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Chaitanya Swamy University of Waterloo Joint work with Deeparnab ChakrabartyPowerPoint Presentation

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 Approximation Algorithms for Minimum Latency Problems

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

facility

client

Two well-studied problems Approximation Algorithms for Minimum Latency 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 Approximation Algorithms for Minimum Latency 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 Approximation Algorithms for Minimum Latency 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 Approximation Algorithms for Minimum Latency 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) separately.

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) separately.

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 separately.

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 separately.

- 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.) separately.

- 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 separately.

- 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 separately.

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)

Rounding algorithm (overview) separately.

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.) separately.

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 separately.

- 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. separately.

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