1. Approximation Algorithms for Regret-Bounded Vehicle Routing and Applications
Chaitanya Swamy
University of Waterloo
Joint work with Zachary Friggstad
University of Alberta

2. Vehicle routing problems (VRPs)
Metric space
starting depot r
client
Typical setup: visit all clients via route(s) starting from depot so as to minimize client delays: e.g., max client delay (TSP)
But this does not differentiate between clients close to the depot and those far away from it
Nearer clients may face more delay than further-away clients – source of dissatisfaction

3. Regret-bounded VRP
Metric space
starting depot r
client
Adopt a more client-centric view: seek bounded client regret client regret  measure of waiting time of a client relative to its shortest-path distance from depot

4. Regret-bounded VRP
cP(v)
Metric space r
starting depot P v
client Dv
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
cP(v) = time to reach v along P
Dv (min. possible waiting time of v)

5. Regret-bounded VRP
cP(v)
Metric space r
starting depot P v
client Dv
Two natural ways to measure regret:
additive regret of v = cP(v) – Dv
multiplicative regret of v = cP(v) / 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

6. Additive RVRP turns out to be more fundamental.
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

7. Main result We devise an O(1)-approx. algorithm for (additive) RVRP.
≈ 30
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

8. 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

9. 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

10. 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

11. 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

12. Building some intuition
Suppose that all nodes were at the same distance from r
V = nodes other than r r

13. Building some intuition
Suppose that all nodes were at the same distance from r
V = nodes other than r 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

14. Building some intuition
Suppose that all nodes were at the same distance from r
V = nodes other than r 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

15. Building some intuition
Suppose that all nodes were at the same distance from r
V = nodes other than r 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
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.

16. 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.

17. So, suffices to find O(OPT) paths of total creg-length O(R.OPT).
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).
path P Dv
Lemma (Blum et al.): total c-cost of red edges on P ≤ 1.5 creg(P).

18. 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 .

19. 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

20. 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 (c-)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)

21. 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 (c-)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

22. 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 (c-)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*
Idea: pick edges to cover all sets S s.t. ∑P: red(P)d(S)≠ x*P ≥ 0.5
(x*  fractional solution of cost O(k*R) that covers all such S)
Do not know how to do this

23. 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 (c-)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*
Idea: pick edges to cover all sets S s.t. ∑P: red(P)d(S)≠ x*P ≥ 0.5
Settle for: cover all sets S s.t. vS, ∑P: red(P, v)d(S)≠ x*P ≥ 0.5
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

24. Application to DVRP
RECALL Distance-constrained VRP (DVRP): Cover all nodes with min no. of rooted paths s.t. waiting time of each node v ≤ D.
cP(v) (where v lies on path P)
Simple O(log D)-approximation using RVRP Dv D

25. Application to DVRP
RECALL Distance-constrained VRP (DVRP): Cover all nodes with min no. of rooted paths s.t. waiting time of each node v ≤ D.
cP(v) (where v lies on path P)
Simple O(log D)-approximation using RVRP -7 -1 Dv
-D/2
-2i
-2i-1 -3
D  0
Nodes v with Dv(D-2i, D-2i-1] have regret < 2i under OPT
 can cover these using O(OPT) paths, each of regret ≤ 2i-1
 waiting time of each such v ≤ D
O(log D) intervals  O(log D)-approximation

26. Application to DVRP Improved O( )-approximation using RVRP
Let N*i = no. of paths of OPT having regret < 2i
Fi = no. of (feasible) paths we use to cover Si
log D
log log D
These paths cover Si = {v: Dv > D-2i} (#) Si -7 -1 Dv -3
-D/2
-2i
-2i-1
D  0
For any k<i, can cover Si using Fk + O(N*k+2i-k(N*i – N*k)) paths
So Fi ≤ min0≤k<i [Fk+ O(N*k+2i-k(N*i – N*k))]
Base case F0 ≤ N*0 recurrence solves to give Fi = O(i/log i) N*i
i goes up to log D  get O(log D/log log D)-approximation

27. 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)?

28. Thank You.