Approximation Algorithms

Approximation Algorithms Department of Mathematics and Computer Science Drexel University

Today’s Lecture: • Simple Algorithm for Bin-Packing • Maximum Cut • Approximation algorithm for graph coloring • Approximation algorithm for set cover

The Bin-Packing Problem: • INSTANCE: Finite set I of rational numbers {a1,…,an} with ai (0,1] • SOLUTION: Partition {B1,…,Bk} of I into k bins such that the sum of the numbers in each bin is at most 1 • MEASURE: Cardinality of the partition, i.e., k

The Algorithm: • Polynomial-time 2-approximation algorithm for MINIMUM BIN PACKING (Next Fit algorithm) begin for each number a if a fits into the last open bin then assign a to this bin else open new bin and assign a to this bin return f end.

Theorem: • Number of bins used by the algorithm is at most 2A, where A is the sum of all numbers. • Proof: • For each pair of consecutive bins, the sum of the number included in these two bins is greater than 1. • Each feasible solution uses at least A bins • Best case each bin is full (i.e., the sum of its numbers is 1). • Performance ratio is at most 2.

Tightness: • Let I={1/2,1/2n,1/2,1/2n,…,1/2,1/2n} contain 4n items.

Maximum Cut: • INSTANCE: Graph G=(V,E) • SOLUTION: Partition of V into disjoint sets V1 and V2 • MEASURE: Cardinality of the cut, i.e., the number of edges with one endpoint in V1 and one endpoint in V2

The Algorithm: • Polynomial-time 2-approximation algorithm for MAXIMUM CUT begin V1:=ø; repeat if exchanging one node between V1 and V2 =V- V1 improves the cut then perform the exchange; until a local optimum is reached; return f end.

Theorem: • We prove that any local optimum contains at least half of the m edges • Notation: • c= # of edges of the cut • i= # of edges inside V1 • o= # of edges outside V1 • m=c+i+o, that is, i+o=m-c • For any node v, • i(v)= # of edges between v and a node in V1 • o(v)= # of edges between v and a node not in V1

Proof: • V1 is a local optimum: for any vV1, i(v)-o(v)  0 and, for any v not in V1, o(v)-i(v)  0. • Summing over all nodes in V1, we have 2i-c  0 • Summing over all nodes not in V1, we have 2o-c  0 • That is, i+o-c  0 • That is, m-2c  0 • That is, c m/2

Graph Vertex Coloring: • Given a graph G=(V,E), on n vertices and m edges, find a mapping from vertex set to k-colors such that no two end-points of an edge will receive the same color. • The Decision problem: • Given an instance of GVC, and an integer k* , is it possible to k* color the vertices of the graph? • Optimization problem: • Given an instance of GVC, find the smallest possible value for k* , for which a vertex coloring of G exists. • Input-dependent Optimization problem: • Same assumptions as optimization version, except we would like to use as few colors as possible to find the actual mapping.

The Algorithm: • The following is a polynomial-time n/logn-approximation algorithm for MINIMUM GRAPH COLORING begin i:=0; U:=V; while U ødo begin i:=i+1;V[i]:=ø;W:=U;H:=graph induced by W; while W  ø do begin v=node of minimum degree in H; insert v inV[i]; delete v and its neighbors from W; U:=U-V[i] end end end.

What is the Algorithm doing? • During each iteration, we will look at the left over from the graph U (initially U denotes the vertex set of G) and try to use a new colors to color a sub graph W. • Once we are done with sub-graph W, it will be removed from the graph, i.e. the graph is shrinking after this step, (U=U-V[i]). • But we will need to use new colors for the rest of U. • The inner loop will identify all vertices receiving a particular color “i”. • The outer-loop will identify number of required colors for whole G.

Performance Guarantee: • To begin with, if we know the graph is k-colorable, and we show the algorithm uses at most 3|V|/logk|V| colors: • At any iteration of the inner loop, H the induced graph on vertex set W is k-colorable (since the whole graph is k-colorable). • By pigenwhole principle, it contains an independent set of at least |W|/k nodes of degree at most |W|(k-1)/k. • As a result at least |W|/k nodes will be carried to next iteration of inner loop, i.e., during j-the iteration of inner loop the size of the set is at least |W|/kj . Consequently, there is at least logk|W|iterations for inner loop, i.e. |V[i]| is at least logk|U|

Continued: • Observe that U is a shirking set, starts with the whole V and eventually becomes empty. • At each step, the size of U goes down by at least logk|U| (the vertices receiving color i). • During the iterations for which |U|  |V|/logk|V|, we have logk|U|(1 /2)logk|V|. • As a result the first time |U| < |V|/logk|V| we have only used at most 2|V|/logk|V| in outer-loop. • To color the left-over|V|/logk|V|,we will use at most |V|/logk|V| new color. • The is the algorithm uses at most 3|V|/logk|V| colors for G.

Finally: • The algorithms uses 3n/logoptn, that is equal to 3nlog OPT/log n. • The perfoamnce gurantee of the algorithm is: 3n/log n.

Minimum Set-Cover: • INSTANCE: Collection C of subsets of a finite setS. • SOLUTION: A set cover for S, i.e., a subset C’ of C such that every element in S belongs to at least one member of C’ • MEASURE: |C’|

Johnson’s algorithm : • Polynomial-time logarithmic approximation algorithm for MINIMUM SET COVER begin U:=S; C’:=ø; for any ci do c’i := ci ; repeat i:=index of c’ with maximum cardinality; insert ci in C’; U := U-{elements of c’i }; delete all elements of ci from all c’; untilU:=ø end.

Performance Guarantee: • We prove that the performance ratio of the algorithm is at most ∑1 ik (1/i) where k is the maximum cardinality of the sets in C • Let a1 ,...,a|C’| be the sequence of indices obtained by the algorithm • Let cji be the surviving part of ci before index aj has been chosen • The intersection of ci and cjaj is equal to cji - cj+1i • li denote largest index j such that cji is not empty

Theorem 1: • For any i, • H(|ci|) 1 j |C’|(|ci Icjaj |/|cjaj|) whereH(n)= 1 i n(1/i). • Proof:

Theorem 2: For any set cover C’’, ci C’’1 j  |C’|(|ciIcjaj|/|cjaj|)  |C’| Proof:

Finally:

Approximation Algorithms