University of Delaware Computer and Information Sciences CIS621 Algorithm Design and Analysis Homework Set #5 Due Tuesday, December 5, 1995

18. How hard is hard?

a) 36.4-5

b) 36.4-7

Grader: Sakir Yucel

19. Hamilton

a) Widget A of figure 36.15 introduces 12 new vertices. Design a widget that has the same effect as widget A but requires as few new vertices as you can manage (fewer than 9). Remark: 36.5-7 is a related question.

b) 36.5-5

Grader: Jyotsa Laxminarayanan

20. Independence An independent set I of an undirected graph G(V,E), is a subset of V such that no two elements of I share an edge in E. Note that independent sets are roughly the opposite of cliques! It should come as no surprise that the following problem is NP-complete: INDEPENDENT-SET (reference: Problem 36-1): Given an undirected graph G(V,E), and integer k, decide if there exists in G an independent set of cardinality at least k?

To describe the problem we consider, we first need a definition: For a vertex v, the neighborhood of v, denoted N(v), consists of v, and all of the nodes adjacent to v. A fat independent set F of G is a subset of V such that the neighborhoods of the nodes in F are pairwise disjoint. Now, show the following problem is NP-complete, using a reduction from independent set: FAT-INDEPENDENT-SET: Given Undirected graph G(V,E), and integer k, decide if there exists in G a fat independent set of cardinality at least k?

Grader: Marjeta Cedilnik

21. Carpenter’s Ruler A Carpenter’s Ruler consists of line segments of integral lengths, hinged together consecutively. The line segments (called links) can rotate freely about the hinges. Such a ruler models a robot arm. In moving a robot arm it is found convenient to fold the arm as compactly as possible. In such a folding, we allow the links to cross each other, but the angle at a hinge is restricted to be either 0 or 180 degrees. Prove that given a carpenter’s ruler in which the successive links have lengths w1, w2, ..., wn, and an arbitrary bound D, it is NP-complete to determine whether the ruler can be folded into a length of at most D.

Grader: Earl Ferry EC-4. Reductions, reductions Problems to show various problems are NP-complete: 36.5-1, 36.5-2, 36.5-4, 36.5-6, 36.1(a). Do 4 of the 5. Grader: Marjeta Cedilnik Remember to put your solutions to separate problems on separate pages and to put your name on each.