CSE 6311 – Spring 2009ADVANCED COMPUTATIONAL MODELS

AND ALGORITHMS

Lecture Notes – March 12, 2009

ILP – Integer Linear ProgrammingApproximate algorithm for ILP

Instructor: Dr. Gautam Das

notes by Walter Wilson

ILP – Integer Linear Programming

set of integer variables

linear constraints

linear goal function

(same as LP except vars are integers)

Integer Linear Programming Example

Factory product material profit

Product1 w1 grams metal per unit p1 dollars per unit

Product2 w2 " p2 "

Constraints:

total # of products per day <= P

total amount of material per day <= W

Unknowns:

x1 - # of units of Product1

x2 - " Product2

Goal function:

maximize profit: x1 p1 + x2 p2

Constraint equations:

x1 + x2 <= P

x1 w1 + x2 w2 <= W

x1 >= 0, x2 >= 0, x1,x2 integers!

ILP Example 2 – Shortest PathsGiven weighted directed graph and start, end nodes s & f,

find weight of shortest path from start to end.Let wj >= 0 be integer weight for each edge j

ui >= 0 be weight of shortest path from s to node uiConsider shortest path from s to node v:

s

u1

u2

w1

w2v

u1

u2

Shortest path to v in terms of path to preceding node uk:

v <= uk + wk (edge (uk,v) has weight wk)

v >= 0

Goal: minimize f (shortest path to f)

ILP Decision ProblemIs there a var asnmt s.t. goal <= C

Proof that ILP (decision problem) is NP-Complete –Reduction from Vertex Cover:Given unweighted graph G and k does there exist a vertex cover of size <= k?for each node vi, make constraints xi>=0, xi<=1 for each edge (vi,vj) make constraint xi + xj >= 1goal: minimize sum of xi

-- xi == 1 means xi selected for vertex cover-- goal <= k means V.C. <= k

Approximation Algorithm for ILP Vertex Cover• Algorithm:

– Treat as LP problem• Will get values 0.0 to 1.0 for vertices (VChypoth)

– VChypoth <= VCopt – LP solution more optimal than ILP

• One possible algorithm: take vertices in decreasing order

– Stop when cover achieved

• Simpler: round to 0 or 1 (.5 rounds up)– Is this a vertex cover?

» Yes since sum of values for each edge >= 1– Approximation bound:

» VChypoth = sum of x's < .5 + sum of x's >= .5» (left vars round to 0, right round to 1)» VCapprox = sum of rounded vars <= 2 * unrounded» Thus: VCapprox <= 2 * VChypoth» Thus: VCapprox <= 2 * VCopt

Weighted Vertex Cover

• Graph with weighted nodes

• Find vertex cover that touches all edges but minimizes the sum of the weights

• ILP problem– Goal: sum i=1..n xi wi– How to do rounding? -- do same way– How to prove approximation ratio?

• VCapprox <= 2 VCopt