Facility Location with Client Latencies: LP-based

Approximation Algorithms for Minimum Latency Problems

Chaitanya SwamyUniversity of Waterloo

Joint work with Deeparnab Chakrabarty

University of Pennsylvania

Two well-studied problems

client

facility

1) Facility location problems (e.g., uncapacitated FL (UFL))

Two well-studied problems

client

facility

1) Facility location problems (e.g., uncapacitated FL (UFL))

Open facilities and connect clients to open facilities to: minimize (facility-opening cost) + (client-connection cost)

open facility

Two well-studied problems

client

2) Vehicle routing problems (e.g., minimum latency (ML), TSP)

starting depot

Find a route that visits all clients starting from depot to:

Two well-studied problems

client

starting depot

Find a route that visits all clients starting from depot to: minimize (sum of arrival times)

minimum latency

2) Vehicle routing problems (e.g., minimum latency (ML), TSP)

Two well-studied problems

client

starting depot

Find a route that visits all clients starting from depot to: minimize (sum of arrival times)

OR (maximum arrival time)

minimum latency

TSP

2) Vehicle routing problems (e.g., minimum latency (ML), TSP)

• 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 decidingwhich 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

Minimum latency UFL (MLUFL)

client

facility

Facilities with opening costs {fi}Clients with connection cost cij : cost of assigning client j to fac. iRoot (depot) node rTime metric d on {facilities}∪{r}

root r

We want to:

Minimum latency UFL (MLUFL)

client

facility

Facilities with opening costs {fi}Clients with connection cost cij : cost of assigning client j to fac. iRoot (depot) node rTime metric d on {facilities}∪{r}

root r

We want to:

– open facilities– connect each client j to an open facility

i(j)– find a path P starting at r, spanning

open facilitiesGoal: min ∑(i opened) fi + ∑clients j (ci(j)j + dP(r, i(j))

facility opening cost

connection cost latency cost

open facility

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 min-sum set cover

• min-max version of MGL: min maxi (cover time of Gi) is essentially Group Steiner tree (GST)

minimum group latency (MGL)

Our Results

• Give an O(log2 max(n, m))-approx. for MLUFL

– result is “tight” in that a -approx. algorithm (even) for MGL O(.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

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) can handle lp-version of MLUFL

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

LP-relaxation for MLUFL

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 ≥ 1 for all j

xij,t ≤ yi,t for all i, j, t

∑e deze,t ≤ t for all t

∑e ∈(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)

F: set of facilities D: set of clients T: UB on max. activation time

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 , (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≤ (j) xij,t ≥ 2/3

1. 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.

2. Concatenating these O(log m) tours gives the final solution.

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≤ (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:

a)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

b)if t(j) ≤ 2r, then Pr[tree contains some i ∈N(j)] ≥ 5/9

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?

Thank You.