Approximation Algorithms for Prize-Collecting Forest Problems with Submodular Penalty Functions Chaitanya Swamy University of Waterloo Joint work with Yogeshwer Sharma David Williamson Cornell University

Prize-collecting Steiner tree (PCST) Given: graph G=(V,E), edge costs c e ≥ 0, root r  V, penalties p v ≥ 0 on vertices Goal: choose a set of edges F   E so as to minimize∑ e  F c e + ∑ v not connected to r p v cost of edges picked + penalty of nodes disconnected from r

Prize-collecting Steiner tree (PCST) Given: graph G=(V,E), edge costs c e ≥ 0, root r  V, penalties p v ≥ 0 on vertices Goal: choose a set of edges F   E so as to minimize∑ e  F c e + ∑ v not connected to r p v r cost of edges picked + penalty of nodes disconnected from r

Prize-collecting Steiner tree (PCST) Given: graph G=(V,E), edge costs c e ≥ 0, root r  V, penalties p v ≥ 0 on vertices Goal: choose a set of edges F   E so as to minimize∑ e  F c e + ∑ v not connected to r p v 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 c e ≥ 0, root r  V, penalty is given by a set-function p : 2 V   ≥ 0 p(A): penalty if set A  V is disconnected from r p 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 ∑ e  F c e + 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 c e ≥ 0, source-sink pairs s i -t i penalties p i ≥ 0 on each s i -t i pair Goal: choose a set of edges F   E so as to minimize∑ e  F c e + ∑ i: s i not connected to t i in F p i

Prize-collecting Steiner forest (PCSF) Given: graph G=(V,E), edge costs c e ≥ 0, source-sink pairs s i -t i penalties p i ≥ 0 on each s i -t i pair Goal: choose a set of edges F   E so as to minimize∑ e  F c e + ∑ i: s i not connected to t i in F p i 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 c e ≥ 0, connectivity function f: 2 V   {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: 2 2 V   ≥ 0 p( S ): penalty if collection S of subsets is violated Goal: choose a set of edges F   E so as to minimize∑ e  F c e + p({S  V: f(S)= 1, F  (S)=  }) Example: Prize-collecting Steiner forest f(S) = 1 iff there exists some i s.t. exactly one of s i, t i  S p( S ) = ∑ i:  S  S that separates s i -t i p i violated subsets

PCF: properties of p(.) p(  )=0 Monotonicity: if S  T then p( S ) ≤ p( T ) Submodularity: p( S ) + p( T ) ≥ p( S   T ) + p( S   T ) Complement property: for A  V, p({A, A c }) = 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, p f ( S ) = M (very large #) if  S  S with f(S)= 1 ; and 0 o/w. Solving PCF with (f, p f )  solving network design problem with connectivity f’n. f  need certain restrictions on p(.) If f(  )=0, then f is 0- 1 proper iff p f 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 2 n constraints and 2 2 n 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 2 n  2 2 n –overcome difficulty by proving certain structural properties; also required for the rounding procedure

An Integer Program x e : indicates if edge e is picked z S : indicates if penalty is incurred for collection S   2 V Minimize ∑ e c e x e + ∑ S p( S )z S subject to∑ e  (S) x e + ∑ S :S  S z S ≥ f(S) for each S  V x e, z S  {0, 1 }for each e, S

A Linear Program x e : indicates if edge e is picked z S : indicates if penalty is incurred for collection S   2 V Minimize ∑ e c e x e + ∑ S p( S )z S (PCF-LP) subject to∑ e  (S) x e + ∑ S :S  S z S ≥ f(S) for each S  V x e, z S  {0, 1 }for each e, S x e, z S ≥ 0for each e, S LP has 2 2 n variables and 2 n constraints Not clear if even a basic solution has a polynomial- size description – what does “solving the LP” mean?

A Compact Formulation x e : indicates if edge e is picked z S : indicates if penalty is incurred for collection S   2 V Minimizeh(x):=∑ e c e x e + g(x)s.t.0 ≤ x e ≤ 1 for each e (PCF-CP) where,g(x):=min ∑ S p( S )z S (Pen-P) s.t. ∑ S :S  S z S ≥f(S) – ∑ e  (S) x e for each S  V z S ≥0for each e, S g(x) is convex, so (PCF-CP) is a convex program Equivalent 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 Method Start with ball containing polytope P. y i = center of current ellipsoid. Min h(x) subject to x  P. P

The Ellipsoid Method P New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid. Min h(x) subject to x  P. If y i is infeasible, use violated inequality to chop off infeasible half-ellipsoid. Start with ball containing polytope P. y i = center of current ellipsoid.

The Ellipsoid Method New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid. P Min h(x) subject to x  P. If y i is infeasible, use violated inequality to chop off infeasible half-ellipsoid. Start with ball containing polytope P. y i = center of current ellipsoid. If y i  P – how to make progress?

The Ellipsoid Method Min h(x) subject to x  P. P Start with ball containing polytope P. y i = center of current ellipsoid. If y i is infeasible, use violated inequality. If y i  P – how to make progress? add inequality h(x) ≤ h(y i )? Separation becomes difficult. yiyi h(x) ≤ h(y i )

Let d = subgradient at y i. use subgradient cut d. (x–y i ) ≤ 0. Generate new min. volume ellipsoid. The Ellipsoid Method Min h(x) subject to x  P. P Start with ball containing polytope P. y i = center of current ellipsoid. If y i  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(y i )? Separation becomes difficult. If y i is infeasible, use violated inequality. d yiyi h(x) ≤ h(y i )

The Ellipsoid Method Min h(x) subject to x  P. P Start with ball containing polytope P. y i = center of current ellipsoid. If y i  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 y i. use subgradient cut d. (x–y i ) ≤ 0. Generate new min. volume ellipsoid. x 1, x 2, …, x k : points in P. Can show, min i= 1 …k h(x i ) ≤ OPT+ . x*x* x1x1 x2x2 add inequality h(x) ≤ h(y i )? Separation becomes difficult. If y i is infeasible, use violated inequality.

Computing a subgradient h(x) := ∑ e c e x e + g(x) g(x):=min. ∑ S p( S )z S s.t.∑ S:S  S z S ≥ f(S) – ∑ e  (S) x e  S  V z S ≥ 0  S

Computing a subgradient h(x) := ∑ e c e x e + g(x) g(x):=min. ∑ S p( S )z S = max.∑ S (f(S) – ∑ e  (S) x e ) y S s.t.∑ S:S  S z S ≥ f(S) – ∑ e  (S) x e s.t. ∑ S  S y S ≤ p( S )  S  S  V z S ≥ 0  S y S ≥ 0  S Consider point u  m. Let y  optimal dual solution to g(u). Soh(u) = ∑ e c e u e + ∑ S (f(S) – ∑ e  (S) u e ) y S = ∑ e d e u e + ∑ S f(S)y S where d e = c e – ∑ S:e  (S) y S. At any point v  m, y is a feasible solution to dual of g(v), so h(v) ≥ ∑ e c e v e + ∑ S (f(S) – ∑ e  (S) v e ) y S = ∑ e d e v e + ∑ S f(S)y S Lemma: For any point v  m, 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))]y S (Pen-D) s.t.∑ S  S y S ≤ p( S )for all S  2 V y S ≥ 0for all S Bad: Dual has 2 n variables and 2 2 n 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 S i, if [f(S i )-x(  (S i ))] > 0, set y S i = p {S 1,…S i- 1 } (S i ) Bad: Reduces complexity to 2 n, but still not polytime Good: Show that  optimal solution where the sets S with y S > 0 form a laminar family – key structural lemma Notation: x(  (S))= ∑ e  (S) x e p S (A) = p( S  {A}) – p( S )

Useful properties of p(.) If A, B  S, then p S (T) = p S (T c ) = 0 for all sets T in {A  B, A  B, A\B, B\A, A c, B c } – due to complementarity and union properties If p({A}) = 0, then for any B  V, p S  {A} ({B}) = p S ({B}) – due to submodularity  ordering of sets A with f(A)=0 is irrelevant If p S  {A} ({B}) = p S  {B} ({A}) = 0, then for any set T  V, p S  {A} ({T}) = p S  {B} ({T}) – by submodularity

Solving the dual (contd.) Initialize y S = 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 y T = p L ({T}), L   L  {T} Theorem: y is an optimal solution to (Pen-D). Let L ' = {T  L : y T >0} = {T 1,…,T k }, T i = maximal superset of {T 1,…,T i } s.t. p( T i ) = p({T 1,…,T i }) Theorem: Setting z T i = x(  (T i+1 )) – x(  (T i )) ( x(  (T k+1 )) := 1 ) for i= 1,…,k, and z S = 0 for all other S, yields an optimal solution to (Pen-P). Structural lemma yields following algorithm:

Rounding procedure Given: fractional solution x, sets T 1,…, T k – gives succinct description of collections T 1,…, T k, 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 :S  S z S <  ; 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 bounded Penalty is at most p({S   V: ∑ S :S  S z S ≥  }) ≤ [∑ S p( S )z S ]/ 

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.