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### Minimum WeightCycle

Liam Roditty

Bar-Ilan University

- Based on:
- Approximating the girth, (SODA`11) with Roei Tov (BIU)
- Minimum Weight Cycles and Triangles:
- Equivalences and Algorithms, with Virginia V. Williams (UC Berkeley)

Problem Definition

Given a graph G=(V,E) find the Minimum Weight

Cycle(Min-Cycle)in the graph.

Variants:

- Undirected / Directed
- Unweighted / Weighted
- Exact / Approximation
- With or without Fast Matrix Multiplication (FMM)

u

The Trivial Algorithm- Directed graphs:
- Compute All-Pairs of Shortest Paths (APSP).
- Min-Cycle = min(u,v)Ed[v,u] + w(u,v)

The Trivial Algorithm

- Undirected graphs:
- Dijkstra/BFS from every vertex of the graph
- Pick the minimum cycle detected

Example (unweighted):

v

u

s

In both cases the running time is Õ(n³).

Previous Work (1)

Itai & Rodeh (SICOMP 1978) considered only

unweighted graphs:

- Min-Cycle in undirected graphs in O(nω) time.
- Min-Cycle in directed graphs in O(nωlog n) time.
- Approximation in undirected graphs in O(n²) time.
- Return a min-cycle or a cycle whose length is larger by one from the minimum.

Previous Work (2)

Yuster & Zwick (SIDMA 1997) presented an algorithm

that finds the shortest even cycle in unweighted

undirected graphs in O(n²) time.

Previous Work (3)

Lingas & Lundell (IPL 2009) presented two

approximation algorithms for undirected graphs:

- For unweighted graphs an Õ(n3/2) algorithm that returns a 2.666…-approximation.
- For weighted graphs with integral edge weights [1,M] an O(n² log n log nM) algorithm that returns a

2-approximation.

Previous Work (4)

V. Williams & Williams (FOCS 2010) showed

that if a Min-Cycle in an undirected

graph with weights [1,M] can be found in

O(n3-δ log M) for some δ>0

then

it is possible to compute APSP in a directed graph

with weights [-M,M] in O(n3-δ’ log M) for some

δ’>0.

Results (SODA'11)

- For undirected graphs with weights from [1,M]

a 4/3-approximation algorithm with a running time of

O(n² log n log nM).

- For undirected graphs with real edge weights a

(4/3+ε)-approximation with a running time of Õ(1/ε n²).

- An algorithm with (4/3-ε)-approximation can multiply two Boolean matrices. (Using V.W & W ideas).

New Results

“A related problem is finding a minimum weighted circuit in a

weighted graph. It is unclear to us whether our methods can be

modified to answer this problem too.”SICOMP 1978

For undirected and directed graphs with weights from [1,M] we show that the Min-Cycle problem can be efficiently reduced to finding a minimum weight triangle in graphs with weights [1,Θ(M)].

This implies an Õ(Mnω) algorithm for Min-Cycle problem both in undirected and directed graphs.

We resolve the open problem raised by Itai & Rodeh:

New Results

Notice!

APSP in directed graphs can be computed in

O(M0.681n2.575) time (Zwick JACM`02).

APSP in undirected graphs can be computed in

Õ(Mnω) time (Shoshan & Zwick FOCS`99).

We show that Min-Cycle in directed graphs can be

solved in Õ(Mnω) time!

In this talk

- For undirected graphs with weights from [1,M]

a 4/3-approximation algorithm with a running time of

O(n² log n log nM) (SODA’11)

- For undirected graphs with weights from [1,M] we show that the Min-Cycle problem can be efficiently reduced to finding a minimum weight triangle in graph with weights [1,Θ(M)]. (New result)

Itai & Rodeh’s Additive 1-Approximataion

s

0

1

x

y

z

2

w

3

The trivial algorithm: Alledges of x are scanned in an arbitrary order.

and in particular (x,y). A Min-Cycle is detected.

Running time: O(m). We repeat it from every vertex. Total running time is O(mn).

Itai & Rodeh’s Additive 1-Approximataion

Unweighted undirected.

s

0

1

x

y

z

2

w

3

Itai & Rodeh: Stop with the first cycle discovered. Running time O(n²).

Assume xis out from the queue after z and before y.

Lucky

scan (x,y) and stop.

Output: Min-Cycle

Unlucky

scan (x,w) and stop.

Output: Min-Cycle+1

Lingas & Lundell 2-Approximataion

Weighted [1,M] undirected.

s

0

1

1

1

1

1

1

1

1

1

x

y

z

2

100

1

1

w

3

The same approach as IR with Dijkstra instead of BFS will not work.

Lingas & Lundell: Run Dijkstra with a bounding parameter t, that is, grow a shortest paths tree up to depth t and report the first cycle detected.

Reminder: Dijkstra’s algorithm

- Dijkstra(G,s)
- Q←{s}
- While Q is not empty do

u ← Extract-Min(Q)

Relax(u)

- Relax(u)
- foreach (u,v) E

if d[s,v]>d[s,u]+w(u,v)

d[s,v] ← d[s,u]+w(u,v)

Lingas & Lundell 2-Approximataion

- Bounded Dijkstra(G,s,t)
- Q←{s}
- While Q is not empty do

u ← Extract-Min(Q)

Controlled-Relax(u,t)

- Controlled-Relax(u,t)
- (u,v) ← Extract-Min(Qu)
- While d[u]+w(u,v) ≤ t do

RelaxOrStop(u,v)

(u,v) ← Extract-Min(Qu)

s

- RelaxOrStop(u,v)
- if d[v] then

Repot a cycle and stop

else

Relax(u,v)

50

t=100

u

4

1

3

X2

X3

X1

d[X3] =70

d[X2] =92

d[X1] =

STOP: Min-Cycle≤145

d[X1] = 51

Lingas & Lundell 2-Approximataion

?

s

t=100

50

50

50

50

1

50

50

Bounded Dijkstra with t=100 might find the red cycle and not the blue one.

May report

½t*

Lingas & Lundell 2-Approximataion

The algorithm: Binary search the interval [1,nM] for the smallest value of t for which Bounded Dijkstra reports a cycle.

Always reports

t*=Min-Cycle

Taking the smallest value of t for which a cycle is reported ensures that t≤t*and a 2-approximation.

Running time:O(n² log n (log n+log M))

⅓t*

⅔t*

t*

Our 4/3-Approximataion

- Let Min-Cycle=t* and Max-Edge(Min-Cycle)=w*. Let tt* .
- We present two algorithms:
- Alg1: Reports a cycle of weight at most t* + w*.
- Alg2: Given wl and wusuch that w*[ wl , wu ] reports a cycle of weight at most max{2t* - 2wl , t* + wu - wl }.
- Notice! max{2t* - 2wl , t* + wu - wl } is 2t* - 2wl as long as wu ≤ t* - wl

w*

w*

w*

Alg1

t* + w*≤1⅓t*

Alg2[⅓t* , ⅔t*]

2t* - 2w*≤1⅓t*

Alg2[⅔t*, t*]

t* + t*- ⅔t*≤1⅓t*

Our 4/3-Approximataion: Alg1 (t* + w*)

MiddleEdgeLemma: Let C={v1,v2,…vℓ} be a cycle of weight t and let

sC. Then there exists an edge (vi,vi+1) such that:

s

d[s,vi]≤½t

d[s,vi+1]≤½t

vi

vi+1

We have:

1. dC[s,vi]≤½t

2. dC[vi+1,s]≤½t

1. d[s,vi]≤½t

2. d[s,vi+1]≤½t

Our 4/3-Approximataion: Alg1 (t* + w*)The edge (vi,vi+1) is the middle edge.

s

vi+1

vi

Our 4/3-Approximataion: Alg1 (t* + w*)

And now, the algorithm!

- Approx-Min-Cycle(G,s,t)
- R ←ф
- Q←{s}
- While Q is not empty do

u ← Extract-Min(Q)

Controlled-Relax(u,½t)

Let (u,v) be the smallest edge of u that was not relaxed

- R←{(u,v)}
- While R is not empty do

(u,v) ← Extract-Min(R)

RelaxOrStop(u,v)

(u,v) ← Extract-Min(Qu)

- R← R {(u,v)}

Phase 1

Phase 2

Our 4/3-Approximataion: Alg1 (t* + w*)

Phase 1:

- Approx-Min-Cycle(G,s,t)
- R ←ф
- Q←{s}
- While Q is not empty do

u ← Extract-Min(Q)

Controlled-Relax(u,½t)

Let (u,v) be the smallest edge of u that was not relaxed

- R← R {(u,v)}
- …

s

- Bounded Dijkstra(G,s,t)
- Q←{s}
- While Q is not empty do

u ← Extract-Min(Q)

Controlled-Relax(u,t)

R = {(u0,v0)}

R = {(u0,v0) , (u1,v1) , (u2,v2) }

u0

u1

u2

½t

v0

w

v1

v2

Our 4/3-Approximataion: Alg1 (t* + w*)

Phase 2:

- Approx-Min-Cycle(G,s,t)
- …
- While R is not empty do

(u,v) ← Extract-Min(R)

RelaxOrStop(u,v)

(u,v) ← Extract-Min(Qu)

- R← R {(u,v)}

R = {(u0,v0) , (u1,v1) , (u2,v2) }

R = {(u0,v0) , (u1,v2) , (u2,v2) }

R = {(u0,v0) , (u1,v2) }

R = {(u0,v0)}

s

u0

u1

u2

½t

v0

w

v1

v2

Our 4/3-Approximataion: Alg1 (t* + w*)

- We need to prove that:
- We report a cycle of weight t*+w* (Weight bound).
- The running time from a given vertex is Õ(n) (Running time).
- A bound on the weight:

Phase 1:

Phase 2:

- …
- While R is not empty do

(u,v) ← Extract-Min(R)

RelaxOrStop(u,v) (2)

(u,v) ← Extract-Min(Qu)

- R←{(u,v)}

- Approx-Min-Cycle(G,s,t)
- R ←ф
- Q←{s}
- While Q is not empty do

u ← Extract-Min(Q)

Controlled-Relax(u,½t) (1)

R←{(u,v)}

A cycle can be reported either in line (1) or (2).

Easy case (line 1): We stopped in line (1). The weight of the reported

cycle is at most t.

Hard case (line 2): Next slide.

Phase 1

Our 4/3-Approximataion: Alg1 (t* + w*)Hard case (line 2): If we stopped at phase 2 before the middle edge (vi,vi+1) of s got

out then any edge (x,y) that was popped out from R satisfies:

d[s,x]+w(x,y) ≤ min {d[s,vi]+w*,d[s,vi+1]+w*}

In particular, if we stopped with the edge (h,g) then:

d[s,h]+w(h,g)+d[s,g] ≤ 2 min {d[s,vi]+w*,d[s,vi+1]+w*} ≤ d[s,vi]+2w*+d[s,vi+1] ≤ t* +w*

s

s

½t*

≤½t*

g

h

≤w*

vi

vi+1

Our 4/3-Approximataion: Alg1 (t* + w*)

Running time:

The first time that a vertex beyond the ½t threshold is added to R for the second

time we stop and report a cycle.

Hence, we can charge any edge of R to its endpoint beyond the ½t threshold and the

running time is Õ(n).

R = {(u0,v0) , (u1,v1) , (u2,v2) }

s

u0

u1

u2

½t

v0

w

v1

v2

Our 4/3-Approximataion: Alg1 (t* + w*)

- Summary:
- We have shown that given tt* in Õ(n²) time it is possible to return a cycle
- of weight t* + w*
- This algorithm is a natural generalization of the additive 1-approximation of Itai and Rodeh for unweighted graphs.
- Recall that this is just one ingredient of our algorithm.

Our 4/3-Approximataion: Alg2(max{2t* - 2wl , t* + wu - wl }).

Sketch:

Analysis: More complicated.

Some intuition: when a cycle is reported it is because we reach some vertex u

twice. The bound is either 2t* - 2wl or t* + wu - wl.

- Approx-Min-Cycle(G,s,t,wl,wu)
- d[s] = 0
- Controlled-Relax(s,wu)
- While Q is not empty do

u ← Extract-Min(Q)

Controlled-Relax(u,t-wl)

=w*

vi+1

s=vi

w*[wl,wu]

In this talk

- For undirected graphs with weights from [1,M]

a 4/3-approximation algorithm with a running time of

O(n² log n log nM). (SODA’11)

- For undirected graphs with weights from [1,M] we show that the Min-Cycle problem can be efficiently reduced to finding a minimum weight triangle in graph with weights [1,Θ(M)]. (New result)

Min-Cycle to Min-Triangle

Theorem:

Given an undirected graph G with weights [1,M] there is an

O(n² log n log nM) time algorithm that either find a Min-Cycle

in G or compute log n graphs:

G’1 , G’2 ,G’3 , ... G’log n

With Θ(n) nodes and edge weights [1,O(M)] such that:

The minimum weight triangle among all G’igraphs has the same

weight as the Min-Cycle of G.

Min-Cycle to Min-Triangle

Simple (and important) observation:

Assume that we run just the first phase of Alg1 (t*+w*) for a given t.

We either compute all distances of length at most ½t from s or report a

cycle of weight at most t.

- Approx-Min-Cycle(G,s,t)
- R ←ф
- Q←{s}
- While Q is not empty do

u ← Extract-Min(Q)

Controlled-Relax(u,½t)

Let (u,v) be the smallest edge of u that was not relaxed

- R←{(u,v)}
- While R is not empty do

(u,v) ← Extract-Min(R)

RelaxOrStop(u,v)

(u,v) ← Extract-Min(Qu)

- R←{(u,v)}

- Approx-Min-Cycle(G,s,t)
- Q←{s}
- While Q is not empty do

u ← Extract-Min(Q)

Controlled-Relax(u,½t)

Min-Cycle to Min-Triangle

First stage:

Binary search for the first t such that Approx-Min-Cycle(G,2t) does not

report on a cycle and Approx-Min-Cycle(G,2t+2) reports.

Report a cycle ≤2t+2

t

t+1

No cycle is reported.

All distances up to t are computed

Need to show: If the Min-Cycle ≤ 2t+1 we can find it.

vj

Min-Cycle to Min-Triangle

vi

vi+1

Main Lemma:

Let C={v1,v2, … ,vℓ} and assume that Approx-Min-Cycle(G,t) does

not report a cycle then there are three distinct vertices

vi,vi+1, and vjC such that:

dC[vj,vi]+w(vi,vi+1)>t and dC[vi+1,vj]+w(vi,vi+1)>t

≤ 2t+1

Min-Cycle to Min-Triangle

Main Lemma:

Let C={v1,v2, … ,vℓ} and assume that Approx-Min-Cycle(G,t) does

not report a cycle then there are three distinct vertices

vi,vi+1, and vjC such that:

dC[vj,vi]+w(vi,vi+1)>t and dC[vi+1,vj]+w(vi,vi+1)>t

vj

dC[vj,vi] ≤ t

dC[vi+1,vj] ≤ t

and we have computed

these distances !

As this is a Min-Cycle

d[vj,vi] = dC[vj,vi]

d[vi+1,vj]=dC[vi+1,vj]

≤ 2t+1

vi

vi+1

v1j

v2i+1

Min-Cycle to Min-Triangle

Create a new graph G’:

(V1,ø)

Distances

(V2,E)

vj

v2i

vi

vi+1

Min-Cycle to Min-Triangle

There are two problems with G’.

- We may create a triangle that corresponds to a path rather than a cycle
- The weights in G’ may be Ω(nM) (not covered in the talk)

x1

y

y2

z2

Min-Cycle to Min-Triangle

Fixing the first problem:

The last vertex on the path to z is y.

We use color coding (Alon, Yuster and Zwick JACM`95 ).

We use two colors.

x

(V1,ø)

Distances

(V2,E)

y

z

We add an edge (x1,z2) only if

the last vertex before z is red

y cannot be red and blue at the

same time!

Min-Cycle to Min-Triangle

We avoid the bad case and

remain with the good case

vj

vi

vi+1

Summary

- We have settled many variants of a fundamental open problem in algorithmic graph theory.
- But, what about approximation in directed graphs? Can we get O(n3-δ) for some δ>0 and constant approximation? Or maybe we can show a “lower” bound.

We search for two vertices that satisfy:

1. dC[s,vi]≤½t

2. dC[s,vi+1]>½t

1. d[s,vi]≤dC[s,vi]≤½t

2. d[s,vi+1]≤½t

Our 4/3-Approximataion: Alg1 (t* + w*)Proof: Let (vi ,vi+1) be an edge of C with weight strictly more

than ½t. In this case the claim holds for every vertex using this edge.

Thus, we can assume that w*≤½t.

s=v1

v2

v3

vi

vi+1

1. dC[s,vℓ-1] ≤ ½t

2. dC[s,vℓ] ≤ ½t

A contradiction as w* ≤ ½t

Our 4/3-Approximataion: Alg1 (t* + w*)Proof (cont): To complete the proof we need to show that these

vertices exist. Assume on the contrary that we have not found two

such vertices in the process.

> ½t

s=v1

This means that:

v2

vℓ

v3

vℓ-1

vi

vi+1

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