P NP. Approximation Algorithms. Computational hardness. Vertex cover. Vertex cover. Vertex cover. Plan: Vertex Cover Metric TSP 3SAT

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1 Great Theoretical Ideas In Coputer Science Victor Adachi CS - Carnegie Mellon University Approxiation Algoriths P NP Plan: Vertex Cover 3SAT Coputational hardness Suppose we are given an NP-coplete proble to solve. Can we develop polynoial-tie algoriths that always produce a good enough" solution? Given G=(V,E), find the sallest S V s.t. every edge is incident on a vertex in S. NP-copete proble. Lea. Let M be a atching in G, and S be a vertex cover, then S M. Proof. S ust cover at least one vertex for each edge in M. Def. A atching M is axial if there is no atching M such that M M. Which of the following algos. would find a axial atching: a) Greedily add edges that are disjoint fro the edges added so far, while such edges exist b) Copute a axiu atching c) Both d) Neither

2 Approxiation Vertex Cover Approx-VC(G): M axial atching on G S tae both endpoints of edges in M Return S Theore. Let OPT(G) be the size of the optial vertex cover and S = Approx-VC(G). Then S OPT(G) Proof. S = M OPT(G) Approxiation Vertex Cover Theore. Let OPT(G) be the size of the optial vertex cover and S = Approx- VC(G). Then S OPT(G) Can we do better than?? Fact. Nobody nows any algorith with approxiation ratio.9 Approxiation Vertex Cover Is a tight bound for this algorith? Consider a coplete bipartite graph K n,n What is the size of the optial solution OPT(K n,n )? n What is the size of any axial atching M(K n,n )? n Approx-VC(K n,n ) = n Foral Definition Let P be a iniization proble, and I be an instance of P. Let ALG(I) be a solution returned by an algorith, and let OPT(I) be an optial solution. Then ALG(I) is said to be a c-approxiation algorith, if for I, ALG(I) c OPT(I). These notions allow us to circuvent NP-hardness by designing polynoial-tie algos with foral worst-case guarantees! Traveling Salesan Proble Given a coplete undirected graph G=(V,E) with edge cost c:e R +, find a in cost Hailtonian cycle (HC). Clai: TSP is NP-hard. Proof by reduction fro a HC which is NP-Coplete. Given the input G=(V,E) to HC, we odify it to construct a coplete graph G =(V, E ) and cost function as follows: c(u,v) = 0, if edge (u,v) E c(u,v) =, otherwise. G has a HC iff TSP(G ) = 0 We are allowed to visit vertices ultiple ties. We construct a new graph with an edge between every pair of nodes with length equal to the length of the shortest path between the. The shortest path fors a etric: c(u, v) 0, c(v, v) = 0 c(u, v) = c(v, u), c(u, v) c(u, w) + c(w, v) Clai: is NP-hard.

3 Traveling salesan proble Approxiation Algorith Approx-TSP(G): ) Find a MST of G ) Coplete an Euler tour by doubling edges 3) Reove ultiply visited edges (shortcuts) The largest solved TSP (as of 03), an,900-vertex route calculated in 00. The graph corresponds to the design of a custoized coputer chip created at Bell Laboratories, and the solution exhibits the shortest path for a laser to follow as it sculpts the chip. 0 3 Approxiation Metric-TSP Theore. Approx-TSP is a -approxiation algorith for a etric TSP. Proof. Approx-TSP Euler Tour = MST OPT shortcutting decreases the cost. doubling edges we can get a spanning tree fro HC by reoving edges Observe that a factor in the approxiation ratio is due to doubling edges; we did this in order to obtain an Eulerian tour. But any graph with even degrees vertices has an Eulerian tour. Thus we have to add edges only between odd degree vertices Approx-C(G): T MST of G S vertices of odd degree in T M in-cost atching on S Return: Euler Tour T M Theore. Christofides is 3/ approxiation for The algo has been nown for over 30 years and yet no iproveents have been ade since its discovery Proof. ALG = c(m) + c(t) We now that c(t) OPT. It reains * Not for to show the exa c(m) ½ OPT. 3

4 Traveling Salesan Proble Lea. c(m) ½ OPT Proof. Consider two feasible atching: M and M. Note, S is even. Thus, c(m) ½ (c(m ) + c(m )) Since c(m ) + c(m ) OPT It follows, c(m) ½ OPT * Not for the exa M 3 M 9 M M Theore: If P NP, then for c> there is NO a poly-tie c-approxiation of general TSP. Proof. To show Ha-cycle p c-approx TSP. Start with G and create a new coplete graph G with the cost function c(u,v) =, if (u,v) E c(u,v) = c n, otherwise (where n = V ) If G has HC, then TSP(G ) = n. If G has no HC, then TSP(G ) (n-) + c n c n Since the TSP differs by a factor c, our approx. algorith can be able to distinguish between two cases, thus decide if G has a ha-cycle. MAX-SAT Given a CNF forula (lie in SAT), try to axiize the nuber of clauses satisfied. CNF is a conjunction of clauses, where each clause is a disjunction of literals (X X X ). Faous NP-coplete proble. satisfy at least a / fraction of clauses. Proof. Try a rando assignent to the variables. Pr[clause is false] =? Since there is only one out of cobinations that can ae it false, the probability of the clause being false is /. satisfy at least a / fraction of clauses. Proof. (cont) So if there are clauses total, the expected nuber satisfied is (/). If the assignent satisfies less, just repeat. With high probability it won't tae too any tries before you do at least as well as the expectation. With high probability it won't tae too any tries before you do at least as well as the expectation. Proof. (cont) Let Z be the rando variable denoting the nuber of clauses satisfied by a rando assignent. p p Let p = Pr[Z = ] ( - ) p E[Z] It follows, 0 p p p 0 / 0 / / p / ( - ) p is the probability that a rando assignent satisfies at least / clauses. p

5 satisfy at least a / fraction of clauses. Approxiation Algoriths for: Vertex Cover 3SAT Theore (Hastad, 99). If there is an c-approxiation with c >/, then P = NP. Here s What You Need to Know