Solving the Stable Set Problem in Terms of the Odd Cycle Packing Number
نویسندگان
چکیده
The classic stable set problem asks to find a maximum cardinality set of pairwise non-adjacent vertices in an undirected graph G. This problem is NP-hard to approximate with factor n1−ε for any constant ε > 0 [10, 24], where n is the number of vertices, and therefore there is no hope for good approximations in the general case. We study the stable set problem when restricted to graphs with bounded odd cycle packing number ocp(G), possibly by a function of n. This is the largest number of vertex-disjoint odd cycles in G. Equivalently, it is the logarithm of the largest absolute value of a sub-determinant of the edge-node incidence matrix AG of G. Hence, if AG is totally unimodular, then ocp(G) = 0. Therefore, ocp(G) is a natural distance measure of AG to the set of totally unimodular matrices on a scale from 1 to n/3. When ocp(G) = 0, the graph is bipartite and it is well known that stable set can be solved in polynomial time. Our results imply that the odd cycle packing number indeed strongly influences the approximability of stable set. More precisely, we obtain a polynomial-time approximation scheme for graphs with ocp(G) = o(n/ logn), and an α-approximation algorithm for any graph where α smoothly increases from a constant to n as ocp(G) grows from O(n/ logn) to n/3. On the hardness side, we show that, assuming the exponential-time hypothesis, stable set cannot be solved in polynomial time if ocp(G) = Ω(log1+ε n) for some ε > 0. Finally, we generalize a theorem by Györi et al. [9] and show that graphs without odd cycles of small weight can be made bipartite by removing a small number of vertices. This allows us to extend some of our above results to the weighted stable set problem. 1998 ACM Subject Classification G.2.1 Combinatorics, G.2.2 Graph Theory
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