Jie Gao, Su Jia, Joseph S.B. Mitchell Stony Brook University Exact and Approximation Algorithms for Time Window TSP and Time Window PC Jie Gao, Su Jia, Joseph S.B. Mitchell Stony Brook University Full paper: Exact and Approximation Algorithms for Time Window TSP and Time Window PC (accepted by WAFR’16)

Problem Description Time Window Prize Collecting (TWPC): Unit speed robot; must visit each site i during given time window, (ri , di). Goal: max # sites visited (or total “prize”) Time Window Travelling Salesman (TWTSP): Robot with speed s; must visit each site i during given time window, (ri , di). Goal: min distance robot travels to visit all sites (in TW) Assume each TW has length at least one (often called “TWTSP”) (may not be feasible for small s)

Some Prior Work

Our Contributions (1D) 1D TWTSP/PC: in time, we can compute a path collecting at least OPT prize (or at most OPT length), but visiting each point in relaxed time window Assuming the max length is bounded by some poly(n), then this is a dual QPTAS. Direct comparison with previous results: in time, we can compute a path collecting at least OPT prize but visiting each point in relaxed time window . Compare with Bansal et al's result: visit

Visualize

Dyadic Interval Interval [x,y] is said to be dyadic if (y-x) is a power of 2, and x is an integer multiple of its length. e.g. [25,26], [64,96],...

Theorem. The TWTSP problem with infinite speed in 1D is polytime solvable for dyadic instances. Hint: DP, Vertical Range Constraints (VRC).

Thm. For h-dyadic instance, the 1D TWTSP problem with infinite speed can be solved in O(n^O(h)) time. Mimic dyadic case. Encode the north/southmost position that P visits in each subinterval (i.e. vertical ranges)

1D TWTSP, general instance, proof outline (1) h-dyadic instances in O(n^O(h)) time (2) general instance can be rounded to an h-dyadic instance by stretching each time window for at most (1+ ) times, where h= O(log L/ log (1+ )). (1)+(2) => dual QPTAS for 1D TWTSP with infinite speed

TWTSP & TWPC: Need to encode more parameters!

Dual QPTAS for Monotone TSPN in 2D

TWTSP in Euclidean Space

Why can’t we generalize 1d to higher dimension? “for-free” property

Can’t just find MST and traverse (even in 1D) TW-matching

Unit length time window: O(1)-apx (subroutine for log-factor apx.)

Dual approximations (a,b)-dual apx for TWTSP with speed s: Define OPT(s). Travel with speed a*s Distance b*OPT(s)

Thank You! Open Problems: Q1: TWTSP with infinite speed in Euclidean space Q2: TWPC and TWTSP with finite speed in Euclidean space Q3: Other problems involving time