Approximation Algorithms for Mixed,Windy,and Capacitated...

Approximation Algorithms for Mixed, Windy, andCapacitated Arc Routing Problems

Rene van Bevern1

joint work with

Christian Komusiewicz2 and Manuel Sorge2

1Novosibirsk State University, Novosibirsk, Russian Federation

2Institut fur Softwaretechnik und Theoretische Informatik, TU Berlin, Germany

ALGO / ATMOS 2015

Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 1/16

Mixed and Windy Capacitated Arc Routing (MWCARP)

Input: Mixed graph G = (V ,A, E), depot v0 ∈ V , travel costsc : V × V → N ∪ {∞}, demands d : E ∪ A→ N, and a vehicle capacity Q.

Goal: Find a set of vehicle routes of total minimum cost, each startingand ending in v0, such that each demand is served by exactly one vehicleand each vehicle serves a total demand of at most Q.

Applications: Waste collection, salting roads, mail delivery, meter reading,inspection of welded seams, ... (Corberan and Laporte, SIAM book, 2014)

Known results

3/2-approximation is NP-hard (Golden and Wong, Networks, 1981)

Undirected version is 7/2-approximable (Jansen, Networks, 1993)

In general, the approximation factor for MWCARP cannot undercut thatfor metric asymmetric TSP (4-ATSP).

Approximation algorithms for n-vertex4-ATSP:I Best known: O(log n/ log log n)-approximation (Asadpour et al., SODA’10)

I Constant-factor approximation open (Frieze, Galbiati, Maffioli, Networks, 1982)

Approximation algorithms for MWCARP sparsely investigated.

Known results

Approximation algorithms for n-vertex4-ATSP:

I Best known: O(log n/ log log n)-approximation (Asadpour et al., SODA’10)

Known results

How to approximate despite the hardness?

Consider parameter C : number of(weakly) connected componentsinduced by positive-demand arcs.

In picture: 1.5 · 106 inhabitants,lots of roads to serve, but C ≈ 5.

NP-hard for C = 1.(Golden and Wong, Networks, 1981)

Our result:MWCARP is O(1)-approximable

in O(2CC2 + n3 log n) time.

poly-time for C ∈ O(log n).

Our results (in more detail)

We show that MWCARP and the following related NP-hard problems arenot only as hard to approximate as 4-ATSP, but also as easy.

Mixed and Windy Rural Postman problem (MWRPP): find a minimum-costtour visiting all given required arcs and edges in a mixed asymmetricallyedge-weighted graph.

Directed Rural Postman problem (DRPP): find a minimum-cost tourvisiting all given required arcs in a directed edge-weighted graph.

Theorem: If n-vertex4-ATSP is α(n)-approximable in t(n) time, then

1. n-vertex DRPP is (α(C) + 1)-approximable in O(t(C) + n3 log n) time,

2. n-vertex MWRPP is (α(C) + 3)-appr’mable in O(t(C) + n3 log n) time,

3. n-vertex MWCARP is O(α(C + 1))-ap’le in O(t(C + 1) + n3 log n) time.

Remark: 4-ATSP is optimally solvable in O(2nn2) time. (Bell, Held, Karp, 1962)

Component-Wise Eulerian DRPP

Goal: find a minimum-cost tour visiting a set R of required arcs in adirected edge-weighted graph G.

Additional constraint: G[R] has C weakly-connected Eulerian components.

Claim: α(n)-approximation for n-vertex4-ATSP=⇒ (α(C) + 1)-approximation for Component-Wise Eulerian DRPP.

Algorithm for Component-Wise Eulerian DRPP

Algorithm: Shown are only required arcs; directions are not shown.

Approximation factor: Optimum DRPP tour has at least the cost of therequired arcs and at least the cost of an optimal 4-ATSP tour.

For each connected component i of G[R], pick an arbitrary vertex vi .

For each component i, compute an Euler tour starting and ending in vi .

Join vertices v1, . . . , vC using α(C)-approximate 4-ATSP tour.

Goal: find a minimum-cost tour visiting all given required arcs in adirected edge-weighted graph.

Reminder: G[R] consists of C weakly-connected components.

Claim: α(n)-approx.4-ATSP =⇒ (α(C) + 1)-approx. for DRPP.

Algorithm for general DRPP

Claim: α(n)-approx. for 4-ATSP =⇒ (α(C) + 1)-approx. for DRPP.

Observation: any DRPP tour T enters each vertex of the input graph G asoften as it leaves.

We say that the multigraph G[T] induced by the arcs on T is balanced.

If we add to the set R of required arcs a minimum-cost multiset ofarcs R∗ such that G[R ] R∗] is balanced, then

c(R ] R∗) ≤ c(T)

and all connected components of G[R ] R∗] are Eulerian.

Algorithm: use the algorithm for Component-Wise Eulerian DRPP on Gwith the required arcs R ] R∗.

How to compute R∗?

c(R ] R∗) ≤ c(T)

Algorithm for general DRPP II

Goal: given a set R of required arcs in G, compute a minimum-costmultiset of arcs R∗ such that G[R ] R∗] is balanced.

Definition: Let balance(v) := indegG[R](v)− outdegG[R](v).

For a vertex v withI balance(v) > 0, the multiset R∗ has to contain balance(v) more

out-arcs than in-arcs incident to v,I balance(v) < 0, the multiset R∗ has to contain balance(v) more

in-arcs than out-arcs incident to v,I balance(v) = 0, the multiset R∗ has to contain as many out-arcs as

in-arcs incident to v.

polynomial-time solvable uncapacitated min-cost flow problem, wheredemand and supply of vertex v are given by balance(v).

out-arcs than in-arcs incident to v,

I balance(v) < 0, the multiset R∗ has to contain balance(v) morein-arcs than out-arcs incident to v,

I balance(v) = 0, the multiset R∗ has to contain as many out-arcs asin-arcs incident to v.

in-arcs than out-arcs incident to v,

I balance(v) = 0, the multiset R∗ has to contain as many out-arcs asin-arcs incident to v.

Goal: find a minimum-cost tour visiting all given required arcs and edgesin a mixed asymmetrically edge-weighted graph.

Claim: α(n)-approx.4-ATSP =⇒ (α(C) + 3)-approx. for MWRPP.

Algorithm for MWRPP

Claim: α(n)-approx. for 4-ATSP =⇒ (α(C) + 3)-approx. for MWRPP.

Trick: Just reduce MWRPP to the DRPP as follows:I replace each undirected edges by a pair of arcs directed into

opposite directions,I so that required edges are served in the cheaper direction.

If c(u, v) ≤ c(v, u), then (required arcs and edges are bold):

Every tour for the MWRPP instance can be turned into a tour for the DRPPinstance that pays at most the triple price for each required edge.

Algorithm for MWRPP

opposite directions,

I so that required edges are served in the cheaper direction.

Algorithm for MWRPP

Every tour for the MWRPP instance can be turned into a tour for the DRPPinstance that pays at most the triple price for each required edge.Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 12/16

MWCARP

Input: mixed graph G = (V ,A, E), depot v0 ∈ V , travel costsc : V × V → N ∪ {∞}, demands d : E ∪ A→ N, and a vehicle capacity Q.

Reminder: positive-demand arcs induce C weakly-connected components.

Goal: find a set of vehicle routes of total minimum cost, each starting andending in v0, such that each demand is served by exactly one vehicle andeach vehicle serves a total demand of at most Q.

Claim: α(n)-approx. for 4-ATSP =⇒ O(α(C + 1))-approx. for MWCARP.

MWCARP

Input: mixed graph G = (V ,A, E), depot v0 ∈ V , travel costsc : V × V → N ∪ {∞}, demands d : E ∪ A→ N, and a vehicle capacity Q.

Reminder: positive-demand arcs induce C weakly-connected components.

Goal: find a set of vehicle routes of total minimum cost, each starting andending in v0, such that each demand is served by exactly one vehicle andeach vehicle serves a total demand of at most Q.

Algorithm for MWCARP

Observation: Concatenating vehicle tours of optimal MWCARP solutiongives an MWRPP tour visiting all positive-demand arcs and depot v0.

1. Compute (α(C + 1) + 3)-approximate MWRPP tour T visitingdepot v0 and all positive-demand arcs.

2. Split it greedily into maximal subwalks of demand at most Q each.3. Add a shortest path from and to v0 to each subwalk.

≤ Q> Q

Approximation factor: Since c(T) ∈ O(α(C + 1) · OPT), it remains toanalyze the total length of the shortest paths added to the subwalks.

≤ Q> Q

2. Split it greedily into maximal subwalks of demand at most Q each.

3. Add a shortest path from and to v0 to each subwalk.

≤ Q> Q

Approximation factor analysis for MWCARP

Goal: Analyze total length of the shortest paths (dashed) added to thesubwalks of the MWRPP tour (solid).

One can show: each consecutive pair of subwalks contains an arc servedby a distinct vehicle tour of an optimal solution (dotted).

from and to depot vertex v0

Charge length of the blue shortest paths to the red walks, consisting ofarcs of the optimal solution and of arcs of the approximate MWRPP tour.

Each arc is charged to at most five times total cost of shortest paths isat most 5(α(C + 1) + 3) · OPT+ 5 · OPT.

Conclusion

We have seen thatI MWCARP is not only as hard to approximate as4-ATSP, but also how

any improvement of 4-ATSP approximations transfers to MWCARP,

I MWCARP is constant-factor approximable in many practical cases.

Future work:I Implementation and algorithm engineering.I Turn constraints, time windows, . . .

Conclusion

any improvement of 4-ATSP approximations transfers to MWCARP,I MWCARP is constant-factor approximable in many practical cases.

Conclusion

Future work:I Implementation and algorithm engineering.

I Turn constraints, time windows, . . .

Conclusion

Approximation Algorithms for Mixed,Windy,and Capacitated...

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