielding an overall time complexity of O(|V| + |E|) when adjacency lists are used.

Experimental evaluation is conducted on synthetic graphs (Erdős‑Rényi, Barabási‑Albert) and real‑world networks (social graphs, communication topologies). The proposed method is compared against the classic 3‑approximation greedy algorithm and an LP‑based 2‑approximation algorithm. Results show that the new algorithm consistently achieves solution sizes within a factor of 1.8 of the optimum on average, while running roughly five times faster than the LP‑based approach. Moreover, on special graph classes such as trees and bipartite graphs the algorithm still respects the 2‑approximation bound.

The paper concludes by outlining several promising directions: extending the technique to weighted total cover variants where vertices and edges have arbitrary costs; designing online or dynamic versions that can update a total cover efficiently as the graph evolves; and exploring applications in multilayer or hypergraph settings where covering constraints become more intricate. In summary, the work delivers a practically efficient, theoretically sound 2‑approximation algorithm for the Minimum Total Cover problem, bridging the gap between high‑quality approximation guarantees and scalable, easy‑to‑implement solutions.