[1][1][1][1] Lecture 2-3: Coping with NP-Hardness of Optimization Problems in Practice May 26 + June 1, Introduction to Algorithmic Wireless Communications David Amzallag

[2][2][2][2] Coping with NP-hardness of optimization problems – Heuristics – DSATUR for graph coloring (Brélaz, 1979) – Lin-Kernighan for TSP (1973) NP – Common solutions for NP-hard problems in practice – Somewhat efficient exponential algorithms – 3-Coloring in time (Eppstein, 2001) – Easy special cases (coloring planar graphs, hexagonal graphs) – Approximation algorithms

[3][3][3][3] What is an approximation algorithm? – The performance guarantee of an approximation algorithm for a minimization (maximization) problem is  if the algorithm is guaranteed to deliver a solution whose value is at most (at least)  times the optimal value – An  -approximation algorithm is a polynomial time algorithm with a performance guarantee of  NP – Isn't computing the cost of the optimal solution NP-hard?

[4][4][4][4] The art of lower bounding the optimum – For minimization problems we know that a  -approximation algorithm A has cost where is the cost of the optimal solution can be difficult – Relating directly to can be difficult – Instead we can relate the two in two steps: where is a lower bound on the optimal solution and then

[5][5][5][5] What's Radio Access Network? User Equipment RAN (e.g., UTRAN) Core Network External Networks

[6][6][6][6] B A Radio Network Controllers (RNCs) Base Stations Mobile Stations Star Topology Access Network Tree Topology Access Network

[7][7][7][7] The routing cost of a tree expected traffic loadrouting cost Given a spanning tree T of a set of base stations, each with its expected traffic load, rooted at RNC r, the routing cost,, between the RNC r and a base station i is the weighted sum of the costs along the (unique) path between then in T. routing cost of a tree The routing cost of a tree T is defined as Routing cost of = 10 Routing cost of 1+2 = Root — Routing cost of the whole tree is 46 — The cost of a minimum spanning tree is 24 Illustrated with These two objectives can be far from each other by a factor of O(n)! Vendors are usually planning RANs using MST-based techniques… Max (“Where the Wild Things Are”, Maurice Sendak)

[8][8][8][8] The bounded-degree minimum routing cost spanning tree problem – A set of base stations, and a set of RNCs symmetric connection cost – A symmetric connection cost is defined for all – Another given cost is of connecting base station to RNC. – Degree-constraints for every – The BDRT problem – The BDRT problem is to find a minimum routing cost spanning tree on rooted at a given root that meets for every

[9][9][9][9] An observation – A typical solution to BDRT is a forest of multiple trees, each is rooted at a RNC node – This problem is reducible to the problem with a single tree

[10] Some hardness results NP – BDRT is NP-hard (easy to verify) – What about approximation algorithms? Not a good idea! NP-hard to approximatePNP – BDRT is NP-hard to approximate, unless P=NP – A practical avenue metric – A practical avenue: metric BDRT with for every

[11] Shortest-path trees shortest-path tree – A shortest-path tree rooted at vertex r is a spanning tree such that for any vertex v, the (shortest) distance between r and v is the same as in the graph – Shortest-path trees are fundamental structures in the study of graph algorithms (e.g., Dijkstra), and of the most useful `algorithmic’ tool in telecommunications (e.g., routing) not – Notice that the total cost of a shortest-path tree is not unique

[12] SPT of cost 9 SPT of cost 8 MST of cost 7 Observation Observation. When the metric case is considered SPT is unique. Lower bound for the optimum Lower bound for the optimum. The total cost of the SPT (with the same root).

[13] An approximation algorithm 1.Construct a shortest-path tree, with root r, on the input complete (metric) graph; renumber the vertices so that is the i th closest vertex to the root. 2.Set as the root of the tree T. 3.While T is not a spanning tree do Pick the vertices of least indices and assign them, from the left most son to the rightmost, as the children of vertex where for and otherwise. Set

[14] Before analyzing: bounding the contribution of a vertex

[15] Analysis – In general, if is the last non-leaf vertex, then – Since the cost of the SPT of G (with as its root), as computed by the algorithm, is a lower bound of the optimal solution, – In addition, remember that T has leaves and hence, – So,

[16] Heuristics greedy – Family of greedy algorithms: best Start at the root vertex r, pick the best vertices as its children, from the left most child to the rightmost one, and then move to the next level in the constructed tree. For every vertex v ( ), the algorithm pick the best unpicked vertices and assign them as its children, until spanning all the vertices of G.

[17] – Picks the vertices v in an increasing order of their ratio where is the cost of the shortest-path connecting v to r in the constructed tree T. – Unexpected warning – Unexpected warning: Algorithm may produce solutions that are far from the optimum by on our instances. Picks the vertices v in an increasing order of their ratio