A polynomial time $frac 3 2$ -approximation algorithm for the vertex cover problem on a class of graphs

We develop a polynomial time 3/2-approximation algorithm to solve the vertex cover problem on a class of graphs satisfying a property called ``active edge hypothesis’’. The algorithm also guarantees an optimal solution on specially structured graphs. Further, we give an extended algorithm which guarantees a vertex cover $S_1$ on an arbitrary graph such that $|S_1|\leq {3/2} |S^|+\xi$ where $S^$ is an optimal vertex cover and $\xi$ is an error bound identified by the algorithm. We obtained $\xi = 0$ for all the test problems we have considered which include specially constructed instances that were expected to be hard. So far we could not construct a graph that gives $\xi \not= 0$.

💡 Research Summary

The paper addresses the classic NP‑hard Vertex Cover Problem (VCP) by developing a polynomial‑time algorithm that achieves a 3/2 approximation ratio on a broad class of graphs. The authors begin by formulating the standard integer programming model for VCP and then introduce an extended linear programming (ELP) relaxation that augments the usual edge constraints with odd‑cycle inequalities: for every odd cycle ω of length 2s+1, the sum of the variables on its vertices must be at least s+1. Although the number of odd cycles can be exponential, a polynomial‑time separation oracle exists, so the ELP can be solved efficiently.

The central conceptual tool is the “active‑edge hypothesis.” Given an optimal basic feasible solution x⁰ of the ELP on a graph G, the hypothesis asserts that at least one of the following holds: (i) G contains a triangle (3‑cycle); (ii) there exists an edge (i,j) with x⁰_i + x⁰_j = 1 (an active edge); (iii) there exists an edge (i,j) with x⁰_i + x⁰_j ≥ 4/3 (an over‑active edge); or (iv) some vertex i has x⁰_i = 1. When the hypothesis is satisfied, the algorithm can apply one of four reduction operations that strictly decrease the ELP objective value while simplifying the graph:

1. {0,1}‑reduction – vertices with x⁰_i = 0 are deleted; those with x⁰_i = 1 are added to the cover and then deleted.
2. 3‑cycle reduction – a triangle is removed entirely.
3. Active‑edge reduction – for an active edge (i,j), vertices i and j are removed, and every neighbor of i (except j) is connected to every neighbor of j (except i), forming a new graph G(i,j).
4. Over‑active‑edge reduction – both endpoints of an over‑active edge are simply removed.

Each reduction guarantees a drop in the ELP objective: at least 1 for an active‑edge reduction, 4/3 for an over‑active‑edge reduction, and 2 for a 3‑cycle reduction. Lemma proofs show that the reduced graph’s ELP solution can be constructed from the original solution, preserving feasibility.

The algorithm proceeds iteratively: solve the ELP on the current graph, apply any applicable reduction, and repeat until the ELP solution becomes integral. At that point a back‑tracking phase reconstructs a vertex cover for the original graph by reversing the reductions. For active‑edge reductions the algorithm chooses either endpoint based on whether all neighbors of the opposite endpoint are already in the partial cover; for over‑active reductions both endpoints are added; for 3‑cycle reductions all three vertices are added; and for {0,1} reductions the vertices with value 1 are already included.

The key analytical result (Theorem 1) demonstrates that, under the active‑edge hypothesis, the total number of vertices added during back‑tracking is at most 3/2 times the total decrease in the ELP objective across all reductions. Since the initial ELP objective is a lower bound on the size of an optimal vertex cover |S*|, the final cover S₁ satisfies |S₁| ≤ 3/2·|S*|. The algorithm runs in polynomial time because each reduction removes at least one vertex, leading to O(n) LP solves, each solvable in polynomial time.

The authors identify graph families for which the active‑edge hypothesis is guaranteed. If the graph contains a triangle, the hypothesis trivially holds. More generally, if the graph has fewer than |V| linearly independent odd cycles (i.e., the incidence vectors of its odd cycles are not full rank), then the odd‑cycle constraints are strong enough to force the existence of an active or over‑active edge, or a variable equal to 1. This includes many sparse and structured graphs.

To handle graphs where the hypothesis might fail, the authors propose an extended version of the algorithm that still terminates with a solution S₁ satisfying |S₁| ≤ 3/2·|S*| + ξ, where ξ ≥ 0 is a computable error bound produced during execution. In all computational experiments conducted—covering random graphs, dense graphs, and specially crafted hard instances—the observed ξ was zero, meaning the algorithm achieved the pure 3/2 guarantee even without a formal proof that the hypothesis always holds.

Experimental results confirm the theoretical claims: the algorithm consistently produced vertex covers no larger than 1.5 times optimal, often matching the optimum exactly, and ran comfortably within polynomial time limits. The combination of 3‑cycle and active‑edge reductions proved especially powerful, as each reduction can create new triangles that are subsequently eliminated, leading to a cascade of simplifications.

In conclusion, the paper contributes a novel LP‑based framework that leverages odd‑cycle inequalities and a carefully designed reduction scheme to break the long‑standing 2‑approximation barrier for VCP on a substantial class of graphs. While the active‑edge hypothesis remains unproven for all graphs, the empirical evidence suggests it may hold universally, opening avenues for future work to either prove the hypothesis or to characterize graphs where ξ > 0. Moreover, the methodology may be adaptable to related combinatorial optimization problems such as Maximum Independent Set or Minimum Dominating Set, where odd‑cycle constraints also play a pivotal role.