(b) In the relaxation we can get many fractional paths between each demand pair of total unit value. These paths can be read out of the solution in polynomial time in the following way. We find an edge e of the smallest positive flow value c, and subtract flow of this value from a cycle or a path going through e over edges of positive flow value. Repeating this procedure we will decompose the f...

Problem 1. this needs to be fleshed out We will use 3 arrays A, B, and C. A will represent that actual array and hold the stored items in their stored positions. Arrays B and C will act as “checks” for which entries of A are truly initialized. B will contain a list, built up sequentially from position 0, of all entries of A that have been filled. We use a “length” variable i to keep track of th...

As usual, we are only guaranteed the attacks find a key consistent with the input-output examples rather than finding the target key itself, but empirically we estimate that with three input-output examples the target key will be the only one consistent with the input-output examples and hence will be the one found by the attack. The same attacks using only two input-output examples will also t...

We will remove vertices from a given graph G of treewidth k in a good elimination ordering, and we can assume w.l.o.g. that G is connected. Some cliques in a current graph will be active cliques. For each active clique C and for each subset S of vertices in C, we store information if there exists a vertex cover of the original graph G that chooses exactly vertices in S out of the vertices in C....

I state without proof a few results concerning the local properties of smooth maps. Section 11, beginning at page 89, of our class notes contains more information, examples, and proofs. All functions and maps will be assumed here to be smooth (infinitely continuously differentiable) but the discussion in the notes is more general. By a neighborhood of a point I mean an open set containing the p...

Solution. In a finite connected graph with n ≥ 2 vertices, the domain for the vertex degrees is the set {1, 2, . . . , n − 1} since each vertex can be adjacent to at most all of the remaining n−1 vertices and the existence of a degree 0 vertex would violate the assumption that the graph be connected. Therefore, treating the n vertices as the pigeons and the n−1 possible degrees as the pigeonhol...

Many word processors and keyword search engines have a spelling correction feature. If you type in a misspelled word x, the word processor or search engine can suggest a correction y. The correction y should be a word that is close to x. One way to measure the similarity in spelling between two text strings is by “edit distance.” The notion of edit distance is useful in other fields as well. Fo...