TTIC 31010 and CMSC 37000 Algorithms
Winter 2017
Homework set 4
Due: Tuesday March 7, noon
• Please read the homework policy on the course webpage before preparing your homework.
• An extra 5/100 points will be added to your grade if the solution is completely typed.
• You can rely on statements in sub-questions (even if you cannot prove them) when solving
subsequent sub-questions. You can also use all results that were proven in class without reproving them. Additionally, you can use efficient algorithms for solving Linear Programs as
black-boxes.
• Please do not write code. A clear description of algorithms in plain English, or in pseudo-code, is
sufficient. If you choose to write pseudo-code, please keep it as close to plain English as possible.
Question 1 (20 points) Suppose we are given a satisfiable system of linear inequalities over the
variables x1 , . . . , xn :
a1,1 x1 + a1,2 x2 + · · · + a1,n xn ≤ b1
a2,1 x1 + a2,2 x2 + · · · + a2,n xn ≤ b2
..
.
am,1 x1 + am,2 x2 + · · · + am,n xn ≤ bm
(that is, we are guaranteed that the system has a feasible solution).
We say that the jth inequality is forced-equal iff in every
P feasible solution (x1 , . . . , xn ) to the system,
the inequality is satisfied with equality. Equivalently, i aji xi ≤ bP
j is not forced-equal if there is a
feasible solution (x1 , . . . , xn ) to the system of inequalities in which i aji xi < bj .
A solution x to the system is called characteristic if, for every inequality I that is not forced-equal, x
satisfies I without equality.
a. Prove that any satisfiable system has a characteristic solution.
b. Show an efficient algorithm, that, given a satisfiable system of linear inequalities, determines
which inequalities are forced-equal, and finds a characteristic solution.
Question 2 (25 points) In the minimum-cost perfect bipartite matching problem, we are given a
bipartite graph G = (U, V, E) with non-negative P
weights we on its edges e ∈ E. The goal is to find a
perfect matching M ⊆ E of minimum total cost e∈M we . For each vertex v ∈ U ∪ V , we denote by
δ(v) the set of all edges incident to v. Consider the following linear program:
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(LP)
min
P
e∈E
we x e
s.t.
P
e∈δ(v) xe
= 1 ∀v ∈ V ∪ U
xe ≥ 0
∀e ∈ E
a. Show that (LP) is a relaxation of the minimum-cost perfect bipartite matching problem.
b. We say that a solution x to (LP) is strictly fractional iff there is at least one edge e ∈ E with
0 < xe < 1. Prove that for every strictly fractional solution x, there is a cycle C in G with an
even number of edges, such that for each edge e of C, 0 < xe < 1.
c. Let x be a strictly fractional solution for (LP), and let C be a cycle like in the previous subquestion. Assume that C contains 2k edges e1 , e2 , . . . , e2k that appear on C in this order. Prove
that there is a value 0 < < 1, such that, if we add to xei for all even indices i, and subtract
from xei for all odd indices i, then we will obtain another feasible solution to (LP).
d. Show an efficient algorithm that, given a strictly fractional feasible solution x to (LP), transforms
it into another feasible solution x0 , such that: (i) x0 has fewer fractional coordinates than x; and
(ii) the objective function value of x0 is no greater than that of x.
e. What is the integrality gap of (LP)? Explain your answer.
f. Compute the dual of (LP). Give the necessary and the sufficient conditions under which the dual
has a finite feasible optimal solution. Explain your answer.
Question 3 (15 points) A SAT formula ϕ is called monotone, iff each clause in ϕ only contains
non-negated variables. For example, (x1 ∨ x3 ∨ x2 ) ∧ (x1 ∨ x5 ∨ x7 ) ∧ (x1 ∨ x4 ∨ x6 ) is a monotone
formula, while (x1 ∨ x2 ∨ x3 ) ∧ (x1 ∨ x4 ) is not. Notice that a monotone SAT formula always has a
satisfying assignment — the one in which the values of all variables are set to TRUE. In the Restricted
Monotone Satisfiability problem, we are given a monotone SAT formula ϕ and an integer k, and we
need to establish whether there is a satisfying assignment to ϕ, in which exactly k of the variables
are assigned the value TRUE, and the rest of the variables are set to FALSE. Prove that Restricted
Monotone Satisfiability is NP-complete.
Question 4 (20 points) Consider the following solitaire game. The puzzle consists of an (n × m)
grid of squares, where each square may be empty, occupied by a red stone, or occupied by a blue stone.
The goal of the puzzle is to remove some of the given stones so that the remaining stones satisfy two
conditions: (1) every row contains at least one stone, and (2) no column contains stones of both colors.
For some initial configurations of stones, reaching this goal is impossible. Prove that the problem of
determining, given an initial configuration of red and blue stones, whether the puzzle can be solved,
is NP-complete.
Question 5 (20 points) In the Maximum Cut problem, the input is an undirected graph G =
(V, E), and the goal is to find a partition (A, B) of V (called a cut), maximizing the size of the cut
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— the number of edges with one endpoint in A and another in B. Unlike the Minimum Cut problem,
this problem is NP-hard. Let m denote the number of edges in G.
a. Consider the following simple randomized algorithm: for each vertex v ∈ V , we independently
decide to add v to A or to B, with probability 1/2 each. What is the expected size of the
resulting cut? What lower bound does it give on the size of the maximum cut in G?
b. Consider the following deterministic algorithm for the problem. Start with an arbitrary partition
(A, B) of V , and then iterate. While there is a vertex v, such that moving v from its current
side of the cut to the opposite side increases the cut value, move v to the opposite side of the
cut.
(i) Prove that the algorithm terminates after O(m) iterations.
(ii) Prove that the algorithm obtains a factor-2 approximation for the Maximum Cut problem.
Question 6 (10 points extra credit) A subset S of vertices in an undirected graph G is called
triangle-free if, for every triple of vertices u, v, w ∈ S, at least one of the three edges (uv), (uw), (vw)
is absent from G. Prove that finding the size of the largest triangle-free subset of vertices in a given
undirected graph is NP-hard.
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