lp-based techniques for minimum latency problems

LP-based Techniques for Minimum Latency

Problems

Chaitanya SwamyUniversity of Waterloo

Joint work with Deeparnab Chakrabarty

Microsoft Research, India

Facility Location with Client Latencies: LP-based Techniques for Minimum

Latency Problems

Chaitanya SwamyUniversity of Waterloo

Joint work with Deeparnab Chakrabarty

Microsoft Research, India

Two well-studied problems

client

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

starting depot

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


client

starting depot

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

minimum latency



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

(path) TSP



client

facility

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


client

facility

2) 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

• These two problem classes have mostly been studied separately.

• UFL has a rich history of LP-based algorithms;Algorithms for ML use combinatorial arguments – use k-MST as a lower bound and rely on good algorithms for k-MST.

• 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, from 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 in what order to replenish the opened outlets

• Propose a model that abstracts such settings and generalizes UFL and ML

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 MLUFLMLUFL 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 uGi 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)

Approximation Algorithm

Hard to solve the problem exactly. Settle for approximate solutions. Give polytime algorithm that always finds near-optimal solutions.

A is a -approximation algorithm if,

•A runs in polynomial time.

•A(I) ≤ .OPT(I) on all instances I,

is called the approximation ratio of A.

Theorem: There is an O(log2 max(n, m))-approximation algorithm for MLUFL.

• result is “tight” in that a -approx. algorithm (even) for MGL O(.log m)-approx. for GST with m groups (longstanding open problem to improve the O(log2 n.log m) approx. ratio for GST [GKR00])

• O(1)-approx. for: (a) related-metrics (c = M.d; M

≥ 1);

(b) uniform MLUFL with metric connection costs

n = no. of facilities m = no. of clients

Our algorithms and techniques are LP-based. So• Get some interesting LP-based insights into ML:

– obtain promising LP-relaxations for ML and can upper bound integrality gap by a constant.

– Rounding algorithm only relies on integrality-gap of TSP being O(1) (as opposed to an O(1)-approximation for k-MST)

• Algorithms easily extend to handle various generalizations– k-route MLUFL (can use k paths to span open facilities)– setting when latency-cost of j is f(time taken to reach

i(j)), where f is increasing and has growth-rate at most p: 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 ≥ ∑iS, t’≤t xij,t’ for all j, t, S⊆F

x, y, z≥ 0, yi,t = 0 for all i, t: di,t>TAssume 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)

This talk: assume d is a tree metric (with facilities as leaves).

Consider first special case of MGL: recall fi=0, cij

{0,} i,j so for each j, have a group G(j) of facilities that can serve j,find a path starting at r (r-path) visiting all groups to min ∑j (first time when some facility in G(j) is visited)

xij,t only defined for iG(j) and t such that di,t ≤ T

Minimize ∑j,iG(j),t t xij,t

subject to, ∑iG(j),t xij,t ≥ 1 for all j

∑e deze,t≤ t for all t

∑e(S), t ze,t ≥ ∑iS, t’≤t xij,t’ for all j, t, S⊆F

x, z ≥ 0.

Rounding algorithm (contd.)

Let (x, y, z): optimal solution to LP, L*

j = ∑j,i,t txij,t , (j) = 3.L*j ∑i, t≤ (j) xij,t ≥ 2/3

1. At each time T(k) = 2k, suppose (ideally) we get an r-tour of cost O().T(k) that covers every G(j) for j s.t. (j) ≤ T(k).Note: {ze,T(k)} is a fractional group Steiner tree (GST) that 2/3-covers each such G(j) can use LP-based =O(log2

n)-approx. algorithm of GKR for GST to get r-tour of cost O().T(k)

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

Latency of each j is O()(j), so total cost is O().OPT.






Improvement: Suffices to get an r-tour that covers every G(j) with (j) < T(k) with probability > ½; GKR analysis actually shows that this can be done at cost O(log n).T(k)






Improvement: Suffices to get an r-tour that covers every G(j) with (j) < T(k) with probability > ½; GKR analysis actually shows that this can be done at cost O(log n).T(k)

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

Latency of each j is O()(j), so total cost is O().OPT.E[Latency of j] = O(log n).(j), so total E[cost] is

O(log n).OPT.

Rounding for general MLUFL

Let (x, y, z): optimal solution to LPC*

j = ∑j,i,t cijxij,t , L*j = ∑j,i,t txij,t ,

(j) = 12.L*j

For every j, create group N(j) = {i: cij ≤ 4C*j}. Then we

have (i) ∑iN(j), t xij,t ≥ ¾; and (ii) ∑iN(j), t≤ (j) xij,t ≥ 2/3.Now use MGL rounding: at every time T(k)=2k

–extend tree by adding facility edges (i,v(i)) for every facility i–extend {ze,T(k)} by setting zi,v(i) = ∑t≤ T(k) yi,t

–again ({zi,v(i)}, {ze,t}) is a fractional GST that ≥ 2/3-covers the v(i)-group obtained from N(j) for each j with (j) ≤ T(k); so can use GKR to obtain an r-tour tree such that:a) for every j with (j) ≤ T(k), Pr[tour contains some i

N(j)] > ½b)with high probability– d-cost of tour = O(log n).T(k)– cost of facilities in tour = O(log n). ∑i fizi,v(i) = O(log

n). ∑i fiyi

Insights for MLLP for MLUFL gives a (compact) LP-relaxation for ML Can also formulate the following huge LP. Let P(t) = all r-paths of length at most t.zP, t: indicates if path PP(t) is used to visit clients

with latency ≤ tMinimize ∑j, t t xij,t (LP2)

subject to, ∑t xj,t ≥ 1 for all j

∑PP(t) zP, t ≤ 1 for all t

∑PP(t) zP, t ≥ ∑t’≤t xj,t’ for all j, t

x, z ≥ 0.

(Can also use tree-variables; can write a similar LP for MGL.)

Insights for ML (contd.)ML has not been attacked (directly) using LP-based methods, and these LPs open up promising new venues of attack.

Both compact LP and huge LP have O(1) integrality gap. Rounding uses nice ideas from scheduling, polyhedral insights from TSP.

Separation oracle for dual of (LP2) is an orienteering problem: given rewards {Rj} on the clients and a budget B find an r-path of length at most B that collects maximum reward.Theorem: An (even bicriteria) approximation algorithm for orienteering can be used to find “approximate” solution to (LP2).

Coupled with above, this gives a new proof that -approx. for orienteering O()-approx. algorithm for ML

Open Questions

•What is the integrality gap for ML? (We prove an upper bound of 10.78 = 3(3.59), but we suspect the LPs are much better.)

•What is the integrality gap for trees?– For unweighted trees, ML can be solved

optimally; is our compact LP exact on trees?

•What is the integrality gap for TSP?

How good are these LP-relaxations?

Thank You.

lp-based techniques for minimum latency problems

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