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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. Two well-studied problems.

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Chaitanya Swamy University of Waterloo Joint work with Deeparnab Chakrabarty

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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


Two well-studied problems

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

facility

client


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)


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:


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)


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


  • 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


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:


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


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)


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


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


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

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 = 0for all i, t: di,t>T

Assume T= poly(m:=|F|) for simplicity (handled by scaling)


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.


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


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.


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