Traveling Salesman Problems Motivated by Robot Navigation based on joint work with Avrim Blum, Shuchi Chawla, David Karger, Terran Lane, Adam Meyerson

A Robot Navigation Problem Robot delivering packages in a building Goal to deliver as quickly as possible Classic model: Traveling Salesman Problem find a tour of minimum length Additional constraints: some packages have higher priority robot’s lifetime is uncertain power loss catastrophic failure

Exponential Discounting Markov Decision Process approach assign reward ri to package i robot receives discounted reward gt ri for delivering package i at time t Motivates to deliver high-priority packages quickly Inflation: reward collected in distant future decreases in value due to uncertainty at time t robot loses power with fixed probability probability of being alive at t is exponentially distributed discounting reflects value of reward in expectation

Discounted-Reward TSP Given undirected graph G=(V,E) edge weights (travel times) de ≥ 0 weights on nodes (rewards) rv ≥ 0 discount factor   (0,1) root node s Goal find a path P starting at s that maximizes total discounted reward (P) = vP rv dP(v)

Approximation Algorithms Discounted-Reward TSP is NP-hard reduction from minimum latency TSP So intractable to solve exactly Goal: approximation algorithm that is guaranteed to collect at least some constant fraction of the best possible discounted reward

Related Problems Goal of Discounted-Reward TSP seems to be to find a “short” path that collects “lots” of reward k-TSP and k-path Find a tour (path) of minimum length that starts at a given node s and visits at least k vertices (2+)-approximation algorithm for k-TSP [AK’00] (2+)-approximation algorithm for k-path follows from work of Chaudhuri et al. [CGRT’03] Mismatch: constant factor approximation on length doesn’t exponentiate well

Orienteering Problem Find a path of length at most D that maximizes net reward collected Complement of k-path approximates reward collected instead of length, so exponentiation doesn’t hurt unrooted case can be solved via k-TSP or k-path Drawback no constant factor approximation for rooted non-geometric version previously known

Our Results Using -approximation for k-path as subroutine 3/2 +1/2 -approximation for Orienteering e(3/2  + 1/2)-approximation for Discounted-Reward TSP constant-factor approximations for tree- and multiple-path versions of the problems

Our Results Using -approximation for k-path as subroutine substitute = 2+ from Chaudhuri et al. 3/2 +1/4-approximation for Orienteering e(3/2  + 8.5 -approximation for Discounted-Reward TSP constant-factor approximations for tree- and multiple-path versions of the problems

Eliminating Exponentiation Let dv = shortest path distance (time) to v Define prize at v as pv=gdv rv max discounted reward possibly collectable at v If given path reaches v at time tv, define excess ev = tv – dv difference between shortest path and chosen one Then discounted reward at v is gev pv Idea: if excess small, prize ~ discounted reward Fact: excess only increases as traverse path excess reflects lost time; can’t make it up

Property of optimum path assume g = ½ (can scale edge lengths) Claim: at least 1/2 of optimum path’s discounted reward R is collected before path’s excess reaches 1 s 0.5 Proof : shortcut to u reduces all excesses after u by at least 1 so “undiscounts” rewards by factor g-1=2 so doubles discounted reward collected u 1 1.5 0.5 2 1 Algorithm idea: Guess u Find a path to u of small excess that spans  R/2 prize 3 2

New problem: Approximate Min-Excess Path Suppose there exists an s-t path P* with prize value  of length l(P*)=dt+e Optimization: find s-t path P with prize value ≥  that minimizes excess l(P)-dt over shortest path to t equivalent to minimizing total length, e.g. k-path Approximation: find s-t path P with prize value ≥  that approximates optimum excess over shortest path to t, i.e. has length l(P) = dt + ce better than approximating entire path length obtain (2.5+)-approximation algorithm via dynamic programming using k-path algorithm Remind what k-TSP is

Decompose optimum path distance from s s t monotone monotone monotone wiggly wiggly Divides into independent problems exactly solvable case: monotonic paths approximable case: wiggly paths > 2/3 of each wiggly path is excess approximate excess by approximating length

Summary Algorithm for approximately maximizing E[reward] over uncertain lifetime Our techniques also give 4-approximation for previously open Orienteering problem of maximizing reward over fixed period of time Open questions non-uniform discount factors each vertex v has its own v non-uniform deadlines each vertex specifies its own deadline by which it has to be visited in order to collect reward d irected graphs