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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.
3
3
2
1
1
6
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)
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 2/16
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.
3
3
2
1
1
6
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)
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 2/16
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.
3
3
2
1
1
6
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)
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 2/16
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 3/16
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 3/16
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 3/16
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 3/16
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 3/16
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 3/16
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 3/16
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).
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 4/16
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).
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 4/16
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).
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 4/16
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).
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 4/16
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).
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 4/16
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)
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 5/16
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)
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 5/16
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)
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 5/16
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)
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 5/16
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)
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 5/16
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)
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 5/16
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)
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 5/16
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 6/16
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 6/16
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 6/16
Algorithm for Component-Wise Eulerian DRPP
Claim: α(n)-approximation for n-vertex4-ATSP=⇒ (α(C) + 1)-approximation 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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 7/16
Algorithm for Component-Wise Eulerian DRPP
Claim: α(n)-approximation for n-vertex4-ATSP=⇒ (α(C) + 1)-approximation 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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 7/16
Algorithm for Component-Wise Eulerian DRPP
Claim: α(n)-approximation for n-vertex4-ATSP=⇒ (α(C) + 1)-approximation for Component-Wise Eulerian DRPP.
Algorithm: Shown are only required arcs; directions are not shown.
For each connected component i of G[R], pick an arbitrary vertex vi .
Approximation factor: Optimum DRPP tour has at least the cost of therequired arcs and at least the cost of an optimal 4-ATSP tour.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 7/16
Algorithm for Component-Wise Eulerian DRPP
Claim: α(n)-approximation for n-vertex4-ATSP=⇒ (α(C) + 1)-approximation for Component-Wise Eulerian DRPP.
Algorithm: Shown are only required arcs; directions are not shown.
For each component i, compute an Euler tour starting and ending in vi .
Approximation factor: Optimum DRPP tour has at least the cost of therequired arcs and at least the cost of an optimal 4-ATSP tour.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 7/16
Algorithm for Component-Wise Eulerian DRPP
Claim: α(n)-approximation for n-vertex4-ATSP=⇒ (α(C) + 1)-approximation for Component-Wise Eulerian DRPP.
Algorithm: Shown are only required arcs; directions are not shown.
Join vertices v1, . . . , vC using α(C)-approximate 4-ATSP tour.
Approximation factor: Optimum DRPP tour has at least the cost of therequired arcs and at least the cost of an optimal 4-ATSP tour.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 7/16
Algorithm for Component-Wise Eulerian DRPP
Claim: α(n)-approximation for n-vertex4-ATSP=⇒ (α(C) + 1)-approximation for Component-Wise Eulerian DRPP.
Algorithm: Shown are only required arcs; directions are not shown.
Join vertices v1, . . . , vC using α(C)-approximate 4-ATSP tour.
Approximation factor: Optimum DRPP tour has at least the cost of therequired arcs and at least the cost of an optimal 4-ATSP tour.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 7/16
DRPP
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 8/16
DRPP
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 8/16
DRPP
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 8/16
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∗?
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 9/16
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∗?
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 9/16
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∗?
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 9/16
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∗?
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 9/16
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∗?
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 9/16
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∗?
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 9/16
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∗?
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 9/16
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).
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 10/16
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).
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 10/16
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) 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.
polynomial-time solvable uncapacitated min-cost flow problem, wheredemand and supply of vertex v are given by balance(v).
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 10/16
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 asin-arcs incident to v.
polynomial-time solvable uncapacitated min-cost flow problem, wheredemand and supply of vertex v are given by balance(v).
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 10/16
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).
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 10/16
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).
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 10/16
MWRPP
Goal: find a minimum-cost tour visiting all given required arcs and edgesin a mixed asymmetrically edge-weighted graph.
Reminder: G[R] consists of C weakly-connected components.
Claim: α(n)-approx.4-ATSP =⇒ (α(C) + 3)-approx. for MWRPP.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 11/16
MWRPP
Goal: find a minimum-cost tour visiting all given required arcs and edgesin a mixed asymmetrically edge-weighted graph.
Reminder: G[R] consists of C weakly-connected components.
Claim: α(n)-approx.4-ATSP =⇒ (α(C) + 3)-approx. for MWRPP.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 11/16
MWRPP
Goal: find a minimum-cost tour visiting all given required arcs and edgesin a mixed asymmetrically edge-weighted graph.
Reminder: G[R] consists of C weakly-connected components.
Claim: α(n)-approx.4-ATSP =⇒ (α(C) + 3)-approx. for MWRPP.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 11/16
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):
u
v
u
v
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
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):
u
v
u
v
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
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):
u
v
u
v
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
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):
u
v
u
v
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 13/16
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 13/16
Algorithm for MWCARP
Claim: α(n)-approx. for 4-ATSP =⇒ O(α(C + 1))-approx. 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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 14/16
Algorithm for MWCARP
Claim: α(n)-approx. for 4-ATSP =⇒ O(α(C + 1))-approx. 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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 14/16
Algorithm for MWCARP
Claim: α(n)-approx. for 4-ATSP =⇒ O(α(C + 1))-approx. 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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 14/16
Algorithm for MWCARP
Claim: α(n)-approx. for 4-ATSP =⇒ O(α(C + 1))-approx. 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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 14/16
Algorithm for MWCARP
Claim: α(n)-approx. for 4-ATSP =⇒ O(α(C + 1))-approx. 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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 14/16
Algorithm for MWCARP
Claim: α(n)-approx. for 4-ATSP =⇒ O(α(C + 1))-approx. 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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 14/16
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 15/16
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 15/16
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 15/16
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 15/16
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 15/16
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 15/16
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 15/16
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 15/16
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.
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 15/16
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, . . .
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 16/16
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, . . .
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 16/16
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, . . .
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 16/16
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, . . .
Mixed, Windy, and Capacitated Arc Routing Problems Rene van Bevern (Novosibirsk State University) 16/16