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Homework #4 CS675 Fall 2019 Homework #4 CS675 Fall 2019 Due Monday Dec 2 by 4:10pm General Instructions The following assignment is meant to be challenging. Feel free to discuss with fellow students, though please write up your solutions independently and acknowledge everyone you discussed the homework with on your writeup. I also expect that you will not attempt to consult outside sources, on the Internet or otherwise, for solutions to any of these homework problems. Finally, please provide a formal mathematical proof for all your claims. Problem 1. (9 points) n Let X = fv1; : : : ; vmg where vi 2 R for i = 1; : : : ; m. As explained in class, the set system over X whose feasible sets are the linearly independent subsets of X is a matroid. In this problem, we will explore whether other forms of independence also yield matroids. (a) [3 points]. Suppose instead that we define S ⊆ X to be feasible if the vectors in S are independent with respect to convex combinations | i.e. no vector v 2 S can be written as a convex combination of the remaining vectors S n v. Is the resulting set system a matroid? Prove or disprove. (b) [3 points]. What if we define S ⊆ X to be feasible if the vectors in S are affinely independent | i.e. no vector v 2 S can be written as an affine combination of the remaining vectors S n v. Is the resulting set system a matroid? Prove or disprove. (c) [3 points]. Finally, what if we say S ⊆ X is feasible if the vectors in S are conically independent | i.e. no vector v 2 S can be written as a conic combination of the remaining vectors S n v. Is the resulting set system a matroid? Prove or disprove. Problem 2. (15 points) In this problem, we will examine some miscellaneous properties of matroids. We use M = (X ; I) to denote an arbitrary matroid. X (a) [3 points]. Prove that the rank function rankM : 2 ! N of a matroid is monotone, normalized, and submodular. (b) [4 points]. Prove the strong exchange property of matroids: If B and B0 are bases of matroid M, then for every i 2 B nB0 there is some j 2 B0 nB such that (B nfig)[fjg is a basis of M. Show that the exchange property is implied by the strong exchange property for every downwards-closed set system. (i.e. we could have equivalently defined matroids using the strong exchange property 1 instead). (c) [2 points]. Show that matroids are closed under truncation. Namely, if M = (X ; I) is a ma- troid, and k is a nonnegative integer, then the set system Mk = (X ; Ik) with Ik = fS 2 I : jSj ≤ kg is also a matroid. (d) [3 points]. Recall that we call a set C ⊆ X a circuit of the matroid M if it is a minimal dependent set in M. Note that a circuits of a graphical matroid correspond to cycles in the graph. X Prove the circuit property of matroids: if M = (X ; I) is a matroid with distinct weights w 2 R assigned to its ground set, and C is a circuit of M, then the element i 2 C of minimum weight is not in any maximum-weight basis of M. (e) [3 points]. We call a set C ⊆ X a cut of the matroid M if every basis of M intersects C, and no proper subset of C has this property. Note that a cut of a graphical matroid correspond to our notion of a cut in a graph | namely, a family of edges who's removal disconnects some connected component of the graph. Prove the cut property of matroids: if M = (X ; I) is a matroid with X distinct weights w 2 R assigned to its ground set, and C is a cut of M, then the element i 2 C of maximum weight is in every maximum-weight basis of M. Problem 3. (12 points) Given two matroids M1 = (X ; I1) and M2 = (X ; I2) with a common ground set X , and a weight X vector w 2 R , the maximum-weight common independent set of M1 and M2 is the set S 2 I1 \I2 P maximizing i2S w(i). We discussed in class how the maximum-weight common independent set can be computed in time poly(jX j), given only independence-oracle access to the matroids M1 and M2. Recall that this followed from the integrality of the polytope P(M1) \P(M2), where P(M1) and P(M2) are the two matroid polytopes. We will now consider a related problem, the minimum-weight common-basis problem, show how to solve it efficiently, and show that it can be viewed as a generalization of the shortest path X problem in directed graphs. Given two matroids M1 = (X ; I1) and M2 = (X ; I2), let B1 ⊆ 2 denote the family of all bases (maximum-cardinality independent sets) of M1, and similarly B2 X for M2. Given a weight vector w 2 R , the minimum-weight common basis of M1 and M2, if it P exists, is the set B 2 B1 \B2 minimizing i2B w(i). Note that for a common basis to exist, both matroids must have the same rank (though this is not sufficient). (a) [6 points]. Show that the minimum-weight common-basis problem can be solved in polynomial time in jX j, given only independence-oracle access to the two matroids M1 and M2. Your algorithm should either output a minimum-weight common-basis of M1 and M2 if one exists, or else assert that there is no common basis. (Hint: You can start by showing that that the convex hull of common bases is a face of the matroid intersection polytope) (b) [6 points]. Show that the shortest path problem in weighted directed graphs can be reduced, in linear time, to the minimum-weight common-basis problem when the graph has no negative cycles. (Hint: Add self-loops to all nodes other than the source and destination, and define two partition 2 matroids such that their common bases are precisely a path combined with some of the self-loops.) Problem 4. (15 points) In the lecture devoted to submodular function maximization subject to a matroid constraint, your instructor neglected the implementation details of the Pipage rounding procedure. At issue were steps (1) and (3). Step (1) requires computing the minimum-size tight set of a matroid M = (X ; I) with respect to a given point x 2 P(M) subject to including a fractional element. Step (3) requires optimally trading off xi and xj while remaining in the matroid polytope. In this problem, you will show how to fill in these blanks. (a) [2 points]. Let x be in the matroid base polytope Pbase(M), and recall that Pbase(M) ⊆ P(M). Show that the family of tight sets fS ⊆ X : x(S) = rankM(S)g forms a lattice. Conclude that, for each i 2 X , there is a tight set Ti with i 2 Ti such that every tight set containing i is a superset of Ti; we say Ti is the minimum tight set containing i. (b) [8 points]. Show how to efficiently implement step (1) of the Pipage rounding procedure. Your algorithm should compute T in time polynomial in jX j and the bit complexity hxi of the solution x. You should assume that you have access to an independence oracle for the matroid M. (Hint: Reduce this to computing Ti for each i 2 X . To compute Ti, use a greedy algorithm which starts with R = X , and iteratively removes elements from R until it ends up with Ti. Your algorithm will use submodular minimization as a subroutine.) (c) [5 points]. Show how to efficiently implement step (3) of the Pipage rounding procedure. Given x 2 P(M) and i; j 2 X , your algorithm should compute µ 2 [0; 1] maximizing F (x + µ(ei − ej)), where F is the multilinear extension of a submodular function, in time polynomial in jX j and hxi. As usual, you may assume access to an independence oracle for M. Moreover, for purposes of this problem you may assume that you can exactly evaluate F in constant time. Problem 5. (8 points) The following is a (simplification of) a task encountered in speech recognition. Here, we wish to select a small vocabulary (a set of words) which express many common phrases. Specifically, you are given a set P of phrases which, collectively, use only the words in the set W . Moreover, each phrase p 2 P is given a nonnegative weight w(p) indicating its importance. For a vocabulary V ⊆ W , we use P (V ) ⊆ P to denote the phrases which use only words in V . Your goal is to choose a vocabulary optimizing a tradeoff between the total weight of expressed phrases and the size of the P vocabulary; specifically, find V ⊆ W maximizing p2P (V ) w(p) − jV j. Show that this problem is solvable in poly(jP j; jW j) time. You may assume that you can test whether a particular word occurs in a particular phrase in constant time. Problem 6. (12 points) Sensor networks are frequently deployed for sensing the environment (e.g. temperature or pressure). Consider the following simplified scenario: There are n sensors S and m reference locations L. For ` 2 L and s 2 S, you are given a coverage quality q(s; `) 2 R+. We assume there are T time periods, and your goal is to select a subset of the sensors Xt ⊆ S to \activate" during each time period t 2 f1;:::;T g.
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