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Minimum Weight Cycle

Minimum Weight Cycle. 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.

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Minimum Weight Cycle

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  1. 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)

  2. 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)

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

  4. 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).

  5. 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.

  6. Previous Work (2) Yuster & Zwick (SIDMA 1997) presented an algorithm that finds the shortest even cycle in unweighted undirected graphs in O(n²) time.

  7. 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.

  8. 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.

  9. 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).

  10. 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:

  11. 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!

  12. 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)

  13. Unweighted undirected. 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).

  14. 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

  15. 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.

  16. Weighted [1,M] undirected. 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)

  17. Weighted [1,M] undirected. 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

  18. Weighted [1,M] undirected. 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.

  19. Weighted [1,M] undirected. 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))

  20. Weighted [1,M] undirected. ⅓t* ⅔t* t* Our 4/3-Approximataion • Let Min-Cycle=t* and Max-Edge(Min-Cycle)=w*. Let tt* . • 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*

  21. Weighted [1,M] undirected. Our 4/3-Approximataion: Alg1 (t* + w*) MiddleEdgeLemma: Let C={v1,v2,…vℓ} be a cycle of weight t and let sC. Then there exists an edge (vi,vi+1) such that: s d[s,vi]≤½t d[s,vi+1]≤½t vi vi+1

  22. Weighted [1,M] undirected. Our 4/3-Approximataion: Alg1 (t* + w*) For a given s there is a mid point in on the cycle s =t*/2 =t*/2

  23. Weighted [1,M] undirected. Our 4/3-Approximataion: Alg1 (t* + w*) For a given s there is a mid point in on the cycle s =t*/2 t*/2=

  24. Weighted [1,M] undirected. 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

  25. Weighted [1,M] undirected. 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

  26. Weighted [1,M] undirected. 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

  27. Weighted [1,M] undirected. 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

  28. Weighted [1,M] undirected. 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.

  29. Weighted [1,M] undirected. 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

  30. Weighted [1,M] undirected. 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

  31. Weighted [1,M] undirected. Our 4/3-Approximataion: Alg1 (t* + w*) • Summary: • We have shown that given tt* 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.

  32. Weighted [1,M] undirected. 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]

  33. 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)

  34. Weighted [1,M] undirected. 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.

  35. Weighted [1,M] undirected. 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)

  36. Weighted [1,M] undirected. 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.

  37. Weighted [1,M] undirected. 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 vjC such that: dC[vj,vi]+w(vi,vi+1)>t and dC[vi+1,vj]+w(vi,vi+1)>t ≤ 2t+1

  38. Weighted [1,M] undirected. 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 vjC 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

  39. Weighted [1,M] undirected. v1j v2i+1 Min-Cycle to Min-Triangle Create a new graph G’: (V1,ø) Distances (V2,E) vj v2i vi vi+1

  40. Weighted [1,M] undirected. 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)

  41. Weighted [1,M] undirected. 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!

  42. Weighted [1,M] undirected. Min-Cycle to Min-Triangle We avoid the bad case and remain with the good case vj vi vi+1

  43. 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.

  44. Thank you!

  45. Weighted [1,M] undirected. 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

  46. Weighted [1,M] undirected. 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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