By Georgiev P., Pardalos P., Theis F.

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**Additional info for A bilinear algorithm for sparse representations**

**Example text**

In the Edge Clique Cover problem, we are given a graph G and a nonnegative integer k, and the goal is to decide whether the edges of G can be covered by at most k cliques. In this section we give an exponential kernel for Edge Clique Cover. 3, we remark that this simple kernel is essentially optimal. Let us recall the reader that we use N (v) = {u : uv ∈ E(G)} to denote the neighborhood of vertex v in G, and N [v] = N (v) ∪ {v} to denote the closed neighborhood of v. , we always use the lowest-numbered rule that modiﬁes the instance).

1, we greedily take into a solution an edge that participates in k + 1 otherwise disjoint forbidden structures (here, triangles). 2, we discard vertices that do not participate in any forbidden structure, and should be irrelevant to the problem. 2 is not immediate: we need to verify that deleting v and its incident edges does not make make a yes-instance out of a no-instance. Note that after applying any of the two rules, the resulting graph is again a tournament. The ﬁrst rule is safe because if we do not reverse e, we have to reverse at least one edge from each of k + 1 triangles containing e.

11), we infer that |X| ≤ k. If no vertex of X is in VM , then X ⊆ I. We claim that X = I. For a contradiction assume that there is a vertex w ∈ I \ X. Because G has no isolated vertices there is an edge, say wz, incident to w in GI,VM . Since GI,VM is bipartite, we have that z ∈ VM . However, X is a vertex cover of GI,VM such that X ∩ VM = ∅, which implies that w ∈ X. This is contrary to our assumption that w ∈ / X, thus proving that X = I. But then |I| ≤ |X| ≤ k, and G has at most |I| + |VM | ≤ k + 2k = 3k vertices, which is a contradiction.

### A bilinear algorithm for sparse representations by Georgiev P., Pardalos P., Theis F.

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