Approximation Algorithms: Solving NP-Hard Problems with Suboptimal Solutions - Prof. Eric , Study notes of Computer Science

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Approximation Algorithms • Main issues – Finding a good lower or upper bound for the optimal solution value – Relating your algorithm’s solution value to this bound – Examples: Scheduling and Vertex Cover • Basic Setting – Working with an optimization problem, not a decision problem, where you are trying to minimize or maximize some value – Examples: ∗ Vertex cover · Input: Graph G = (V, E) · Task: Find a vertex cover of minimum size · Value to be minimized: vertex cover size ∗ Independent Set · Input: Graph G = (V, E) · Task: Find an independent set of maximum size · Value to be maximized: independent set size – The problems are NP-hard ∗ The decision version of all these problems is NP-complete ∗ Thus, we probably cannot find a polynomial time algorithm that solves the problem optimally for all input instances. – Notation ∗ Let Π be the problem under consideration ∗ Let I be an input instance of Π ∗ let OPT denote the optimal algorithm ∗ Let A denote the algorithm under consideration ∗ For any algorithm A and any input instance I, let A(I) denote the value of A’s solution for input instance I – Absolute approximation of c (c-absolute-approximation algorithm) ∗ ∃c such that ∀IA(I) ≤ OPT (I) + c for minimization problem Π · A is a 2-absolute-approximation algorithm for vertex cover if A can always find a vertex cover at most 2 nodes larger than the optimal size. ∗ ∃c such that ∀IA(I) ≥ OPT (I)− c for maximization problem Π · A is a 2-absolute-approximation algorithm for independent set if A can always find an independent set at most 2 nodes smaller than the optimal size. ∗ Very few NP-hard problems have polynomial-time absolute approximation algorithms 1 – Relative approximation (c-approximation algorithm) ∗ ∃c such that ∀IA(I) ≤ c×OPT (I) for minimization problem Π · For example, A is a 2-approximation algorithm for vertex cover if A can always find a vertex cover of at most twice the optimal size. ∗ ∃c such that ∀IA(I) ≥ (1/c)×OPT (I) for maximization problem Π · For example, A is a 2-approximation algorithm for independent set if A can always find an independent set of at least half the optimal size. ∗ Most of our focus is finding good relative approximation algorithms; unless explicitly stated otherwise, when I talk about approximation algorithms, I mean relative approximation algorithms • Example: Multiprocessor scheduling to minimize makespan – Input ∗ Set of n jobs with processing times xn ∗ Number of machines m – Task ∗ Schedule the jobs on the m machines with the goal of minimizing the makespan (maximum completion time of any job) of the schedule. – Initial thoughts about this problem ∗ Suppose m = 1: is this problem hard? ∗ We know this problem is hard for m ≥ 2 because of a reduction from the partition problem. • Graham’s list scheduling algorithm (Greedy) – Take the items in an arbitrary order – Place each item on the currently least loaded machine – This is a greedy algorithm • Proof of (2− 1/m)-approximation factor – Greedy’s cost ∗ Let job j be the last job to complete by Greedy ∗ Let h be the load of the machine j is placed onto before job j is placed ∗ Greedy’s cost is h + xj – Bounds on OPT (I) ∗ OPT (I) ≥ maxj xj ∗ OPT (I) ≥ 1 m ∑n i=1 xi – Relating the two costs ∗ h ≤ 1 m (( ∑n i=1 xi)− xj) · This follows because greedy places j onto the least loaded machine available 2 – OPT (I) ≥ M ∗ This follows because each edge in the matching must be covered by at least one node ∗ Since no two edges in the matching share a node, at least one of the two nodes from each edge must be chosen. • Example: Bin Packing – Input ∗ Set of n objects of sizes si ∗ Infinite number of bins of size B – Task ∗ Find a packing of the n objects into the minimum possible number of bins – Bin packing is NP-complete ∗ Can reuse the reduction from Partition to Makespan Scheduling • Algorithms – First Fit Algorithm ∗ Arbitrarily order the items ∗ Number the bins by the order they are opened (initially no bins are open) ∗ When working with item i, place it into the first open bin it fits into. If it does not fit into any open bin, open a new bin and place it into that bin. – Best Fit Algorithm ∗ Arbitrarily order the items ∗ When working with item i, place it into the open bin it fits most tightly into. If it does not fit into any open bin, open a new bin and place it into that bin. – First Fit Decreasing Algorithm ∗ Sort the items into non-decreasing order by size ∗ Number the bins by the order they are opened (initially no bins are open) ∗ When working with item i, place it into the first open bin it fits into. If it does not fit into any open bin, open a new bin and place it into that bin. – Best Fit Decreasing Algorithm ∗ Sort the items into non-decreasing order by size ∗ When working with item i, place it into the open bin it fits most tightly into. If it does not fit into any open bin, open a new bin and place it into that bin. – Proof that all algorithms are at least 2-approximations ∗ There is at most 1 open bin that is more than half empty · If not, any of the algorithms would have combined the items in the two at least half-empty bins into 1 bin ∗ Thus, an upper bound on the number of bins (ignoring the one possibly half- empty bin) used by any of these algorithms is 2 1 B ∑n i=1 si 5 ∗ Clearly, OPT (I) ≥ 1 B ∑n i=1 si ∗ The result of 2 follows. – FF and BF have approximation ratios of 17/10 ignoring a constant additive factor – FFD and BFD have approxmation ratios of 11/9 ignoring a constant additive factor – The proofs of these results are very long and involved. • Example: Traveling Salesperson – Input ∗ List of n cities ∗ Distances d(i, j) between each pair of cities – Task ∗ Find a tour of the cities of minimum possible total length – No polynomial-time c-approximation algorithm exists for this problem unless P = NP ∗ If there is a polynomial-time c-approximation algorithm for this problem, then Hamiltonian Cycle can be solved in polynomial time. ∗ Take an arbitrary input instance (graph G = (V, E)) of Hamiltonian Cycle and turn it into a TSP problem as follows: · For each node in V , create a city · For each edge (i, j) ∈ E, set d(i, j) = 1 · For each pair of vertices (i, j) 6∈ E, set d(i, j) = nc · If G has a Hamiltonian cycle, then the optimal tour in the TSP instance is n · If G does not have a Hamiltonian cycle, then the optimal tour in the TSP instance has length at least n− 1 + nc ∗ Apply our c-approximation algorithm to the TSP instance · If our approximation algorithm for TSP returns an answer of at most nc, then we know our original graph had a Hamiltonian cycle. · If our approximation algorithm for TSP returns an answer greater than nc, then we know our original graph did not have a Hamiltonian cycle. · Thus, we can solve the Hamiltonian cycle problem in polynomial time using our c-approximation algorithm for TSP. • Example: Metric Traveling Salesperson – Input ∗ List of n cities ∗ Distances d(i, j) between each pair of cities · Distances satisfy triangle inequality: d(i, j) + d(j, k) ≤ d(i, k) · Note that the distances in our proof above that unrestricted TSP is hard to approximate do not satisfy this constraint 6 – Task ∗ Find a tour of the cities of minimum possible total length • Greedy algorithm – Best guarantee is O(log n)OPT (I) • MST algorithm – Find a minimum spanning tree T . – Double all the edges in T to create an Eulerian graph. – Take an Euler tour of the graph using shortcuts to avoid visiting cities more than once. ∗ Shortcuts cannot increase cost because of triangle inequality • Proof of 2-approximation ratio – Let C(T ) be the total weight of the minimum spanning tree T – OPT (I) ≥ C(T ) ∗ Remove an edge from the optimal tour. ∗ We now have a path which is one possible spanning tree. ∗ The minimum spanning tree must have cost no more than this path. – MST (I) ≤ 2C(T ) ∗ The Eulerian tour has cost exactly 2C(T ) ∗ When taking shortcuts, this cost may decrease but cannot increase because of triangle inequality • Christofides’ matching improvement – Find a minimum spanning tree T . – Find all the nodes in T with odd degree – Find a minimum weight matching M of these nodes – Create an Eulerian graph by adding the edges in M to those in T – Take an Euler tour of the graph using shortcuts to avoid visiting cities more than once. • Proof of 3/2-approximation ratio – Let C(T ) be the total weight of the minimum spanning tree T and let C(M) be the total weight of M – OPT (I) ≥ C(T ) ∗ Remove an edge from the optimal tour. ∗ We now have a path which is one possible spanning tree. ∗ The minimum spanning tree must have cost no more than this path. – 1/2OPT (I) ≥ C(M) 7