On (In)Tractability of Movement Problems

1. On (In)Tractability of Movement Problems Course Project for CMSC 858F Team : Rajesh Chitnis, M. Reza Khani and VahidLiaghat

2. Outline of our presentation • Rajesh – present the framework of movement problems introduced by Demaine et al. [SODA 07] and show that various polytimesolvable problems become hard if we allow movement. • Vahid – present some results we obtained in our project in the framework of Demaine et al. [SODA 07] • Reza – define a new model of Firefighter problems and present some new results in that framework.

3. Introduction – Movement Problems • Various real-life scenarios involve planning the coordinated motion of mobile agents in order to achieve certain tasks. • Examples are firefighters responding to fires, SWAT teams responding to emergency situations, etc. • In real-life the number of mobile agents is quite small and they have to move in a vast terrain. • We desire the most efficient way to plan the motion of the agents in order to achieve our goals. • Efficiency can be measured in various parameters : time taken, energy spent, etc.

4. Example – Firefighters

5. Example – Firefighters • Consider a group of firefighters responding to a fire where each firefighter has a fixed initial position. • We assume each firefighter has a walkie-talkie and they must communicate with all others to fight the fire effectively. • Hence we need the firefighters to induce a connected graph, i.e., each one has a route to communicate with everybody else. • We want to minimize the time required for the firefighters to form a connected graph, i.e., minimize the maximum movement needed for the firefighters to induce a connected graph assuming that each firefighter has constant speed.

6. General Framework • The previous slide asks us to minimize the maximum movement of firefighters to induce a connected graph. • “Minimize the maximum movement” is the efficiency criterion. • We can look at other efficiency criteria like “minimize the total movement” (which is related to average consumption) or “minimize the number of agents which need to move”. • “Induce a connected graph” is the property we desire the agents to have in final configuration. • We can desire other properties like “induce an independent set”, “graph induced by agents has directed connectivity from each vertex to some root” or “induced graph has a perfect matching”. • This framework was introduced by Demaine et al. [SODA 07]

7. Results of Demaine, Hajiaghayi, Mahini, Sayedi-Roshkar, Oveisgharan and Zadimoghaddam [SODA 07] Constant-factor approximations were given for ConMax and PathMax (and some other problems) by Berman, Demaine and Zadimoghaddam [APPROX 11]

8. Movement makes the problems difficult • Consider the problem ConMax. Just checking if a given set of agents induces a connected subgraph is polytime solvable. Take-home Message: Allowing the movement of agents makes the problems computationally hard • Theorem : ConMax is NP-complete (even to approximate better than factor 2). • Proof : On next slide.

9. ConMax is NP-complete (even to approximate better than 2) • Consider an instance of Hamiltonian Path. • First split each edge into a path of length 3. • Attach a leaf to each vertex of . • Place 2 agents on each vertex of and 1 agent on each leaf. • Consider this instance of ConMax. • We claim that cost of ConMax on is 1 if and only if has a Hamiltonian path.

10. ConMax is NP-complete (even to approximate better than 2) • If has a Hamiltonian path say then we give a solution for ConMax of cost 1 as follows : • For each , the leaf attached to xi shifts its agent to • For each edge in P we make and shift one pebble each to the path of length 3 between them. • This clearly gives a connected graph on the agents such that each agent moves at most 1.

11. ConMax is NP-complete (even to approximate better than 2) • Now suppose there is a solution of ConMax of cost 1. • WLOG let the agent on each leaf move to its corresponding vertex in . Each vertex of now has 3 agents. • So now all vertices of must be connected in the graph induced by the agents. • Therefore there is a tree containing all vertices of . • If a vertex of has degree at least 3 in the tree then that will be a contradiction since this means that he gave all his agents away ( agents of a vertex of cannot reach another vertex of with movement at most 1 ) • And so we have a Hamiltonian Path in .

12. PathMax is NP-hard G has a Hamiltonian path if and only if PathMax has a solution of cost n

13. PathSum is NP-hard G has a Hamiltonian path if and only if PathSum has a solution of cost n(n+2)

14. Mobile Facility Location • We have two types of agents : clients and facilities. • Given an initial location of clients and facilities we want to minimize the maximum movement needed for any agent in arriving at a final configuration such that every client is co-located with some facility. • Simple 2-approximation : Do not move the facilities. Just move each client to his closest facility. [Demaine et al. SODA 07] • Surprisingly this bound is also tight !! [Friggstad and Salvatipour FOCS 08] • Proof on next slide

15. No () approximation for minimizing max. movement unless P=NP Complete Bipartite Graph Facilities = Clients = G has a Vertex Cover of size k if and only if there is a solution with maximum movement 1