Approximating ATSP by Relaxing Connectivity
Approximating ATSP by Relaxing Connectivity Ola Svensson
Motivation of Asymmetric TSP
vs
Asymmetric Traveling Salesman Problem INPUT: an edge-weighted digraph 𝐺=(𝑉,𝐸,𝑤) OUTPUT: a tour of minimum weight that visits each vertex at least once
Asymmetric Traveling Salesman Problem INPUT: an edge-weighted digraph 𝐺=(𝑉,𝐸,𝑤) OUTPUT: a tour of minimum weight that visits each vertex at least once
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Asymmetric Traveling Salesman Problem INPUT: an edge-weighted digraph 𝐺=(𝑉,𝐸,𝑤) OUTPUT: a minimum weight connected Eulerian multigraph (𝑉,𝐸↑′ ) in-degree = out-degree
3 1 5000 1
Asymmetric Traveling Salesman Problem INPUT: an edge-weighted digraph 𝐺=(𝑉,𝐸,𝑤) OUTPUT: a minimum weight connected Eulerian multigraph (𝑉,𝐸↑′ ) Variables:
𝑥↓𝑢𝑣 = #times we traverse arc (𝑢,𝑣) Held-Karp Relaxation
Minimize: 𝑥↓𝑢𝑣
∑𝑢𝑣∈𝐸↑▒𝑤(𝑢,𝑣)
Subject to:
𝒙(𝜹↑+ (𝒗))=𝒙(𝜹↑− (𝒗))
for all 𝑣∈𝑉
𝒙(𝜹↑+ (𝑺))≥𝟏
for all S⊂𝑉
𝑥≥0
Two Approaches Easy to Find Eulerian graph
Easy to Find Connected Graph
Held-Karp Relaxation
Minimize: 𝑥↓𝑢𝑣
∑𝑢𝑣∈𝐸↑▒𝑤(𝑢,𝑣)
Subject to:
𝒙(𝜹↑+ (𝒗))=𝒙(𝜹↑− (𝒗))
for all 𝑣∈𝑉
𝒙(𝜹↑+ (𝑺))≥𝟏
for all S⊂𝑉
𝑥≥0
Find min-cost cycle cover
Repeated Cycle Cover
“Contract“ Repeat until graph is connected
Find min-cost cycle cover
Repeated Cycle Cover
“Contract“ Repeat until graph is connected
Find min-cost cycle cover
Repeated Cycle Cover
“Contract“ Repeat until graph is connected
Find min-cost cycle cover
Repeated Cycle Cover
“Contract“ Repeat until graph is connected
Find min-cost cycle cover
Repeated Cycle Cover
“Contract“ Repeat until graph is connected
Find min-cost cycle cover
Repeated Cycle Cover
“Contract“ Repeat until graph is connected
Find min-cost cycle cover
Repeated Cycle Cover
“Contract“ Repeat until graph is connected
Worst case: all cycles have length 2 so we need to repeat log↓2 𝑛 times (each time cost 𝑂𝑃𝑇↓𝐿
Two Approaches Easy to Find Eulerian graph Repeatedly find cycle-covers to get log↓2 𝑛 -approximation [Frieze, Galbiati, Maffiolo’82]
0.99log↓2 𝑛 -approximation [Bläser’03]
0.84log↓2 𝑛 -approximation [Kaplan, Lewenstein, Shafrir, Sviridenko’05]
Easy to Find Connected Graph
Thin Tree Approach
Find (undirected) spanning tree Obtain Eulerian graph via a circulation that sends one unit through each tree edge
Cost of LP ≈𝟏 red edge
…
…
…
𝑘 vertices
…
Thin Tree Approach
Find (undirected) spanning tree Obtain Eulerian graph via a circulation that sends one unit through each tree edge
Cost of LP ≈𝟏 red edge
…
…
…
𝑘 vertices
…
Thin Tree Approach
Find (undirected) spanning tree Obtain Eulerian graph via a circulation that sends one unit through each tree edge
Cost of LP ≈𝟏 red edge
…
…
…
𝑘 vertices
…
Thin Tree Approach
Find (undirected) spanning tree Obtain Eulerian graph via a circulation that sends one unit through each tree edge
⇒ find a “thin” tree for each cut it has not many more edges than the LP predicts Cost of LP ≈𝟏 red edge
…
…
…
…
𝑘 vertices
Cost of c𝐢𝐫𝐜𝐮𝐥𝐚𝐭𝐢𝐨𝐧≥𝟑 red edges
Two Approaches Easy to Find Eulerian graph
Easy to Find Connected Graph
Repeatedly find cycle-covers to get log↓2 𝑛 -approximation
𝑂(log𝑛 /loglog𝑛) -approximation
[Asadpour, Goemans, Madry, Oveis Gharan, Saberi’10]
[Frieze, Galbiati, Maffiolo’82]
𝑂(1)-approximation for planar and bounded genus graphs
0.99log↓2 𝑛 -approximation
[Oveis Gharan, Saberi’11]
[Bläser’03]
0.84log↓2 𝑛 -approximation [Kaplan, Lewenstein, Shafrir, Sviridenko’05]
O(𝑝𝑜𝑙𝑦 𝑙𝑜𝑔log𝑛) bound on integrality gap (exact formulation for sparsest cut and generalization of Kadison-Singer) [Anari, Oveis Gharan’14]
To summarize Best approximation algorithm: 𝑂(log𝑛 /loglog𝑛 ) Best upper bound on integrality gap: 𝑂(𝑝𝑜𝑙𝑦loglog𝑛) Best lower bound on integrality gap: 2 This is believed to be close to the truth
No better guarantees for shortest path metrics on unweighted graphs for which there was recent improvements for the symmetric TSP • Main difficulty in cycle cover approach is to bound #iterations • In thin-tree approach we reduce ATSP to an unweighted problem
A NEW APPROACH
Application of approach THEOREM:
For ATSP on node-weighted graphs, the integrality gap is at most 15. Moreover, there is a 27-approximation algorithm.
Exist 𝑓:𝑉→𝑅↑+ s.t
𝑤(𝑢,𝑣) = 𝑓(𝑢) for all (𝑢,𝑣)∈𝐸
2 5 1
5 5
1
Application of approach THEOREM:
For ATSP on node-weighted graphs, the integrality gap is at most 15. Moreover, there is a 27-approximation algorithm.
Exist 𝑓:𝑉→𝑅↑+ s.t
𝑤(𝑢,𝑣) = 𝑓(𝑢) for all (𝑢,𝑣)∈𝐸
2 5 1
5 5
Value = 2+5+1+5+1=14
1
Chicago
Lausanne Boston
Puerto Rico
General Metric w.l.o.g. exist laminar family such that for each edge 𝑒 𝑤(𝑒)= ∑𝑆: 𝑒∈𝛿(𝑆)↑▒𝑦↓𝑆
1
2
4 10
3
Do better than 1.5 for symmetric TSP on node-weighted metrics?
Cousin to repeated cycle cover approach Distant relative to thin tree approach
OUR APPROACH
Relaxing Connectivity
Instead of
𝒙(𝜹↑+ (𝒗))=𝒙(𝜹↑− (𝒗))
for all 𝑣∈𝑉
𝒙(𝜹↑+ (𝑺))≥𝟏
for all 𝐒⊂𝑽
𝑥≥0
Do for smart C
𝒙(𝜹↑+ (𝒗))=𝒙(𝜹↑− (𝒗))
for all 𝑣∈𝑉
𝒙(𝜹↑+ (𝑺))≥𝟏
for those 𝐒∈𝑪
𝑥≥0
Local Connectivity ATSP INPUT: an edge-weighted digraph 𝐺=(𝑉,𝐸,𝑤), a partition 𝑉↓1 ∪…∪𝑉↓𝑘 of 𝑉
Local Connectivity ATSP INPUT: an edge-weighted digraph 𝐺=(𝑉,𝐸,𝑤), a partition 𝑉↓1 ∪…∪𝑉↓𝑘 of 𝑉 OUTPUT: an Eulerian multisubset of edges 𝐹 such that each cut (𝑉↓𝑖 , 𝑉↓𝑖 ) is “covered” by 𝐹
Local Connectivity ATSP INPUT: an edge-weighted digraph 𝐺=(𝑉,𝐸,𝑤), a partition 𝑉↓1 ∪…∪𝑉↓𝑘 of 𝑉 OUTPUT: an Eulerian multisubset of edges 𝐹 such that each cut (𝑉↓𝑖 , 𝑉↓𝑖 ) is “covered” by 𝐹
Local Connectivity ATSP An algorithm is 𝜶-light if it always outputs a solution 𝐹 s.t. for each component (𝑽’, 𝑭’) in (𝑽, 𝑭), 𝒘(𝑭↑′ )/𝒍𝒃(𝑽↑′ ) ≤𝜶, where 𝑙𝑏(𝑣)=∑𝑒∈𝛿↑+ (𝑣)↑▒𝑥↓𝑒↑∗ 𝑤(𝑒) and 𝑥↑∗ is optimal solution to LP Note: designing an 𝜶-light algorithm is “easier” than an 𝜶-approximation for ATSP
6/5
1
Our main technical result THEOREM:
If there is an 𝛼-light algorithm A for Local-Connectivity ATSP then the integrality gap for ATSP is at most 5𝛼.
Our main technical result THEOREM:
If there is an 𝛼-light algorithm A for Local-Connectivity ATSP then the integrality gap for ATSP is at most 5𝛼. Moreover, a 9𝛼-approximate tour can be found in polynomial time if A runs in polynomial time.
The problems are equivalent up to a small constant factor There is an easy 3-light algorithm for node-weighted metric (only part where special metric is used)
OUR “TURING” REDUCTION
Initialization
Partition vertices of 𝐺 into Eulerian graphs 𝐻↓1↑∗ =(𝑉↓1↑∗ ,
𝐸↓1↑∗ ), …, 𝐻↓𝑘↑∗ =(𝑉↓𝑘↑∗ , 𝐸↓𝑘↑∗ ) s.t.
1. 𝐻↓𝑖↑∗ is 2𝛼-light, i.e., 𝑤(𝐸↓𝑖↑∗ )/𝑙𝑏(𝑉↓𝑖↑∗ ) 2.
The lexicographic order of is maximized
𝐻↓2↑∗
𝐻↓1↑∗
𝐻↓3↑∗
𝐻↓ 4↑ ∗
𝐻 ↓ 5 ↑ ∗
𝐻 ↓ 6 ↑ ∗
𝐻 ↓ 7 ↑ ∗
Initialization
Partition vertices of 𝐺 into Eulerian graphs 𝐻↓1↑∗ =(𝑉↓1↑∗ ,
𝐸↓1↑∗ ), …, 𝐻↓𝑘↑∗ =(𝑉↓𝑘↑∗ , 𝐸↓𝑘↑∗ ) s.t.
1. 𝐻↓𝑖↑∗ is 2𝛼-light, i.e., 𝑤(𝐸↓𝑖↑∗ )/𝑙𝑏(𝑉↓𝑖↑∗ ) 2.
The lexicographic order of :𝑯is↓𝒊↑ ∗ 𝒊𝒏𝒕𝒆𝒓𝒔𝒆𝒄𝒕𝒔 𝑯) ↑∗ 𝐻↓𝑘↑ ∗ ) maximized
𝐻↓2↑∗
𝐻↓1↑∗
𝐻↓3↑∗
𝐻↓ 4↑ ∗
𝑙𝑜𝑤(𝐻)= 𝐻↓2↑∗ 𝐻 𝐻 ↓ ↓ 6 5 ↑ ↑ ∗ ∗
𝐻 ↓ 7 ↑ ∗
Initialization
Partition vertices of 𝐺 into Eulerian graphs 𝐻↓1↑∗ =(𝑉↓1↑∗ ,
𝐸↓1↑∗ ), …, 𝐻↓𝑘↑∗ =(𝑉↓𝑘↑∗ , 𝐸↓𝑘↑∗ ) s.t.
1. 𝐻↓𝑖↑∗ is 2𝛼-light, i.e., 𝑤(𝐸↓𝑖↑∗ )/𝑙𝑏(𝑉↓𝑖↑∗ ) 2.
The lexicographic order of :𝑯is↓𝒊↑ ∗ 𝒊𝒏𝒕𝒆𝒓𝒔𝒆𝒄𝒕𝒔 𝑯) ↑∗ 𝐻↓𝑘↑ ∗ ) maximized
𝐻↓1↑∗ 𝑙𝑜𝑤(𝐻)=𝐻↓1↑∗
𝐻↓3↑∗
𝐻↓ 4↑ ∗
𝐻↓2↑∗ 𝐻 ↓ 5 ↑ ∗
𝐻 ↓ 6 ↑ ∗
𝐻 ↓ 7 ↑ ∗
Initialization
Partition vertices of 𝐺 into Eulerian graphs 𝐻↓1↑∗ =(𝑉↓1↑∗ ,
𝐸↓1↑∗ ), …, 𝐻↓𝑘↑∗ =(𝑉↓𝑘↑∗ , 𝐸↓𝑘↑∗ ) s.t.
1. 𝐻↓𝑖↑∗ is 2𝛼-light, i.e., 𝑤(𝐸↓𝑖↑∗ )/𝑙𝑏(𝑉↓𝑖↑∗ ) 2.
The lexicographic order of :𝑯is↓𝒊↑ ∗ 𝒊𝒏𝒕𝒆𝒓𝒔𝒆𝒄𝒕𝒔 𝑯) ↑∗ 𝐻↓𝑘↑ ∗ ) maximized Claim 1: For any 2𝛼-light Eulerian subgraph 𝐻, we have 𝑙𝑏(𝐻)≤𝑙𝑏(𝑙𝑜𝑤(𝐻))
𝐻↓2↑∗
𝐻↓1↑∗
𝐻↓3↑∗
𝐻↓ 4↑ ∗
𝐻 ↓ 5 ↑ ∗
𝐻 ↓ 7 ↑ ∗ 𝑙𝑜𝑤(𝐻)=𝐻↓2↑∗ 𝐻 ↓ 6 ↑ ∗
Initialization
Partition vertices of 𝐺 into Eulerian graphs 𝐻↓1↑∗ =(𝑉↓1↑∗ ,
𝐸↓1↑∗ ), …, 𝐻↓𝑘↑∗ =(𝑉↓𝑘↑∗ , 𝐸↓𝑘↑∗ ) s.t.
1. 𝐻↓𝑖↑∗ is 2𝛼-light, i.e., 𝑤(𝐸↓𝑖↑∗ )/𝑙𝑏(𝑉↓𝑖↑∗ ) 2.
The lexicographic order of :𝑯is↓𝒊↑ ∗ 𝒊𝒏𝒕𝒆𝒓𝒔𝒆𝒄𝒕𝒔 𝑯) ↑∗ 𝐻↓𝑘↑ ∗ ) maximized Claim 1: For any 2𝛼-light Eulerian subgraph 𝐻, we have 𝑙𝑏(𝐻)≤𝑙𝑏(𝑙𝑜𝑤(𝐻))
Claim 1I: For any 𝛼-light vertex-disjoint Eulerian subgraphs 𝐻↓1 , 𝐻↓2 , …, 𝐻↓𝑙 , such that 𝑙𝑜𝑤(𝐻↓𝑗 )=𝐻↓𝑖↑∗ we have ∑𝑗=1↑𝑙▒𝑙𝑏(𝐻↓𝑗 ) ≤2𝑙𝑏(𝐻↓𝑖↑∗ )
𝐻↓2↑∗
𝐻↓1↑∗
𝐻↓3↑∗
𝐻↓ 4↑ ∗
𝐻 ↓ 5 ↑ ∗
𝐻 ↓ 7 ↑ ∗ 𝑙𝑜𝑤(𝐻↓𝑗 )=𝐻↓2↑∗ 𝐻 ↓ 6 ↑ ∗
Merging
𝐻↓9 ↑∗
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
𝐻↓9 ↑∗
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
𝐻↓9 ↑∗
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻))
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
temporarily add it and repeat from (*)
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗
𝐻↓3↑∗
temporarily add it and repeat from (*)
𝐻↓5↑ ∗
𝐻↓6 ↑∗
𝐻↓2↑∗
𝐻↓7↑∗ 𝐻 ↓ 1 1 𝐻↓4↑ ↑ 𝐻↓8↑ ∗ ∗ ∗ 𝑤(𝐶)≤𝛼𝑙𝑏(𝐻↓6↑∗ )
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
temporarily add it and repeat from (*)
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
ANALYSIS
Lemma 1: Total cost of blue edges ≤𝛼∑↑▒𝑙𝑏(𝐻↓𝑖↑∗ )
=𝛼𝑂𝑃𝑇↓𝐻𝐾 • Charging scheme
𝐻 𝐻 𝐻 ↓ ↓ 𝐻↓2↑∗ 𝐻↓1↑∗ ↓ 5 9 𝐻 ↑ 1 ↑ 𝐻 ↓ 0 ∗ 𝐻 ∗ 𝐻↓3 𝐻↓7 6 ↑ ↓ ↑∗ ↑∗ ↓↑∗ 4∗ 8 ↑ ↑ Charged to 𝐻 ↓ 6 ↑ ∗ ∗ ∗
𝐻 𝐻 𝐻 ↓ ↓ 𝐻↓1↑∗ ↓ 5 9 1 ↑ 𝐻 ↑ 𝐻0 ↓∗6 ∗ 𝐻↓3 ↑ ↓ ↑∗ ↑∗ 4∗ ↑ Charged ∗ to 𝐻↓2↑∗
𝐻↓2↑∗ 𝐻 𝐻↓7 ↓↑∗ 8 ↑ ∗
• At most one cycle of weight ≤𝛼𝑙𝑏(𝐻↓𝑖↑∗ ) is charged to each 𝐻↓𝑖↑∗
Lemma 2: Total cost of red edges ≤2𝛼∑𝐻↓𝑖↑∗ =2𝛼𝑂𝑃
𝑇↓𝐻𝐾
Charging scheme: the red components with 𝑙𝑜𝑤(𝐻)=𝐻↓𝑖↑∗ is charged to 𝐻↓𝑖↑∗ Charge in one iteration to 𝑯↓𝟓↑∗ is at most 𝟐𝜶𝒍𝒃(𝑯↓𝟓↑∗ )
𝐻↓5↑ ∗ 𝐻↓2
𝐻↓1 𝐻 ↓1 ↓6 ↑∗ 𝐻 0↑ ∗ 𝐻 ↓ 1 1 ↑ ∗
𝑤(𝐻↓1 )+𝑤(𝐻↓2 )+𝑤(𝐻↓3 )
≤𝛼𝑙𝑏(𝐻↓1 )+𝛼𝑙𝑏(𝐻↓2 ) +𝛼𝑙𝑏(𝐻↓3 ) ↓3 𝐻 ≤2𝛼𝑙𝑏(𝐻↓5↑∗ )
𝐻↓7↑∗ By Claim 2
𝐻↓3↑∗
Lemma 2: Total cost of red edges ≤2𝛼∑𝐻↓𝑖↑∗ =2𝛼𝑂𝑃
𝑇↓𝐻𝐾
Charging scheme: the red components with 𝑙𝑜𝑤(𝐻)=𝐻↓𝑖↑∗ is charged to 𝐻↓𝑖↑∗ Charge in one iteration to 𝑯↓𝟓↑∗ is at most 𝟐𝜶𝒍𝒃(𝑯↓𝟓↑∗ )
Lemma 2: Total cost of red edges ≤2𝛼∑𝐻↓𝑖↑∗ =2𝛼𝑂𝑃
𝑇↓𝐻𝐾
Charging scheme: the red components with 𝑙𝑜𝑤(𝐻)=𝐻↓𝑖↑∗ is charged to 𝐻↓𝑖↑∗ Charge in one iteration to 𝑯↓𝟓↑∗ is at most 𝟐𝜶𝒍𝒃(𝑯↓𝟓↑∗ )
𝑯↓𝟓↑∗ charged in at most one iteration
𝐻 ↓1 0↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
Suppose toward contradiction that not
𝐻
𝐻↓7↑∗
𝐻↓3↑∗
𝐻↓ 9↑∗
By Claim 1 𝑤(𝐻)≤𝛼𝑙𝑏(𝐻↓5↑∗ ) ≤𝛼𝑙𝑏(𝐻↓3↑∗ )
𝐻↓8↑∗ Hence, exist cycle C which contradicts that the ifcondition was not satisfied
Lemma 2: Total cost of red edges ≤2𝛼∑𝐻↓𝑖↑∗ =2𝛼𝑂𝑃
𝑇↓𝐻𝐾
Charge in one iteration to 𝑯↓𝟓↑∗ is at most 𝟐𝜶𝒍𝒃(𝑯↓𝟓↑∗ )
𝑯↓𝟓↑∗ charged in at most one iteration
Total Cost: Eulerian partition + Blue edges + Red edges ≤2𝛼𝑂𝑃𝑇↓𝐻𝐾 +𝛼𝑂𝑃𝑇↓𝐻𝐾 +2𝛼𝑂𝑃𝑇↓𝐻𝐾
Summary • New approach that relaxes connectivity • Integrality gap for node-weighted metrics is ≤13 • Q1: Solve Local-Connectivity ATSP for general metrics? • Very flexible choice of lb can be helpful • Primal-dual? • Q2: Tight answer for Node-Weighted? • Q3: What about Node-Weighted Symmetric TSP?
Motivation of Asymmetric TSP
vs
Asymmetric Traveling Salesman Problem INPUT: an edge-weighted digraph 𝐺=(𝑉,𝐸,𝑤) OUTPUT: a tour of minimum weight that visits each vertex at least once
Asymmetric Traveling Salesman Problem INPUT: an edge-weighted digraph 𝐺=(𝑉,𝐸,𝑤) OUTPUT: a tour of minimum weight that visits each vertex at least once
3 1 5000 1
Asymmetric Traveling Salesman Problem INPUT: an edge-weighted digraph 𝐺=(𝑉,𝐸,𝑤) OUTPUT: a minimum weight connected Eulerian multigraph (𝑉,𝐸↑′ ) in-degree = out-degree
3 1 5000 1
Asymmetric Traveling Salesman Problem INPUT: an edge-weighted digraph 𝐺=(𝑉,𝐸,𝑤) OUTPUT: a minimum weight connected Eulerian multigraph (𝑉,𝐸↑′ ) Variables:
𝑥↓𝑢𝑣 = #times we traverse arc (𝑢,𝑣) Held-Karp Relaxation
Minimize: 𝑥↓𝑢𝑣
∑𝑢𝑣∈𝐸↑▒𝑤(𝑢,𝑣)
Subject to:
𝒙(𝜹↑+ (𝒗))=𝒙(𝜹↑− (𝒗))
for all 𝑣∈𝑉
𝒙(𝜹↑+ (𝑺))≥𝟏
for all S⊂𝑉
𝑥≥0
Two Approaches Easy to Find Eulerian graph
Easy to Find Connected Graph
Held-Karp Relaxation
Minimize: 𝑥↓𝑢𝑣
∑𝑢𝑣∈𝐸↑▒𝑤(𝑢,𝑣)
Subject to:
𝒙(𝜹↑+ (𝒗))=𝒙(𝜹↑− (𝒗))
for all 𝑣∈𝑉
𝒙(𝜹↑+ (𝑺))≥𝟏
for all S⊂𝑉
𝑥≥0
Find min-cost cycle cover
Repeated Cycle Cover
“Contract“ Repeat until graph is connected
Find min-cost cycle cover
Repeated Cycle Cover
“Contract“ Repeat until graph is connected
Find min-cost cycle cover
Repeated Cycle Cover
“Contract“ Repeat until graph is connected
Find min-cost cycle cover
Repeated Cycle Cover
“Contract“ Repeat until graph is connected
Find min-cost cycle cover
Repeated Cycle Cover
“Contract“ Repeat until graph is connected
Find min-cost cycle cover
Repeated Cycle Cover
“Contract“ Repeat until graph is connected
Find min-cost cycle cover
Repeated Cycle Cover
“Contract“ Repeat until graph is connected
Worst case: all cycles have length 2 so we need to repeat log↓2 𝑛 times (each time cost 𝑂𝑃𝑇↓𝐿
Two Approaches Easy to Find Eulerian graph Repeatedly find cycle-covers to get log↓2 𝑛 -approximation [Frieze, Galbiati, Maffiolo’82]
0.99log↓2 𝑛 -approximation [Bläser’03]
0.84log↓2 𝑛 -approximation [Kaplan, Lewenstein, Shafrir, Sviridenko’05]
Easy to Find Connected Graph
Thin Tree Approach
Find (undirected) spanning tree Obtain Eulerian graph via a circulation that sends one unit through each tree edge
Cost of LP ≈𝟏 red edge
…
…
…
𝑘 vertices
…
Thin Tree Approach
Find (undirected) spanning tree Obtain Eulerian graph via a circulation that sends one unit through each tree edge
Cost of LP ≈𝟏 red edge
…
…
…
𝑘 vertices
…
Thin Tree Approach
Find (undirected) spanning tree Obtain Eulerian graph via a circulation that sends one unit through each tree edge
Cost of LP ≈𝟏 red edge
…
…
…
𝑘 vertices
…
Thin Tree Approach
Find (undirected) spanning tree Obtain Eulerian graph via a circulation that sends one unit through each tree edge
⇒ find a “thin” tree for each cut it has not many more edges than the LP predicts Cost of LP ≈𝟏 red edge
…
…
…
…
𝑘 vertices
Cost of c𝐢𝐫𝐜𝐮𝐥𝐚𝐭𝐢𝐨𝐧≥𝟑 red edges
Two Approaches Easy to Find Eulerian graph
Easy to Find Connected Graph
Repeatedly find cycle-covers to get log↓2 𝑛 -approximation
𝑂(log𝑛 /loglog𝑛) -approximation
[Asadpour, Goemans, Madry, Oveis Gharan, Saberi’10]
[Frieze, Galbiati, Maffiolo’82]
𝑂(1)-approximation for planar and bounded genus graphs
0.99log↓2 𝑛 -approximation
[Oveis Gharan, Saberi’11]
[Bläser’03]
0.84log↓2 𝑛 -approximation [Kaplan, Lewenstein, Shafrir, Sviridenko’05]
O(𝑝𝑜𝑙𝑦 𝑙𝑜𝑔log𝑛) bound on integrality gap (exact formulation for sparsest cut and generalization of Kadison-Singer) [Anari, Oveis Gharan’14]
To summarize Best approximation algorithm: 𝑂(log𝑛 /loglog𝑛 ) Best upper bound on integrality gap: 𝑂(𝑝𝑜𝑙𝑦loglog𝑛) Best lower bound on integrality gap: 2 This is believed to be close to the truth
No better guarantees for shortest path metrics on unweighted graphs for which there was recent improvements for the symmetric TSP • Main difficulty in cycle cover approach is to bound #iterations • In thin-tree approach we reduce ATSP to an unweighted problem
A NEW APPROACH
Application of approach THEOREM:
For ATSP on node-weighted graphs, the integrality gap is at most 15. Moreover, there is a 27-approximation algorithm.
Exist 𝑓:𝑉→𝑅↑+ s.t
𝑤(𝑢,𝑣) = 𝑓(𝑢) for all (𝑢,𝑣)∈𝐸
2 5 1
5 5
1
Application of approach THEOREM:
For ATSP on node-weighted graphs, the integrality gap is at most 15. Moreover, there is a 27-approximation algorithm.
Exist 𝑓:𝑉→𝑅↑+ s.t
𝑤(𝑢,𝑣) = 𝑓(𝑢) for all (𝑢,𝑣)∈𝐸
2 5 1
5 5
Value = 2+5+1+5+1=14
1
Chicago
Lausanne Boston
Puerto Rico
General Metric w.l.o.g. exist laminar family such that for each edge 𝑒 𝑤(𝑒)= ∑𝑆: 𝑒∈𝛿(𝑆)↑▒𝑦↓𝑆
1
2
4 10
3
Do better than 1.5 for symmetric TSP on node-weighted metrics?
Cousin to repeated cycle cover approach Distant relative to thin tree approach
OUR APPROACH
Relaxing Connectivity
Instead of
𝒙(𝜹↑+ (𝒗))=𝒙(𝜹↑− (𝒗))
for all 𝑣∈𝑉
𝒙(𝜹↑+ (𝑺))≥𝟏
for all 𝐒⊂𝑽
𝑥≥0
Do for smart C
𝒙(𝜹↑+ (𝒗))=𝒙(𝜹↑− (𝒗))
for all 𝑣∈𝑉
𝒙(𝜹↑+ (𝑺))≥𝟏
for those 𝐒∈𝑪
𝑥≥0
Local Connectivity ATSP INPUT: an edge-weighted digraph 𝐺=(𝑉,𝐸,𝑤), a partition 𝑉↓1 ∪…∪𝑉↓𝑘 of 𝑉
Local Connectivity ATSP INPUT: an edge-weighted digraph 𝐺=(𝑉,𝐸,𝑤), a partition 𝑉↓1 ∪…∪𝑉↓𝑘 of 𝑉 OUTPUT: an Eulerian multisubset of edges 𝐹 such that each cut (𝑉↓𝑖 , 𝑉↓𝑖 ) is “covered” by 𝐹
Local Connectivity ATSP INPUT: an edge-weighted digraph 𝐺=(𝑉,𝐸,𝑤), a partition 𝑉↓1 ∪…∪𝑉↓𝑘 of 𝑉 OUTPUT: an Eulerian multisubset of edges 𝐹 such that each cut (𝑉↓𝑖 , 𝑉↓𝑖 ) is “covered” by 𝐹
Local Connectivity ATSP An algorithm is 𝜶-light if it always outputs a solution 𝐹 s.t. for each component (𝑽’, 𝑭’) in (𝑽, 𝑭), 𝒘(𝑭↑′ )/𝒍𝒃(𝑽↑′ ) ≤𝜶, where 𝑙𝑏(𝑣)=∑𝑒∈𝛿↑+ (𝑣)↑▒𝑥↓𝑒↑∗ 𝑤(𝑒) and 𝑥↑∗ is optimal solution to LP Note: designing an 𝜶-light algorithm is “easier” than an 𝜶-approximation for ATSP
6/5
1
Our main technical result THEOREM:
If there is an 𝛼-light algorithm A for Local-Connectivity ATSP then the integrality gap for ATSP is at most 5𝛼.
Our main technical result THEOREM:
If there is an 𝛼-light algorithm A for Local-Connectivity ATSP then the integrality gap for ATSP is at most 5𝛼. Moreover, a 9𝛼-approximate tour can be found in polynomial time if A runs in polynomial time.
The problems are equivalent up to a small constant factor There is an easy 3-light algorithm for node-weighted metric (only part where special metric is used)
OUR “TURING” REDUCTION
Initialization
Partition vertices of 𝐺 into Eulerian graphs 𝐻↓1↑∗ =(𝑉↓1↑∗ ,
𝐸↓1↑∗ ), …, 𝐻↓𝑘↑∗ =(𝑉↓𝑘↑∗ , 𝐸↓𝑘↑∗ ) s.t.
1. 𝐻↓𝑖↑∗ is 2𝛼-light, i.e., 𝑤(𝐸↓𝑖↑∗ )/𝑙𝑏(𝑉↓𝑖↑∗ ) 2.
The lexicographic order of is maximized
𝐻↓2↑∗
𝐻↓1↑∗
𝐻↓3↑∗
𝐻↓ 4↑ ∗
𝐻 ↓ 5 ↑ ∗
𝐻 ↓ 6 ↑ ∗
𝐻 ↓ 7 ↑ ∗
Initialization
Partition vertices of 𝐺 into Eulerian graphs 𝐻↓1↑∗ =(𝑉↓1↑∗ ,
𝐸↓1↑∗ ), …, 𝐻↓𝑘↑∗ =(𝑉↓𝑘↑∗ , 𝐸↓𝑘↑∗ ) s.t.
1. 𝐻↓𝑖↑∗ is 2𝛼-light, i.e., 𝑤(𝐸↓𝑖↑∗ )/𝑙𝑏(𝑉↓𝑖↑∗ ) 2.
The lexicographic order of :𝑯is↓𝒊↑ ∗ 𝒊𝒏𝒕𝒆𝒓𝒔𝒆𝒄𝒕𝒔 𝑯) ↑∗ 𝐻↓𝑘↑ ∗ ) maximized
𝐻↓2↑∗
𝐻↓1↑∗
𝐻↓3↑∗
𝐻↓ 4↑ ∗
𝑙𝑜𝑤(𝐻)= 𝐻↓2↑∗ 𝐻 𝐻 ↓ ↓ 6 5 ↑ ↑ ∗ ∗
𝐻 ↓ 7 ↑ ∗
Initialization
Partition vertices of 𝐺 into Eulerian graphs 𝐻↓1↑∗ =(𝑉↓1↑∗ ,
𝐸↓1↑∗ ), …, 𝐻↓𝑘↑∗ =(𝑉↓𝑘↑∗ , 𝐸↓𝑘↑∗ ) s.t.
1. 𝐻↓𝑖↑∗ is 2𝛼-light, i.e., 𝑤(𝐸↓𝑖↑∗ )/𝑙𝑏(𝑉↓𝑖↑∗ ) 2.
The lexicographic order of :𝑯is↓𝒊↑ ∗ 𝒊𝒏𝒕𝒆𝒓𝒔𝒆𝒄𝒕𝒔 𝑯) ↑∗ 𝐻↓𝑘↑ ∗ ) maximized
𝐻↓1↑∗ 𝑙𝑜𝑤(𝐻)=𝐻↓1↑∗
𝐻↓3↑∗
𝐻↓ 4↑ ∗
𝐻↓2↑∗ 𝐻 ↓ 5 ↑ ∗
𝐻 ↓ 6 ↑ ∗
𝐻 ↓ 7 ↑ ∗
Initialization
Partition vertices of 𝐺 into Eulerian graphs 𝐻↓1↑∗ =(𝑉↓1↑∗ ,
𝐸↓1↑∗ ), …, 𝐻↓𝑘↑∗ =(𝑉↓𝑘↑∗ , 𝐸↓𝑘↑∗ ) s.t.
1. 𝐻↓𝑖↑∗ is 2𝛼-light, i.e., 𝑤(𝐸↓𝑖↑∗ )/𝑙𝑏(𝑉↓𝑖↑∗ ) 2.
The lexicographic order of :𝑯is↓𝒊↑ ∗ 𝒊𝒏𝒕𝒆𝒓𝒔𝒆𝒄𝒕𝒔 𝑯) ↑∗ 𝐻↓𝑘↑ ∗ ) maximized Claim 1: For any 2𝛼-light Eulerian subgraph 𝐻, we have 𝑙𝑏(𝐻)≤𝑙𝑏(𝑙𝑜𝑤(𝐻))
𝐻↓2↑∗
𝐻↓1↑∗
𝐻↓3↑∗
𝐻↓ 4↑ ∗
𝐻 ↓ 5 ↑ ∗
𝐻 ↓ 7 ↑ ∗ 𝑙𝑜𝑤(𝐻)=𝐻↓2↑∗ 𝐻 ↓ 6 ↑ ∗
Initialization
Partition vertices of 𝐺 into Eulerian graphs 𝐻↓1↑∗ =(𝑉↓1↑∗ ,
𝐸↓1↑∗ ), …, 𝐻↓𝑘↑∗ =(𝑉↓𝑘↑∗ , 𝐸↓𝑘↑∗ ) s.t.
1. 𝐻↓𝑖↑∗ is 2𝛼-light, i.e., 𝑤(𝐸↓𝑖↑∗ )/𝑙𝑏(𝑉↓𝑖↑∗ ) 2.
The lexicographic order of :𝑯is↓𝒊↑ ∗ 𝒊𝒏𝒕𝒆𝒓𝒔𝒆𝒄𝒕𝒔 𝑯) ↑∗ 𝐻↓𝑘↑ ∗ ) maximized Claim 1: For any 2𝛼-light Eulerian subgraph 𝐻, we have 𝑙𝑏(𝐻)≤𝑙𝑏(𝑙𝑜𝑤(𝐻))
Claim 1I: For any 𝛼-light vertex-disjoint Eulerian subgraphs 𝐻↓1 , 𝐻↓2 , …, 𝐻↓𝑙 , such that 𝑙𝑜𝑤(𝐻↓𝑗 )=𝐻↓𝑖↑∗ we have ∑𝑗=1↑𝑙▒𝑙𝑏(𝐻↓𝑗 ) ≤2𝑙𝑏(𝐻↓𝑖↑∗ )
𝐻↓2↑∗
𝐻↓1↑∗
𝐻↓3↑∗
𝐻↓ 4↑ ∗
𝐻 ↓ 5 ↑ ∗
𝐻 ↓ 7 ↑ ∗ 𝑙𝑜𝑤(𝐻↓𝑗 )=𝐻↓2↑∗ 𝐻 ↓ 6 ↑ ∗
Merging
𝐻↓9 ↑∗
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
𝐻↓9 ↑∗
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
𝐻↓9 ↑∗
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻))
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
temporarily add it and repeat from (*)
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗
𝐻↓3↑∗
temporarily add it and repeat from (*)
𝐻↓5↑ ∗
𝐻↓6 ↑∗
𝐻↓2↑∗
𝐻↓7↑∗ 𝐻 ↓ 1 1 𝐻↓4↑ ↑ 𝐻↓8↑ ∗ ∗ ∗ 𝑤(𝐶)≤𝛼𝑙𝑏(𝐻↓6↑∗ )
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
temporarily add it and repeat from (*)
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
Merging
Use 𝛼-light algorithm for LC-ATSP to find F w.r.t. current components (*) Select component 𝐻 with smallest 𝑙𝑏(𝑙𝑜𝑤(𝐻)) If exist cycle 𝐶 that connects 𝐻 to another component and
𝑤(𝐶)≤𝛼𝑙𝑏(𝑙𝑜𝑤(𝐻))
temporarily add it and repeat from (*)
Otherwise add new edges belonging to this component
𝐻↓9 ↑∗
𝐻↓1↑∗ 𝐻 ↓1 0↑ ∗ 𝐻↓3↑∗ 𝐻↓4↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
𝐻↓2↑∗
𝐻↓7↑∗
𝐻↓8↑ ∗
ANALYSIS
Lemma 1: Total cost of blue edges ≤𝛼∑↑▒𝑙𝑏(𝐻↓𝑖↑∗ )
=𝛼𝑂𝑃𝑇↓𝐻𝐾 • Charging scheme
𝐻 𝐻 𝐻 ↓ ↓ 𝐻↓2↑∗ 𝐻↓1↑∗ ↓ 5 9 𝐻 ↑ 1 ↑ 𝐻 ↓ 0 ∗ 𝐻 ∗ 𝐻↓3 𝐻↓7 6 ↑ ↓ ↑∗ ↑∗ ↓↑∗ 4∗ 8 ↑ ↑ Charged to 𝐻 ↓ 6 ↑ ∗ ∗ ∗
𝐻 𝐻 𝐻 ↓ ↓ 𝐻↓1↑∗ ↓ 5 9 1 ↑ 𝐻 ↑ 𝐻0 ↓∗6 ∗ 𝐻↓3 ↑ ↓ ↑∗ ↑∗ 4∗ ↑ Charged ∗ to 𝐻↓2↑∗
𝐻↓2↑∗ 𝐻 𝐻↓7 ↓↑∗ 8 ↑ ∗
• At most one cycle of weight ≤𝛼𝑙𝑏(𝐻↓𝑖↑∗ ) is charged to each 𝐻↓𝑖↑∗
Lemma 2: Total cost of red edges ≤2𝛼∑𝐻↓𝑖↑∗ =2𝛼𝑂𝑃
𝑇↓𝐻𝐾
Charging scheme: the red components with 𝑙𝑜𝑤(𝐻)=𝐻↓𝑖↑∗ is charged to 𝐻↓𝑖↑∗ Charge in one iteration to 𝑯↓𝟓↑∗ is at most 𝟐𝜶𝒍𝒃(𝑯↓𝟓↑∗ )
𝐻↓5↑ ∗ 𝐻↓2
𝐻↓1 𝐻 ↓1 ↓6 ↑∗ 𝐻 0↑ ∗ 𝐻 ↓ 1 1 ↑ ∗
𝑤(𝐻↓1 )+𝑤(𝐻↓2 )+𝑤(𝐻↓3 )
≤𝛼𝑙𝑏(𝐻↓1 )+𝛼𝑙𝑏(𝐻↓2 ) +𝛼𝑙𝑏(𝐻↓3 ) ↓3 𝐻 ≤2𝛼𝑙𝑏(𝐻↓5↑∗ )
𝐻↓7↑∗ By Claim 2
𝐻↓3↑∗
Lemma 2: Total cost of red edges ≤2𝛼∑𝐻↓𝑖↑∗ =2𝛼𝑂𝑃
𝑇↓𝐻𝐾
Charging scheme: the red components with 𝑙𝑜𝑤(𝐻)=𝐻↓𝑖↑∗ is charged to 𝐻↓𝑖↑∗ Charge in one iteration to 𝑯↓𝟓↑∗ is at most 𝟐𝜶𝒍𝒃(𝑯↓𝟓↑∗ )
Lemma 2: Total cost of red edges ≤2𝛼∑𝐻↓𝑖↑∗ =2𝛼𝑂𝑃
𝑇↓𝐻𝐾
Charging scheme: the red components with 𝑙𝑜𝑤(𝐻)=𝐻↓𝑖↑∗ is charged to 𝐻↓𝑖↑∗ Charge in one iteration to 𝑯↓𝟓↑∗ is at most 𝟐𝜶𝒍𝒃(𝑯↓𝟓↑∗ )
𝑯↓𝟓↑∗ charged in at most one iteration
𝐻 ↓1 0↑ ∗
𝐻↓5↑ ∗
𝐻↓6 ↑∗ 𝐻 ↓ 1 1 ↑ ∗
Suppose toward contradiction that not
𝐻
𝐻↓7↑∗
𝐻↓3↑∗
𝐻↓ 9↑∗
By Claim 1 𝑤(𝐻)≤𝛼𝑙𝑏(𝐻↓5↑∗ ) ≤𝛼𝑙𝑏(𝐻↓3↑∗ )
𝐻↓8↑∗ Hence, exist cycle C which contradicts that the ifcondition was not satisfied
Lemma 2: Total cost of red edges ≤2𝛼∑𝐻↓𝑖↑∗ =2𝛼𝑂𝑃
𝑇↓𝐻𝐾
Charge in one iteration to 𝑯↓𝟓↑∗ is at most 𝟐𝜶𝒍𝒃(𝑯↓𝟓↑∗ )
𝑯↓𝟓↑∗ charged in at most one iteration
Total Cost: Eulerian partition + Blue edges + Red edges ≤2𝛼𝑂𝑃𝑇↓𝐻𝐾 +𝛼𝑂𝑃𝑇↓𝐻𝐾 +2𝛼𝑂𝑃𝑇↓𝐻𝐾
Summary • New approach that relaxes connectivity • Integrality gap for node-weighted metrics is ≤13 • Q1: Solve Local-Connectivity ATSP for general metrics? • Very flexible choice of lb can be helpful • Primal-dual? • Q2: Tight answer for Node-Weighted? • Q3: What about Node-Weighted Symmetric TSP?
