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Approximation Algorithms for Prize-Collecting Forest Problems with Submodular
Penalty FunctionsChaitanya Swamy
University of Waterloo
Joint work withYogeshwer Sharma David
WilliamsonCornell University
Prize-collecting Steiner tree (PCST)
Given: graph G=(V,E), edge costs ce ≥ 0, root rV,
penalties pv ≥ 0 on vertices
Goal: choose a set of edges F E so as to
minimize ∑eF ce + ∑v not connected to r pvcost of edges picked + penalty of nodes disconnected from r
Prize-collecting Steiner tree (PCST)
Given: graph G=(V,E), edge costs ce ≥ 0, root rV,
penalties pv ≥ 0 on vertices
Goal: choose a set of edges F E so as to
minimize ∑eF ce + ∑v not connected to r pv
r
cost of edges picked + penalty of nodes disconnected from r
Prize-collecting Steiner tree (PCST)
Given: graph G=(V,E), edge costs ce ≥ 0, root rV,
penalties pv ≥ 0 on vertices
Goal: choose a set of edges F E so as to
minimize ∑eF ce + ∑v not connected to r pv
Bienstock et al.: gave a 3-approx. LP-rounding algorithm
Goemans-Williamson (GW): gave a primal-dual 2-approx. algorithm
r
cost of edges picked + penalty of nodes disconnected from r
PCST with submodular penalty f’n.
Given: graph G=(V,E), edge costs ce ≥ 0, root rV,
penalty is given by a set-function p : 2V
≥ 0 p(A): penalty if set AV is disconnected
from rp is submodular: p(A)+p(B) ≥ p(A B)
+p(A B)e.g., p(A) = min(|A|, M)
Goal: choose a set of edges F E so as to minimize
∑eF ce + p({v not connected to r})
r
•Generalizes penalty function of PCST
•Introduced by Hayrapetyan-S-Tardos: gave a 2-approximation algorithm by extending GW primal-dual algorithm
Prize-collecting Steiner forest (PCSF)
Given: graph G=(V,E), edge costs ce ≥ 0,source-sink pairs si-ti penalties pi ≥ 0 on each si-ti pair
Goal: choose a set of edges F E so as to
minimize ∑eF ce + ∑i: si not connected to ti in F pi
Prize-collecting Steiner forest (PCSF)
Given: graph G=(V,E), edge costs ce ≥ 0,source-sink pairs si-ti penalties pi ≥ 0 on each si-ti pair
Goal: choose a set of edges F E so as to
minimize ∑eF ce + ∑i: si not connected to ti in F pi
•Generalizes connectivity function of PCST
•Introduced by Jain-Hajiaghayi: gave a 3-approx. primal-dual algorithm
General framework for Prize-Collecting Forest
Problems
PCST with submodular
penalty function
Prize-collecting
Steiner forest
Prize-Collecting Forest (PCF) – connectivity function: arbitrary 0-1
function– penalty function: submodular function
on collections of sets of vertices
Prize-collecting Steiner tree
Prize-Collecting Forest (PCF)Given: graph G=(V,E) (|V|=n), edge costs ce ≥
0,•connectivity function f: 2V {0,1}
f(S)=1 need an edge from border of S, (S) := {(u,v)E: exactly one of u, v is
in S}
•penalty function p: 22V ≥ 0 p(S): penalty if collection S of subsets is
violated
Goal: choose a set of edges F E so as to
minimize ∑eF ce + p({SV: f(S)=1, F(S)=})
Example: Prize-collecting Steiner forestf(S) = 1 iff there exists some i s.t. exactly one of si, ti S
p(S) = ∑ i:SS that separates si-ti pi
violated subsets
PCF: properties of p(.)
• p()=0
• Monotonicity: if ST then p(S) ≤ p(T)
• Submodularity: p(S) + p(T) ≥ p(S T) + p(S T)
• Complement property: for AV, p({A, Ac}) = p({A})
• Union property: for A,B V, p({A, B, A B})=p({A,B})
• Inactivity property: if f(A)=0, then p({A})=0
For any 0-1 connectivity f’n f, can define penalty function,pf(S) = M (very large #) if SS with f(S)=1; and 0 o/w.
Solving PCF with (f, pf) solving network design problem with connectivity f’n. f need certain restrictions on p(.)
If f()=0, then f is 0-1 proper iff pf satisfies above properties.p(.) will be given as an oracle (ground set has 2|V| elements)
Our Results• Give a primal-dual 3-approximation algorithm
– Requires novel ideas in implementation and analysis, to overcome difficulties caused due to the exponential size of the ground set of p(.)
• Give an LP-rounding 2.54-approximation algorithm– solving the LP relaxation poses a significant challenge– LP has 2n constraints and 22n
variables: not clear if even a basic solution has a polynomial description
– Reformulate LP as a convex program, solve via ellipsoid method; evaluating objective f’n and computing a subgradient both require solving an LP of size 2n22n
– overcome difficulty by proving certain structural properties; also required for the rounding procedure
An Integer Program
xe : indicates if edge e is picked
zS : indicates if penalty is incurred for collection S 2V
Minimize ∑e cexe + ∑S p(S)zS
subject to ∑e(S) xe + ∑S:SS zS ≥ f(S) for each SV
xe, zS {0,1}for each e, S
A Linear Program
xe : indicates if edge e is picked
zS : indicates if penalty is incurred for collection S 2V
Minimize ∑e cexe + ∑S p(S)zS (PCF-LP)
subject to ∑e(S) xe + ∑S:SS zS ≥ f(S) for each SV
xe, zS {0,1}for each e, S
xe, zS ≥ 0 for each e, S
• LP has 22n variables and 2n constraints
•Not clear if even a basic solution has a polynomial-size description – what does “solving the LP” mean?
A Compact Formulationxe : indicates if edge e is picked
zS : indicates if penalty is incurred for collection S 2VMinimize h(x) := ∑e cexe + g(x) s.t. 0
≤ xe ≤ 1 for each e
(PCF-CP)
where, g(x):=min ∑S p(S)zS
(Pen-P)
s.t. ∑S:SS zS≥ f(S) – ∑e(S) xe for each SV
zS ≥ 0 for each e, S
g(x) is convex, so (PCF-CP) is a convex programEquivalent to earlier LP.
The Overall Strategy
1. Get an optimal (or (1+)-optimal solution) x to the convex program using the ellipsoid method.
2. Round fractional solution x to integer solution– need that f is 0-1 proper f’n, or is weakly-submodular– use 2-approx. algorithm for the network-design problem
without penalties (Goemans-Williamson or Jain).
Obtain a 2.54-approximation algorithm for the prize-collecting forest problem.
The Ellipsoid MethodStart with ball containing polytope P.yi = center of current ellipsoid.
Min h(x) subject to xP.
P
The Ellipsoid Method
P
New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid.
Min h(x) subject to xP.
If yi is infeasible, use violated inequality to chop off infeasible half-ellipsoid.
Start with ball containing polytope P.yi = center of current ellipsoid.
The Ellipsoid Method
New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid.
P
Min h(x) subject to xP.
If yi is infeasible, use violated inequality to chop off infeasible half-ellipsoid.
Start with ball containing polytope P.yi = center of current ellipsoid.
If yi P – how to make progress?
The Ellipsoid Method
Min h(x) subject to xP.
P
Start with ball containing polytope P.yi = center of current ellipsoid.If yi is infeasible, use violated inequality.If yi P – how to make progress?
add inequality h(x) ≤ h(yi)? Separation becomes difficult.yi
h(x) ≤ h(yi)
Let d = subgradient at yi.
use subgradient cut d.(x–yi) ≤ 0.Generate new min. volume
ellipsoid.
The Ellipsoid Method
Min h(x) subject to xP.
P
Start with ball containing polytope P.yi = center of current ellipsoid.
If yi P – how to make progress?
d m is a subgradient of h(.) at u, if for every v, h(v)-
h(u) ≥ d.(v-u).
add inequality h(x) ≤ h(yi)? Separation becomes difficult.
If yi is infeasible, use violated inequality.
d
yi
h(x) ≤ h(yi)
The Ellipsoid Method
Min h(x) subject to xP.
P
Start with ball containing polytope P.yi = center of current ellipsoid.
If yi P – how to make progress?
d m is a subgradient of h(.) at u, if for every v, h(v)-
h(u) ≥ d.(v-u).
Let d = subgradient at yi.
use subgradient cut d.(x–yi) ≤ 0.Generate new min. volume
ellipsoid.
x1, x2, …, xk: points in P. Can show, mini=1…k h(xi) ≤ OPT+.
x*
x1
x2
add inequality h(x) ≤ h(yi)? Separation becomes difficult.
If yi is infeasible, use violated inequality.
Computing a subgradienth(x) := ∑e cexe + g(x)
g(x):=min. ∑S p(S)zS
s.t. ∑S:SS zS ≥ f(S) – ∑e(S) xe
SV
zS ≥ 0 S
Computing a subgradienth(x) := ∑e cexe + g(x)
g(x):=min. ∑S p(S)zS = max. ∑S (f(S) – ∑e(S)
xe) yS
s.t. ∑S:SS zS ≥ f(S) – ∑e(S) xe s.t. ∑SS yS ≤ p(S) S
SV
zS ≥ 0 S yS ≥ 0 SConsider point um. Let y optimal dual solution to g(u).So h(u) = ∑e ceue + ∑S (f(S) – ∑e(S) ue) yS = ∑e deue + ∑S
f(S)yS
where de = ce – ∑S:e(S) yS.
At any point vm, y is a feasible solution to dual of g(v), so
h(v) ≥ ∑e ceve + ∑S (f(S) – ∑e(S) ve) yS = ∑e deve + ∑S
f(S)yS
Lemma: For any point vm, we have h(v) – h(u) ≥ d.
(v-u). d is a subgradient of h(.) at point u.
Solving the dual
g(x) =max ∑S [f(S) – x((S))]yS (Pen-D)
s.t. ∑SS yS ≤ p(S) for all S2V
yS ≥ 0 for all S
Bad : Dual has 2n variables and 22n constraints
Good : It is a polymatroid: p(.) is a monotone submodular f’n. Edmonds’ greedy algorithm yields optimal solution
– Sort the sets S in decreasing order of [f(S)-x((S))]
– For the i-th set Si, if [f(Si)-x((Si))] > 0, set ySi =
p{S1,…Si-1}(Si)
Bad : Reduces complexity to 2n, but still not polytimeGood : Show that optimal solution where the sets S with yS > 0 form a laminar family – key structural lemma
Notation: x((S))= ∑e(S) xe
pS(A) = p(S{A}) – p(S)
Useful properties of p(.)
• If A, BS, then pS(T) = pS(Tc) = 0 for all sets T in {AB, AB, A\B, B\A, Ac, Bc} – due to complementarity and union properties
• If p({A}) = 0, then for any BV, pS{A}({B}) = pS({B}) – due to submodularity ordering of sets A with f(A)=0 is irrelevant
• If pS{A}({B}) = pS{B}({A}) = 0, then for any set TV, pS{A}({T}) = pS{B}({T}) – by submodularity
Solving the dual (contd.)
• Initialize yS = 0 for all sets S, laminar family L .
• While set S that does not cross any set of L– find T = argmin {x((S)): S does not cross L}
– if x((T)) ≥ 1 return; else set yT = pL({T}), L L{T}
Theorem: y is an optimal solution to (Pen-D).
LetL' = {TL: yT>0} = {T1,…,Tk},
Ti = maximal superset of {T1,…,Ti} s.t. p(Ti) = p({T1,…,Ti})
Theorem: Setting zTi = x((Ti+1)) – x((Ti)) (x((Tk+1)) :=
1) for i=1,…,k, and zS = 0 for all other S, yields an optimal solution to (Pen-P).
Structural lemma yields following algorithm:
Rounding procedure
Given: fractional solution x, sets T1,…, Tk – gives succinct description of
collections T1,…,Tk, and hence optimal soln. z to (Pen-P)
Let [0,1] be a parameter. – Define 0-1 connectivity function(S) = 1 if f(S) = 1 and ∑S:SS zS < ; 0
otherwise.– Solve network design problem with connectivity
function .
If f is proper or weakly-supermodular, then so is , therefore
cost of edges picked is boundedPenalty is at most p({S V: ∑S:SS zS ≥ }) ≤ [∑S
p(S)zS]/
Open Questions
• Is there a compact description of the LP? Or a more efficient procedure to solve it?
•Obtaining a 2-approximation algorithm: iterative rounding may be the way to go
•Applications to 2-stage stochastic network design: can the second-stage cost be captured by a “nice” penalty function?
•Extensions to higher connectivity reqmts.
Thank You.
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