diff --git a/assignment-4/task2.png b/assignment-4/task2.png new file mode 100644 index 0000000..4b6d290 Binary files /dev/null and b/assignment-4/task2.png differ diff --git a/assignment-4/task2.svg b/assignment-4/task2.svg new file mode 100644 index 0000000..f187ad2 --- /dev/null +++ b/assignment-4/task2.svg @@ -0,0 +1,5 @@ + + +startlet Gundirectedgraphlet kpositiveintegerminimum maximal matchingon G with at most kedgesthere existsmatching in Gthere exists nomatching in Gthere are two outcomes of runningthe algorithm: there is such amatching, or there is no suchmatching.we must prove that this decisionproblem has a polynomial kernel,which would imply that it is fixed-parameter tractable.black-box proof: let M be a maximal matching in G. (1) if |M| > k, then k is too small => no such matching found (2) if |M| <= k, then |M| is within budget let V(M) denote the vertices covered by M similar to vertex cover problem it follows that V(M) <= 2k, since each edge in M covers at most two vertices let u vertex outside of V(M) then N(u) can only contain vertices in V(M), since if v in N(u) and v not in V(M) would mean M is not maximal, thus contradiction. this gives a reduction rule: remove all vertices outside of V(M) then form G[V(M)], the fully connected subgraph induced by V(M) => return G[V(M)] as kernel G[V(M)] is a polynomial kernel because - the number of vertices is at most 2k - the number of edges is 2k(2k-1)/2 = 2k^2 - k (since fully connected, undirected) - the kernel size is thus O(k^2), polynomial in k Q.E.D. \ No newline at end of file