Vehicle Routing Problems With Time Windows and Uncertainty Su Jia Advisor: Joseph Mitchell Stony Brook University Sept 9, 2016

Outline My work: Review of relevant papers from STOC/SODA/FOCS etc: results & methods

My work TWTSP Scheduling with conflict Network optimization on pairs of points

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) A framework for 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

Our Contributions (metric space) Infinite speed: O(log n) apx for TWTSP Finite speed: assuming that there is at least one vertex with time window [t,t+1], for each integer t<=L: (O(1), O(1)) dual apx, when there are only 2 types of TW lengths. (O(log L), O(log L)) dual apx, general dyadic TW The apx factor depends only on the comlexity of time windows, regardless the number of points.

Structure of our paper 1D Metric space

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.

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

Theorem. The TWTSP problem with infinite speed in 1D is polytime solvable for dyadic instances. Hint: DP, Encode north/southmost position visited

h-dyadic Instance

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: what to do next General instance for TWTSP (choose h) TWPC Monotone-TSPN?

Direct Comparison with Previous Work (last slide for 1D)

TWTSP in Metric Space: Simple Case

Cut-Tree Algorithm for TWTSP with speed s (simple case) red nodes: unit length TW black nodes: [0,L] Algo: 1. Find MST on all points 2. Cut into disjoint subtrees (called blocks), each of size O(s) 3. Build unweighted bipartite graph H and find a perfect delta-matching 4. For each red node, traverse the blue blocks assigned to it, and connect the reds.

Cut-Tree Algorithm for TWTSP with speed s (simple case) red nodes: unit length TW black nodes: [0,L] Algo: 1. Find MST on all points 2. Cut into disjoint subtrees (called blocks), each of size O(s) 3. Build unweighted bipartite graph H and find a perfect delta-matching 4. For each red node, traverse the blue blocks assigned to it, and connect the reds.

Given an MST, say T Pick any collection of disjoint subtrees Build MST, say T' on V' compare T' with the sum of the subtrees you chose

H satisfies the generalized Hall's matching condition Intuition: If a set of blocks does not have enough number of red nodes that are within distance s to them (i.e. red neighbors in H), then there is no feasible solution. This is just the Hall's marriage condition!

proof sketch: (what if s too small?) speed: existence of feasible solution (wrt s) => generalized Hall's marriage condition => it is possible to assign blue blocks to red nodes such that each red node is assigned at most const # of blocks => the excess of speed is within const times (note that each block has size O(s).) distance: each blue block is traversed once, and the union of the blocks is a subset of MST(all nodes), which is a lower bound for d*(s) for any s.

My work (explain the other two) TWTSP Scheduling Partition pairs

‘Scheduling with conflict’ Competitive Ratio for Online Scheduling in Software-Defined Optimal WAN. Su Jia, Xin Jin, Golnaz Ghasemesfeh, Jiaxin Ding, Jie Gao. (Submitted to Infocom) Objects: Sender, receiver, jobs, size & release time Rule: Each person can send/receive at most 1 email at each time slot. Allow pmtn. Goal: ‘good’ schedule? Alt. def: matching. coloring Is there a geometric version? (scheduling + routing)

SDN (Online) Algo: 1. Greedy (Smith’s greedy) Competitive ratio for A.C.T Unit Job Size Arbitrary Job Size Zero Release Time 2 Arbitrary Release Time 3 2* (Online) Algo: 1. Greedy (Smith’s greedy) 2. SRPT based algo: 2-apx for min A.C.T, when d_i = 1. hardness, methods Next Page: Literature Review of VRP

Literature Review Time Window Prize Collecting Stochastic orienteering Adaptive TSP Problem Moving-Point TSP

Time Window Prize Collecting Bar-Yehuda, Even, Shahar: on approximating a geometric prize-collecting TSP problem with time windows (ESA’03, Jour of Algo.) Bansal, Blum, Chawla, Meyerson: Approximation Algorithms for Deadline-TSP and Vehicle Routing with Time Windows (STOC’04) -Small margin pts (wrt OPT): pts visited by OPT in [d(v)/(1 + eps), d(v)] -Obsv: when pts have similar deadlines: -O(1)-apx for P2P Orienteering -find P that visits O(1)-fraction small margin pts -generalize ‘small margin’. Classify pts visited by OPT into log groups

Time Window Prize Collecting Chekuri, Pal: A Recursive Greedy Algorithm for Walks in Directed Graphs (FOCS’05) -Recursive greedy. (idea: guess the mid point of P) -log(OPT)-apx when prize func f is submodular -Quasipolytime: (nlogB)^O(logn) -appl (digraph): changing prize orienteering (TW), k-TSP, group Steiner Tree -later: Chekuri, Korula.(SODA’08): improved algorithms for orienteering and related problems

Stochastic orienteering Gupta, Krishnaswamy, Nagarajan, Ravi: Approximation Algorithms for Stochastic Orienteering (SODA’12) Metric space (V,d). Vertex v_i has known reward r_i and unknown size s_i, with distribution f_i. Total Budget B. Goal: max E[prize] Stochastic knapsack + orienteering Relationship to Joe’s problem (StoOri with rel time) Extreme case Stochostic Knapsack

Stochastic orienteering (cont’d) Gupta, Krishnaswamy, Nagarajan, Ravi: Approximation Algorithms for Stochastic Orienteering (SODA’12) O(1)-apx for Knapsack Orienteering: travel budget & waiting-time budget Non-adaptive version: O(1)-apx Adaptivity gap: O(log log B). When rewards are also random: O(logn logB)

Discounted-Reward TSP Blum, Chawla, Karger, Lane, Meyerson, Minkoff: Approximation Algorithms for Orienteering and Discounted-Reward TSP (FOCS’03) Discounted Reward TSP: Prize function P(t) = P(0) * gamma^t Min excess for (ordinary) kTSP: min |P| - SP(s,t) (called ‘excess’) s.t. P is an s-t path collecting prize k O(1)-apx for min excess kTSP => O(1)-apx for Discounted Reward TSP

Moving-Point TSP Chalasani, Motwani: Approximating Capacitate Routing and Delivery Problems (WAFR’96, SIAM J. Comput) Helvig, Robins, Zelikovky: moving-target TSP and related problems (ESA’98) Hammar, Nilson: Approximation Results for Kinetic Variants of TSP (ICALP’99) Moving point: One direction capacitated

Thank you! Doable Open Problems: 1. moving-target TSP/PC (spatial curves, + random walk) 2. other versions of VRP with uncertainty e.g. uncertain TW 3. break the log bound for TWTSP/TWPC 4. security game download my slides from My website: www.ams.stonybrook.edu/~jias