Chaitanya Swamy University of Waterloo Joint work with Deeparnab Chakrabarty

1. Facility Location with Client Latencies: LP-based Approximation Algorithms for Minimum Latency Problems Chaitanya Swamy University of Waterloo Joint work with Deeparnab Chakrabarty University of Pennsylvania

2. Two well-studied problems 1) Facility location problems (e.g., uncapacitated FL (UFL)) facility client

3. Two well-studied problems 1) 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)

4. Two well-studied problems 2) Vehicle routing problems (e.g., minimum latency (ML), TSP) starting depot client Find a route that visits all clients starting from depot to:

5. Two well-studied problems 2) 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)

6. Two well-studied problems 2) 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) TSP

7. These two problem classes have mostly been studied separately. • But 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, form 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 • the order in which to replenish the opened outlets • Propose a model that generalizes UFL and ML and abstracts such settings facility location component vehicle-routing component

8. 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:

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 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

10. 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 u∈Gi 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)

11. Our Results • Give an O(log2 max(n,m))-approx. for MLUFL • result is “tight” in that a r-approx. algorithm (even) for MGL ÞO(r.log m)-approx. for GST with m groups (best approx. ratio for GST has remained at O(log2 n.log m) [GKR00]) • O(1)-approx. for: (a) related-metrics (c = M.d); (b) uniform MLUFL with metric connection costs n = no. of facilities m = no. of clients

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

13. 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

14. 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 ∑e ∈ d(S), t ze,t ≥ ∑i∈S, 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)

15. Rounding algorithm (overview) Assume d is a tree metric (with facilities as leaves) for simplicity. 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 By standard filtering, taking N(j) = {i: cij ≤ 4C*j}, we have (i) ∑i ∈ N(j), t xij,t ≥ ¾;and (ii) ∑i ∈ N(j), t≤ t(j) xij,t ≥ 2/3 At each time T(r) = 2r, we use GKR rounding to obtain a tour of “low cost” starting at r such that for every j with t(j) ≤ T(r), with constant probability, the tour contains a facility from Nj. We open all the facilities in the tour. Concatenating these O(log m) tours gives the final solution.

16. Rounding algorithm (contd.) By standard filtering, taking N(j) = {i: cij ≤ 4C*j}, we have (i) ∑i ∈ N(j), t xij,t ≥ ¾;and (ii) ∑i ∈ N(j), t≤ t(j) xij,t ≥ 2/3 At each time T(r) = 2r, we use GKR rounding to obtain a tour of “low cost” starting at r such that for every j with t(j) ≤ T(r), with constant probability, the tour contains a facility from Nj. • add facility edge (i, v(i)) with cost fi, let zi,v(i) = ∑t≤ T(r) yi,t • Consider j with t(j) ≤ 2r (so ∑i ∈ N(j), t≤ T(r) xij,t ≥ 2/3) • ({zi,v(i)}, {ze,t}) is a fractional group Steiner tree that ≥ 2/3-covers the v(i)-group obtained from Nj, for each such j • Now one can use GKR to obtain a random tree such that: • with high probability • d-cost of tree = O(log n). ∑e deze = O(log n).T(r) • cost of facilities in tree = O(log n). ∑i fizi,v(i) = O(log n). ∑i fiyi • if t(j) ≤ 2r, then Pr[tree contains some i∈ N(j)] ≥5/9

17. Open Questions • What is the integrality gap of our LP relaxations for ML? (The upper bound we prove is 10.78 = 3*3.59, but we suspect the LPs are much better…) • What is the integrality gap for trees?

18. Thank You.