Problem set 2. Problem 1. Problem 2. Problem 3. CS261, Winter Instructor: Ashish Goel.

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1 CS261, Winter Instructor: Ashish Goel. Problem set 2 Electronic submission to Gradescope due 11:59pm Thursday 2/16. Form a group of 2-3 students that is, submit one homework with all of your names. [You may discuss these problems with your classmates, but please do not look for answers to these problems on the Internet. Your submission must be the original work of you and your partners, and you must understand everything that is written on your submission. We strongly suggest that you type up your solutions in LaTeX. A template is provided with the class notes.] 1 Problem 1 We have seen many examples of using linear programs for combinatorial problems. Here is an example of the reverse situation, where we need to use algorithmic thinking to formulate an appropriate linear program. Imagine that you are allocating utility (think of utility as money if you like) to n individuals. Let x i be the amount allocated to the i-th individual. You are required to respect the constraints Ax b and x 0. Show how to write this as a polynomial sized (in n and the number of rows, say m, of A) linear program if your objective function is to maximize 1. min{x 1, x 2,..., x N } 2. P k (x) where k is an integer between 1 and N, and P k (x) is defined to the sum of the k smallest components of the vector x. For example, P 2 ( 1, 7, 3, 4 ) = 4 since the two smallest components of the vector 1, 7, 3, 4 are 1 and 3. Feel free to catch the instructor some time if you want to understand the significance of the functions P k (x); in brief, these functions form a canonical basis for pretty much all measures of fairness in resource allocation. Problem 2 Show that the Gale-Shapley algorithm as specified in class results in a man-optimal stable matching. Hint: Define the set of stable partners S(w) of a woman w as the set of men who are matched with w in some stable matching. Consider the first time in the Gale-Shapley algorithm that a woman rejects a stable partner, and derive a contradiction. Problem 3 You are given a regular bipartite graph with n nodes on each side, with the degree of each node being d, and m = nd being the number of edges. 1 Thanks to Tim Roughgarden for letting us reuse some of the problems from his course. 1

2 1. Assume that d is even. Design an O(m)-time algorithm to partition the edges of the graph into 2 equal parts so that each part is a regular bipartite graph on the same set of nodes. Hint: First show that you can find an Euler decomposition (a partition of the edges of the graph into cycles) in time O(m). 2. Assume that d is a power of 2. Prove that you can find a perfect matching in time O(m). The requirement that d is a power of 2 sounds like a technicality, but it took nearly 20 years to reomve this assumption and find an O(m)-time algorithm for general d. The best known algorithm for this problem runs in time O(n log n), which is sub-linear when d is large. Problem 4 A vertex cover of an undirected graph (V, E) is a subset S V such that, for every edge e E, at least one of e s endpoints lies in S. 2 (a) Prove that in every graph, the minimum size of a vertex cover is at least the size of a maximum matching. (b) Give a non-bipartite graph in which the minimum size of a vertex cover is strictly bigger than the size of a maximum matching. (c) Prove that the problem of computing a minimum-cardinality vertex cover can be solved in polynomial time in bipartite graphs. 3 [Hint: Show how to obtain a minimum-cut after running the Ford-Fulkerson algorithm. Then reduce minimum-cardinality vertex cover to minimum cut. Feel free to use other approaches as well.] (d) Prove that in every bipartite graph, the minimum size of a vertex cover equals the size of a maximum matching. Problem 5 This problem considers approximation algorithms for graph matching problems. (a) For the maximum-cardinality matching problem in bipartite graphs, prove that for every constant ɛ > 0, there is an O(m)-time algorithm that computes a matching with size at most ɛn less than the maximum possible (where n is the number of vertices). (The hidden constant in the big-oh notation can depend on 1 ɛ.) (b) Now for non-bipartite graphs where each edge e has a real-valued weight w e, consider the following greedy algorithm: 2 Yes, the problem is confusingly named. 3 In general graphs, the problem turns out to be NP -hard (you don t need to prove this). 2

3 Greedy Matching Algorithm sort and rename the edges E = {1, 2,..., m} so that w 1 w 2 w m ; M = ; for i = 1 to m do if w i > 0 and e i shares no endpoint with edges in M then add e i to M How fast can you implement this algorithm? (c) Prove that the greedy algorithm always outputs a matching with total weight at least 50% times that of the maximum possible. [Hint: if the greedy algorithm adds an edge e to M, how many edges in the optimal matching can this edge block? How do the weights of the blocked edges compare to that of e?] Problem 6 The goal of this problem is to revisit two problems you studied in CS161 the minimum spanning tree and shortest path problems and to prove the optimality of Kruskal s and Dijkstra s algorithms via the complementary slackness conditions of judiciously chosen linear programs. (a) For convenience, we consider the maximum spanning tree problem (equivalent to the minimum spanning tree problem, after multiplying everything by -1). Consider a connected undirected graph G = (V, E) in which each edge e has a weight w e. For a subset F E, let κ(f ) denote the number of connected components in the subgraph (V, F ). Prove that the spanning trees of G are in an objective function-preserving one-to-one correspondence with the 0-1 feasible solutions of the following linear program (with decision variables {x e } e E ): max e E w e x e x e V κ(f ) for all F E e F x e = V 1 e E x e 0 for all e E. (While this linear program has a huge number of constraints, we are using it purely for the analysis of Kruskal s algorithm.) (b) What is the dual of this linear program? (c) What are the complementary slackness conditions? 3

4 (d) Recall that Kruskal s algorithm, adapted to the current maximization setting, works as follows: do a single pass over the edges from the highest weight to lowest weight (breaking ties arbitrarily), adding an edge to the solution-so-far if and only if it creates no cycle with previously chosen edges. Prove that the corresponding solution to the linear program in (a) is in fact an optimal solution to that linear program, by exhibiting a feasible solution to the dual program in (b) such that the complementary slackness conditions hold. 4 [Hint: for the dual variables of the form y F, it is enough to use only those that correspond to subsets F E that comprise the i edges with the largest weights (for some i).] (e) Now consider the problem of computing a shortest path from s to t in a directed graph G = (V, E) with a nonnegative cost c e on each edge e E. Prove that every simple s-t path of G corresponds to a 0-1 feasible solution of the following linear program with the same objective function value: min e E c e x e x e 1 for all S V with s S, t / S e δ + (S) x e 0 for all e E. While this linear program has exponentially many constraints, we will only use it for analysis. 5 (f) What is the dual of this linear program? (g) What are the complementary slackness conditions? (h) Let P denote the s-t path returned by Dijkstra s algorithm. Prove that the solution to the linear program in (e) corresponding to P is in fact an optimal solution to that linear program, by exhibiting a feasible solution to the dual program in (f) such that the complementary slackness conditions hold. [Hint: it is enough to use only dual variables of the form y S for subsets S V that comprise the first i vertices processed by Dijkstra s algorithm (for some i).] Problem 7 This problem fills in some gaps in our proof sketch of strong linear programming duality. (a) For this part, assume the version of Farkas s Lemma stated in Lecture #8, that given A R m n and b R m, exactly one of the following statements holds: (i) there is an x R n such that Ax = b and x 0; (ii) there is a y R m such that y T A 0 and y T b < 0. 4 You can assume without proof that Kruskal s algorithm outputs a feasible solution (i.e., a spanning tree), and focus on proving its optimality. 5 Here δ + (S) denotes the the set of edges (v, w) such that v is in S and w is not in S. 4

5 Deduce from this a second version of Farkas s Lemma, stating that for A and b as above, exactly one of the following statements holds: (iii) there is an x R n such that Ax b; (iv) there is a y R m such that y 0, y T A = 0, and y T b < 0. [Hint: note the similarity between (i) and (iv). Also note that if (iv) has a solution, then it has a solution with y T b = 1. ] (b) Use the second version of Farkas s Lemma to prove the following version of strong LP duality: if the linear programs max c T x with x unrestricted, and Ax b min b T y A T y = c, y 0 are both feasible, then they have equal optimal objective function values. [Hint: Let γ denote the optimal objective function value of the dual linear program. Add the constraint c T x γ to the primal linear program and use Farkas s Lemma to show that the feasible region is non-empty.] 5