10. EXTENDING TRACTABILITY

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1 Coping with NP-compltnss 0. EXTENDING TRACTABILITY ining small vrtx covrs solving NP-har problms on trs circular arc covrings vrtx covr in bipartit graphs Q. Suppos I n to solv an NP-complt problm. What shoul I o? A. Thory says you r unlikly to in poly-tim algorithm. Must sacriic on o thr sir aturs. Solv problm to optimality. Solv problm in polynomial tim. Solv arbitrary instancs o th problm. This lctur. Solv som spcial cass o NP-complt problms. Lctur slis by Kvin Wayn Copyright 00 Parson-Aison Wsly Last upat on 7//7 :06 AM Vrtx covr 0. EXTENDING TRACTABILITY Givn a graph G = (V, E) an an intgr k, is thr a subst o vrtics S V such that S k, an or ach g (u, v) ithr u S or v S or both? ining small vrtx covrs solving NP-har problms on trs circular arc covrings vrtx covr in bipartit graphs S = {, 6, 7, 0 is a vrtx covr o siz k =

2 Fining small vrtx covrs Fining small vrtx covrs Q. VERTEX-COVER is NP-complt. But what i k is small? Brut orc. O(k n k+ ). Try all C(n, k) = O(n k ) substs o siz k. Taks O(k n) tim to chck whthr a subst is a vrtx covr. Goal. Limit xponntial pnncy on k, say to O( k k n). Ex. n =,000, k = 0. Brut. k n k+ = 0 inasibl. Bttr. k k n = 0 7 asibl. Claim. Lt (u, v) b an g o G. G has a vrtx covr o siz k i at last on o G { u an G { v has a vrtx covr o siz k. lt v an all incint gs P. Suppos G has a vrtx covr S o siz k. S contains ithr u or v (or both). Assum it contains u. S { u is a vrtx covr o G { u. P. Suppos S is a vrtx covr o G { u o siz k. Thn S { u is a vrtx covr o G. Rmark. I k is a constant, thn th algorithm is poly-tim; i k is a small constant, thn it s also practical. Claim. I G has a vrtx covr o siz k, it has k (n ) gs. P. Each vrtx covrs at most n gs. 6 Fining small vrtx covrs: algorithm Fining small vrtx covrs: rcursion tr Claim. Th ollowing algorithm trmins i G has a vrtx covr o siz k in O( k kn) tim. Vrtx-Covr(G, k) { i (G contains no gs) rturn tru $ c i k = 0 & T(n, k) % cn i k = & ' T(n,k )+ ckn i k > T(n, k) k ck n i (G contains kn gs) rturn als k lt (u, v) b any g o G a = Vrtx-Covr(G - {u, k-) b = Vrtx-Covr(G - {v, k-) rturn a or b k- k- P. Corrctnss ollows rom prvious two claims. Thr ar k+ nos in th rcursion tr; ach invocation taks O(kn) tim. k- k- k- k- k - i

3 Inpnnt st on trs 0. EXTENDING TRACTABILITY ining small vrtx covrs solving NP-har problms on trs circular arc covrings vrtx covr in bipartit graphs Inpnnt st on trs. Givn a tr, in a maximum carinality subst o nos such that no two shar an g. Fact. A tr on at last two nos has at last two la nos. gr = Ky obsrvation. I v is a la, thr xists a maximum siz inpnnt st containing v. u P. (xchang argumnt) v Consir a max carinality inpnnt st S. I v S, w r on. I u S an v S, thn S { v is inpnnt S not maximum. I u S an v S, thn S { v { u is inpnnt. 0 Inpnnt st on trs: gry algorithm Wight inpnnt st on trs Thorm. Th ollowing gry algorithm ins a maximum carinality inpnnt st in orsts (an hnc trs). Wight inpnnt st on trs. Givn a tr an no wights w v > 0, in an inpnnt st S that maximizs Σ v S w v. Inpnnt-St-In-A-Forst(F) { S φ whil (F has at last on g) { Lt = (u, v) b an g such that v is a la A v to S Dlt rom F nos u an v, an all gs incint to thm. rturn S P. Corrctnss ollows rom th prvious ky obsrvation. Rmark. Can implmnt in O(n) tim by consiring nos in postorr. Obsrvation. I (u, v) is an g such that v is a la no, thn ithr OPT inclus u or OPT inclus all la nos incint to u. Dynamic programming solution. Root tr at som no, say r. OPT in (u) = max wight inpnnt st o subtr root at u, containing u. OPT out (u) = max wight inpnnt st o subtr root at u, not containing u. OPT in (u) = w u + OPT out (v) v chilrn(u) OPT out (u) = max { OPT in (v), OPT out (v) v chilrn(u) v r u w chilrn(u) = { v, w, x x

4 Wight inpnnt st on trs: ynamic programming algorithm Contxt Thorm. Th ynamic programming algorithm ins a maximum wight inpnnt st in a tr in O(n) tim. can also in inpnnt st itsl (not just valu) Inpnnt st on trs. This structur spcial cas is tractabl bcaus w can in a no that braks th communication among th subproblms in irnt subtrs. Wight-Inpnnt-St-In-A-Tr(T) { Root th tr at a no r orach (no u o T in postorr) { i (u is a la) { M in [u] = w u M out [u] = 0 ls { M in [u] = w u + Σ v chilrn(u) M out [v] nsurs a no is visit atr all its chilrn u s Chaptr 0. (but proc with caution) M out [u] = Σ v chilrn(u) max(m in [v], M out [v]) rturn max(m in [r], M out [r]) Graphs o boun tr with. Elgant gnralization o trs that: Capturs a rich class o graphs that aris in practic. Enabls composition into inpnnt pics. Wavlngth-ivision multiplxing 0. EXTENDING TRACTABILITY ining small vrtx covrs solving NP-har problms on trs Wavlngth-ivision multiplxing (WDM). Allows m communication strams (arcs) to shar a portion o a ibr optic cabl, provi thy ar transmitt using irnt wavlngths. Ring topology. Spcial cas is whn ntwork is a cycl on n nos. circular arc covrings vrtx covr in bipartit graphs Ba nws. NP-complt, vn on rings. Brut orc. Can trmin i k colors suic in O(k m ) tim by trying all k-colorings. c b a Goal. O( (k)) poly(m, n) on rings. n =, m = 6 { c,, { b,, { a, 6

5 Rviw: intrval coloring (Almost) transorming circular arc coloring to intrval coloring Intrval coloring. Gry algorithm ins coloring such that numbr o colors quals pth o schul. c a maximum numbr o strams at on location Circular arc coloring. Wak uality: numbr o colors pth. Strong uality os not hol. b g h j i Circular arc coloring. Givn a st o n arcs with pth k, can th arcs b color with k colors? Equivalnt problm. Cut th ntwork btwn nos v an v n. Th arcs can b color with k colors i th intrvals can b color with k colors in such a way that slic arcs hav th sam color. v v 0 v v colors o a, b, an c must corrspon to colors o a, b, an c v v 0 v v v v v 0 max pth = min colors = 7 8 Circular arc coloring: ynamic programming algorithm Dynamic programming algorithm. Assign istinct color to ach intrval which bgins at cut no v 0. At ach no v i, som intrvals may inish, an othrs may bgin. Enumrat all k-colorings o th intrvals through v i that ar consistnt with th colorings o th intrvals through v i. Th arcs ar k-colorabl i som coloring o intrvals ning at cut no v 0 is consistnt with original coloring o th sam intrvals. ys Circular arc coloring: running tim Running tim. O(k! n). Th algorithm has n phass. Bottlnck in ach phas is numrating all consistnt colorings. Thr ar at most k intrvals through v i, so thr ar at most k! colorings to consir. Rmark. This algorithm is practical or small valus o k (say k = 0) vn i th numbr o nos n (or paths) is larg. c a b b b a c c v 0 v v v v v 0 9 0

6 Vrtx covr 0. EXTENDING TRACTABILITY Givn a graph G = (V, E) an an intgr k, is thr a subst o vrtics S V such that S k, an or ach g (u, v) ithr u S or v S or both? ining small vrtx covrs solving NP-har problms on trs circular arc covrings vrtx covr in bipartit graphs ' ' ' ' ' vrtx covr S = {,,, ', ' Vrtx covr an matching Vrtx covr in bipartit graphs: König-Egrváry Thorm Wak uality. Lt M b a matching, an lt S b a vrtx covr. Thn, M S. Thorm. [König-Egrváry] In a bipartit graph, th max carinality o a matching is qual to th min carinality o a vrtx covr. P. Each vrtx can covr at most on g in any matching. ' ' ' ' ' ' ' ' ' ' matching M: -', -', -', -' matching M: -', -', -', -' vrtx covr S = {,,, ', '

7 Proo o König-Egrváry thorm Proo o König-Egrváry thorm Thorm. [König-Egrváry] In a bipartit graph, th max carinality o a matching is qual to th min carinality o a vrtx covr. Suics to in matching M an covr S such that M = S. Formulat max low problm as or bipartit matching. Lt M b max carinality matching an lt (A, B) b min cut. Thorm. [König-Egrváry] In a bipartit graph, th max carinality o a matching is qual to th min carinality o a vrtx covr. Suics to in matching M an covr S such that M = S. Formulat max low problm as or bipartit matching. Lt M b max carinality matching an lt (A, B) b min cut. Din L A = L A, L B = L B, R A = R A, R B = R B. ' ' Claim. S = L B R A is a vrtx covr. - consir (u, v) E - u L A, v R B impossibl sinc ininit capacity - thus, ithr u L B or v R A or both s ' t ' Claim. M = S. - max-low min-cut thorm M = cap(a, B) - only gs o orm (s, u) or (v, t) contribut to cap(a, B) ' - M = cap(a, B) = L B + R A = S. L R 6

Chapter Finding Small Vertex Covers. Extending the Limits of Tractability. Coping With NP-Completeness. Vertex Cover

Chapter Finding Small Vertex Covers. Extending the Limits of Tractability. Coping With NP-Completeness. Vertex Cover Coping With NP-Compltnss Chaptr 0 Extning th Limits o Tractability Q. Suppos I n to solv an NP-complt problm. What shoul I o? A. Thory says you'r unlikly to in poly-tim algorithm. Must sacriic on o thr