Approximation Algorithms for Regret-Bounded

Regret-bounded VRP Adopt more client-centric view: ensure bounded client regret client regret  measure of waiting time of a client relative to its shortest-path distance from depot
Two natural ways to measure regret:
additive regret of v = cP(v) – Dv
multiplicative regret of v = cP(v) / Dv r v cP(v) P Dv cP(v) = time to reach v along P Dv (min. possible waiting time of v) client starting depot Metric space<br>

Regret-bounded VRP Two natural ways to measure regret:
additive regret of v = cP(v) – Dv
multiplicative regret of v = cP(v) / Dv r v cP(v) P Dv Two problems: Given regret bound R, find minimum no. of paths rooted at r that cover all clients such that:
additive regret of each node ≤ R additive RVRP
multiplicative regret of each node ≤ R multiplicative RVRP client starting depot Metric space<br>

Both additive- and multiplicative- RVRP are NP-hard to approximate to a factor better than 2.
Additive RVRP turns out to be more fundamental.
(RECALL: Cover all nodes with the minimum no. of rooted paths such that additive regret of each node v ≤ R.) cP(v) – Dv (where v lies on path P) Algorithms and techniques developed for additive RVRP also yield algorithms for:
multiplicative RVRP and other regret-based VRPs
other classical vehicle routing problems
In the rest of the talk:
regret  additive regret, RVRP  additive-RVRP
regret-related VRP  VRP under additive regret<br>

Main result We devise an O(1)-approx. algorithm for (additive) RVRP.
Our algorithm is based on LP-rounding: contrasts with our limited understanding of LPs for VRPs (with TSP being the exception)
We write a set-cover style configuration LP (with path variables):
Previously only O(log n)-approximation and integrality gap was known – follows easily from set-cover analysis + orienteering
Our main contribution: we show how to exploit LP-structure and round an LP-solution losing only a constant factor
One of the few results showing how to leverage configuration LPs (other such results are known for bin-packing, Santa Claus problem, min-makespan scheduling, combinatorial auctions)
Near-optimal LP solution can be efficiently obtained: orienteering yields approximate separation oracle for dual LP ≈ 30<br>

Other results Using our algorithm for RVRP and/or our techniques, we obtain:
O(log(R/(R-1))-approx. for multiplicative-RVRP
O(min{log D/log log D, OPTLP, log n})-approx. for distance-constrained VRP: cover all nodes with minimum no. of rooted paths s.t. waiting time cP(v) of each node v is ≤ D; improves upon the previous-best O(min{log D, log n})-guarantee (Nagarajan-Ravi)
O(k2)-approx. for k-RVRP: use k paths to cover nodes and minimize max-regret; previous guarantees were only for k=1 via min-excess path (Blum et al.)
also show that integrality gap of configuration LP is k<br>

Other results (contd.) Also consider directed graphs.
Observe that one can replace regret (in objective or constraint) by cost (in a different asymmetric metric)
Hence, known results give: (a) O(log n)-approx. for RVRP; (b) O(k2 log n)-approx. for k-RVRP
c-approx. for RVRP Þ 2c-approx. for ATSP<br>

Related work Additive-RVRP proposed in Operations Research literature under the generic name “schoolbus problem”
Bock et al. studied RVRP and k-RVRP, and design:
an O(log n)-approximation using set cover + orienteering
a 3-approximation algorithm in tree metrics
a 12.5-approximation for k-RVRP in tree metrics
Additive regret also studied by Blum et al., who used excess to denote regret. They used the min-excess path problem to approximate orienteering.
Nagarajan-Ravi studied distance-constrained VRP:
give an O(min{log D, log n})-approx. (D = distance bound)
2-approximation in tree metrics<br>

A useful transformation Define cregu,v := Du + cuv – Dv for all u, v.
creg is an asymmetric metric – call this the regret metric
cregr,v = 0 for all v
For any path P and any vP,
cregP(v) = cP(v) – Dv = regret of v along P
So RVRP  minimize no. of rooted paths of creg-length at most R that cover all nodes
 distance-constrained VRP in asymmetric creg-metric
c-length and creg-length of any cycle are equal<br>

Building some intuition r Suppose that all nodes were at the same distance from r
V = nodes other than r<br>

Building some intuition V can be grouped into k paths, each of creg-cost = c-cost ≤ R
 can partition V into k components of total cost ≤ kR r Suppose that all nodes were at the same distance from r
V = nodes other than r<br>

Building some intuition Suppose that all nodes were at the same distance from r
V = nodes other than r V can be grouped into k paths, each of creg-cost = c-cost ≤ R
 can partition V into k components of total cost ≤ kR
 by doubling + shortcutting, get k paths of total cost ≤ 2kR
Attach each path to r (pick an end-node v, add rv edge) to get a rooted path. Total creg-length of resulting paths ≤ 2kR r<br>

Building some intuition V can be grouped into k paths, each of creg-cost = c-cost ≤ R
 can partition V into k components of total cost ≤ kR
 by doubling + shortcutting, get k paths of total cost ≤ 2kR
Attach each path to r (pick an end-node v, add rv edge) to get a rooted path. Total creg-length of resulting paths ≤ 2kR
Break each resulting rooted path into segments of creg-length ≤ R and attach each segment to r (this does not increase regret)
This gives ≤ 3k rooted paths covering V, each of creg-length ≤ R. r Suppose that all nodes were at the same distance from r
V = nodes other than r<br>

Lemma: Given at most ak paths of total creg-cost at most bkR, we can efficiently find at most (a+b)k paths, each of creg-cost ≤ R.<br>

Lemma: Given at most ak paths of total creg-cost at most bkR, we can efficiently find at most (a+b)k paths, each of creg-cost ≤ R. So, suffices to find O(OPT) paths of total creg-length O(R.OPT). Lemma (Blum et al.): total c-cost of red edges on P ≤ 1.5 creg(P). Dv path P<br>

Configuration LP Let C(R) = {rooted paths P: cregP(v) = cP(v) – Dv ≤ R for all vP}
Minimize ∑PC(R) xP s.t. ∑PC(R): vP xP ≥ 1 vV, x ≥ 0
Dual separation problem is an orienteering problem
There is a (2+)-approximation for orienteering (Chekuri et al.)
This yields an approximate separation oracle, so a (2+)-approx. solution x* to the configuration LP can be computed efficiently.
Let k* = ∑PC(R) x*P .<br>

Rounding the LP solution x* Easy case: suppose that directing edges of all paths P such that x*P>0 away from r gives an acyclic graph Then x* is (the path-decomposition of) an acyclic flow of value k* and creg-cost ≤ k*R that covers every node
Integrality of flows + acyclicity  can find k* paths of total creg-cost ≤ k*R that cover all nodes<br>

Rounding the LP solution x* Of course, x* need not yield an acyclic flow.
Form a set W of witness nodes, partition V\W into components:
Suitably shortcutting paths P with x*P>0 yields an acyclic flow
that covers every node in W to an extent of at least 0.5
has value at most k* and creg-cost at most k*R
 can obtain O(k*) paths covering W of total creg-cost O(k*R)
The total cost of all components is O(k*R), and
every component contains r or some witness node
 we can attach the V\W to the paths found in step 1 incurring O(k*R) total additional regret (and cost)
So overall, obtain O(k*) paths with total regret ≤ O(k*R)<br>

Rounding the LP solution x* Of course, x* need not yield an acyclic flow.
Form a set W of witness nodes, partition V\W into components:
Shortcutting paths P with x*P>0 yields a suitable acyclic flow that covers each node in W to an extent of at least 0.5
The total cost of all components is O(k*R), and
every component contains r or some witness node
Form components whose cost can be charged to the red edges of the paths in the support of x*
Can be done cleanly by setting up a network design problem with a downwards-monotone cut-requirement function
Also ensures that each component contains a node with large incoming flow on blue edges, which becomes a witness node<br>

Conclusions and open questions We systematically study regret-bounded VRPs
devise a constant-approx. for additive-RVRP
novel rounding method for the configuration LP; new ideas to deal with regret
this yields bounds for various other RVRPs, as well as distance-constrained VRP (DVRP)
Is there an O(1)-approx. for DVRP?
our work is a promising step: improves upon the usual log-guarantee given by set cover, but we use additive-RVRP as a black box; better way of leveraging underlying ideas
Improve upon the O(k2)-approximation for k-RVRP.
What about other regret-based objectives: e.g., minimizing sum of regrets (much stronger than minimizing latency)?<br>

Thank You.<br>