LP-based Techniques for Minimum Latency Problems

1. LP-based Techniques for Minimum Latency Problems Chaitanya Swamy University of Waterloo Joint work with Deeparnab Chakrabarty Microsoft Research, India

2. Facility Location with Client Latencies: LP-based Techniques for Minimum Latency Problems Chaitanya Swamy University of Waterloo Joint work with Deeparnab Chakrabarty Microsoft Research, India

3. Two well-studied problems 1) Vehicle routing problems (e.g., minimum latency (ML), TSP) starting depot client Find a route that visits all clients starting from depot to:

4. Two well-studied problems 1) Vehicle routing problems (e.g., minimum latency (ML), TSP) starting depot client minimum latency Find a route that visits all clients starting from depot to: minimize (sum of arrival times)

5. Two well-studied problems 1) Vehicle routing problems (e.g., minimum latency (ML), TSP) starting depot client minimum latency Find a route that visits all clients starting from depot to: minimize (sum of arrival times) OR (maximum arrival time) (path) TSP

6. Two well-studied problems 2) Facility location problems (e.g., uncapacitated FL (UFL)) facility client

7. Two well-studied problems 2) Facility location problems (e.g., uncapacitated FL (UFL)) facility open facility client Open facilities and connect clients to open facilities to: minimize (facility-opening cost) + (client-connection cost)

8. These two problem classes have mostly been studied separately. • UFL has a rich history of LP-based algorithms; • Algorithms for ML use combinatorial arguments – use k-MST as a lower bound and rely on good algorithms for k-MST. • Various logistics problems have both facility-location and vehicle-routing components. • E.g., opening retail outlets to service customers: • inventory at retail outlets needs to be replenished or ordered (say, from a depot), and delays incurred in getting inventory to outlet adversely affects customers assigned to it • should keep these customer delays in mind when deciding • which outlets to open to service customers, and • in what order to replenish the opened outlets • Propose a model that abstracts such settings and generalizes UFL and ML facility location component vehicle-routing component

9. Minimum latency UFL (MLUFL) Facilities with opening costs {fi} Clients with connection cost cij: cost of assigning client j to fac. i Root (depot) node r Time metricd on {facilities}∪{r} facility root r client We want to:

10. Minimum latency UFL (MLUFL) Facilities with opening costs {fi} Clients with connection cost cij: cost of assigning client j to fac. i Root (depot) node r Time metricd on {facilities}∪{r} facility root r client open facility We want to: • open facilities • connect each client j to an open facility i(j) • find a path P starting at r, spanning open facilities Goal: min ∑(i opened) fi+∑clients j (ci(j)j+dP(r, i(j))) latency cost facility opening cost connection cost

11. Different flavors of MLUFL MLUFL captures various diverse problems of interest • UFL and ML • fi=0 i, {0,¥} cij’s, get interesting generalization of ML: given root r, time-metric d, (disjoint) node-sets G1,…,Gk, find a path starting at r to min ∑i (cover time of Gi)(cover time of Gi = first time when some uÎGi is visited) • MGL where node-sets are sets in set-cover instance, uniform time metric Þmin-sum set cover • min-max version of MGL: min maxi (cover time of Gi) is essentially Group Steiner tree (GST) minimum group latency (MGL)

12. Approximation Algorithm • Hard to solve the problem exactly. • Settle for approximate solutions. Give polytime algorithm that always finds near-optimal solutions. • A is a a-approximation algorithm if, • A runs in polynomial time. • A(I) ≤ a.OPT(I) on all instances I, • ais called the approximation ratio of A.

13. n = no. of facilities m = no. of clients Theorem: There is an O(log2 max(n,m))-approximation algorithm for MLUFL. • result is “tight” in that a r-approx. algorithm (even) for MGL ÞO(r.log m)-approx. for GST with m groups (longstanding open problem to improve the O(log2 n.log m) approx. ratio for GST [GKR00]) • O(1)-approx. for: (a) related-metrics (c = M.d; M≥1); (b) uniform MLUFL with metric connection costs

14. Our algorithms and techniques are LP-based. So • Get some interesting LP-based insights into ML: • obtain promising LP-relaxations for ML and can upper bound integrality gap by a constant. • Rounding algorithm only relies on integrality-gap of TSP being O(1) (as opposed to an O(1)-approximation for k-MST) • Algorithms easily extend to handle various generalizations • k-route MLUFL (can use k paths to span open facilities) • setting when latency-cost of j is f(time taken to reach i(j)), where f is increasing and has growth-rate at most p: f(c.x) ≤ cp.f(x)Þ can handle lp-version of MLUFL

15. Related work • MLUFL and MGL are new problems • Much work on UFL and ML • UFL: Shmoys-Tardos-Aardal, …, Byrka • ML: Blum et al., … Chaudhary et al. • Independently, concurrently Gupta-Nagarajan-Ravi also propose MGL: give O(log2 n)-approx. for MGL, and reduction from GST to MGL (not clear how to extend their combinatorial techniques to handle fi’s) • min-sum set cover: O(1)-approx. by Feige-Lovasz-Tetali; also Bansal et al. gave O(1)-approx. for a generalization • min-max version of MGL is (essentially) GST: Garg-Konjevod-Ravi (GKR) give polylog-approximation

16. LP-relaxation for MLUFL F: set of facilities D: set of clients T: UB on max. activation time yi,t: indicates if facility i is opened at time t xij,t: indicates if client j connects to i at time t ze,t: indicates if edge e is traversed by time t Minimize ∑i fiyi +∑j,i,t (cij + t)xij,t subject to, ∑i,t xij,t ≥ 1for all j, xij,t ≤ yi,tfor all i,j,t ∑e deze,t ≤ t for all t ∑eÎd(S), t ze,t ≥ ∑iÎS, t’≤t xij,t’for all j, t, S⊆F x, y, z ≥ 0, yi,t = 0 for all i, t: di,t>T Assume T= poly(m:=|F|) for simplicity (handled by scaling)

17. Rounding algorithm (overview) This talk: assume d is a tree metric (with facilities as leaves). Consider first special case of MGL: recall fi=0, cij Î{0,¥} i,j so for each j, have a group G(j) of facilities that can serve j, find a path starting at r (r-path) visiting all groups to min ∑j(first time when some facility in G(j) is visited) xij,t only defined for iÎG(j) and t such that di,t ≤ T Minimize ∑j,iÎG(j),t txij,t subject to, ∑iÎG(j),t xij,t ≥ 1for all j ∑e deze,t ≤ t for all t ∑eÎd(S), t ze,t ≥ ∑iÎS, t’≤t xij,t’for all j, t, S⊆F x, z ≥ 0.

18. Rounding algorithm (contd.) Let (x, y, z): optimal solution to LP, L*j = ∑j,i,t txij,t , t(j) = 3.L*j Þ∑i, t≤ t(j) xij,t ≥ 2/3 At each time T(k) = 2k, suppose (ideally) we get an r-tour of cost O(r).T(k) that covers every G(j) for j s.t. t(j) ≤ T(k). Note: {ze,T(k)} is a fractional group Steiner tree (GST) that 2/3-covers each such G(j)Þ can use LP-based r=O(log2n)-approx. algorithm of GKR for GST to get r-tour of cost O(r).T(k) Concatenating these O(log m) tours gives the final solution. Latency of each j is O(r)t(j), so total cost is O(r).OPT.

19. Rounding algorithm (contd.) Let (x, y, z): optimal solution to LP, L*j = ∑j,i,t txij,t , t(j) = 3.L*j Þ∑i, t≤ t(j) xij,t ≥ 2/3 At each time T(k) = 2k, suppose (ideally) we get an r-tour of cost O(r).T(k) that covers every G(j) for j s.t. t(j) ≤ T(k). Note: {ze,T(k)} is a fractional group Steiner tree (GST) that 2/3-covers each such G(j)Þ can use LP-based r=O(log2n)-approx. algorithm of GKR for GST to get r-tour of cost O(r).T(k) Improvement: Suffices to get an r-tour that covers every G(j) with t(j)< T(k) with probability > ½; GKR analysis actually shows that this can be done at cost O(log n).T(k)

20. Rounding algorithm (contd.) Let (x, y, z): optimal solution to LP, L*j = ∑j,i,t txij,t , t(j) = 3.L*j Þ∑i, t≤ t(j) xij,t ≥ 2/3 At each time T(k) = 2k, suppose (ideally) we get an r-tour of cost O(r).T(k) that covers every G(j) for j s.t. t(j) ≤ T(k). Note: {ze,T(k)} is a fractional group Steiner tree (GST) that 2/3-covers each such G(j)Þ can use LP-based r=O(log2n)-approx. algorithm of GKR for GST to get r-tour of cost O(r).T(k) Improvement: Suffices to get an r-tour that covers every G(j) with t(j)< T(k) with probability > ½; GKR analysis actually shows that this can be done at cost O(log n).T(k) Concatenating these O(log m) tours gives the final solution. Latency of each j is O(r)t(j), so total cost is O(r).OPT. E[Latency of j] = O(log n).t(j), so total E[cost] is O(log n).OPT.

21. Rounding for general MLUFL Let (x, y, z): optimal solution to LP C*j = ∑j,i,t cijxij,t , L*j = ∑j,i,t txij,t , t(j) = 12.L*j For every j, create group N(j) = {i: cij ≤ 4C*j}. Then we have (i) ∑iÎN(j), t xij,t ≥ ¾;and (ii) ∑iÎN(j), t≤ t(j) xij,t ≥ 2/3. • Now use MGL rounding: at every time T(k)=2k • extend tree by adding facility edges (i,v(i)) for every facility i • extend {ze,T(k)} by setting zi,v(i) = ∑t≤ T(k) yi,t • again ({zi,v(i)}, {ze,t}) is a fractional GSTthat ≥ 2/3-covers the v(i)-group obtained from N(j) for each j with t(j)≤ T(k);so can use GKR to obtain an r-tour tree such that: • for every j with t(j)≤ T(k), Pr[tour contains some iÎ N(j)] > ½ • with high probability • d-cost of tour = O(log n).T(k) • cost of facilities in tour = O(log n). ∑i fizi,v(i) = O(log n). ∑i fiyi

22. Insights for ML LP for MLUFL gives a (compact) LP-relaxation for ML Can also formulate the following huge LP. Let P(t)= all r-paths of length at most t. zP, t: indicates if path PÎP(t) is used to visit clients with latency ≤ t Minimize ∑j,t txij,t (LP2) subject to, ∑t xj,t ≥ 1for all j ∑PÎP(t) zP, t ≤ 1for all t ∑PÎP(t) zP, t ≥ ∑t’≤t xj,t’for all j, t x, z ≥ 0. (Can also use tree-variables; can write a similar LP for MGL.)

23. Insights for ML (contd.) ML has not been attacked (directly) using LP-based methods, and these LPs open up promising new venues of attack. Both compact LP and huge LP have O(1) integrality gap. Rounding uses nice ideas from scheduling, polyhedral insights from TSP. Separation oracle for dual of (LP2) is an orienteering problem: given rewards {Rj}on the clients and a budget B find an r-path of length at most B that collects maximum reward. Theorem: An (even bicriteria) approximation algorithm for orienteering can be used to find “approximate” solution to (LP2). Coupled with above, this gives a new proof that r-approx. for orienteering ÞO(r)-approx. algorithm for ML

24. Open Questions How good are these LP-relaxations? • What is the integrality gap for ML? (We prove an upper bound of 10.78 = 3(3.59), but we suspect the LPs are much better.) • What is the integrality gap for trees? • For unweighted trees, ML can be solved optimally; is our compact LP exact on trees? • What is the integrality gap for TSP?

25. Thank You.