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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

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

facility

client

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)

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

starting depot

client

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

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)

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

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:

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

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)

- 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 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

- 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

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)

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.

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

- 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?