lp-based techniques for minimum latency problems chaitanya swamy university of waterloo joint work...
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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:
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
1) 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
(path) TSP
1) Vehicle routing problems (e.g., minimum latency (ML), TSP)
Two well-studied problems
client
facility
2) Facility location problems (e.g., uncapacitated FL (UFL))
Two well-studied problems
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.
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)
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)
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)
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.