Iterative Methods in Combinatorial Optimization Mohit Singh McGill University joint works with L.C. Lau, F. Grandoni, T. Kiraly, S. Naor, R. Ravi and M. Salavatipour

Combinatorial Optimization “Easy” Problems : polynomial time solvable (P) Spanning Trees Matchings Matroid Intersection “Hard” Problems : NP-hard Survivable Network Design Facility Location Scheduling Problems Natural dichotomy: easy and hard Easy problems: LP has been the most powerful tool. Hard problems: NP-hard. Unlikely they are solvable in polynomial time. One aims for an approximation algorithm for the problem: define rho approximation. LP has been one of the most generic tools for these problems as well. The general technique to is to formulate a linear programming relaxation for this problem. Do not expect this LP to be integral. General techniques have been developed In this talk, I will talk about the iterative rounding technique and its extensions that we have developed in our work. These extensions allow us to obtain strong results for a large class of problems. We show that iterative methods are well-suited for poly time solvable problems. This will act as basic groundwork for analyzing hard problems as well.

Formulate Linear Program Linear Programming Formulate Linear Program P NP-hard Show Integrality Round the fractional solution to obtain approximation algorithm Natural dichotomy: easy and hard Easy problems: LP has been the most powerful tool. Hard problems: NP-hard. Unlikely they are solvable in polynomial time. One aims for an approximation algorithm for the problem: define rho approximation. LP has been one of the most generic tools for these problems as well. The general technique to is to formulate a linear programming relaxation for this problem. Do not expect this LP to be integral. General techniques have been developed In this talk, I will talk about the iterative rounding technique and its extensions that we have developed in our work. These extensions allow us to obtain strong results for a large class of problems. We show that iterative methods are well-suited for poly time solvable problems. This will act as basic groundwork for analyzing hard problems as well. Randomized Rounding Primal-Dual Schema Iterative Rounding …

Iterative Rounding (Jain’98): ½-element ) 2-approximation Typical Rounding: LP Solver Rounding Procedure Optimal Fractional Solution Integer Solution Problem Instance Iterative Rounding (Jain’98): ½-element ) 2-approximation LP Solver Good Part Part Integer Optimal Fractional Solution Problem Instance Iterative Rounding limitations (SNDP undirected and directed graphs) Lots of techniques (uncrossing) were similar to techniques used in combinatorial optimization for analyzing LP formulations of exact problems. Residual Problem Too Fractional

Minimum Bounded Degree Spanning Tree Design a spanning tree Low cost Low degree at all nodes min c(T) s.t. T is a spanning tree deg (v) · B Checking feasibility is NP-hard problem is simple to state but captures the structure of more complicated problems. Our work shows that this will see that this is indeed true to some extent. Feasibility problem is NP-hard. 8 v 11/12/2018 11/12/2018 5

Base Problem and Constrained Problem MBDST problem Spanning Tree Problem min c(T) s.t. T is a spanning tree deg(v) · B 8 v2 V 11/12/2018

Iterative Rounding and Relaxation Iterative Rounding works for “easy” problems. Iterative proofs of integrality of spanning tree, arborescene, matroid, matching, matroid intersection … Iterative Relaxation Relax complicating constraints and bound violations. Extend these integrality results to approximation algorithms. mainly concentrate on degree bounded network design problems most well-studied being the MBDST problem. Tighter results for 2-criteria problems. Integral formulations. 11/12/2018

Result Theorem [S., Lau ’07]: There exists a polynomial time algorithm for MBDST problem which returns a tree of c(T) · c(OPT) maximum degree · B+1. OPT is the cheapest tree with maximum degree B. Resolved a conjecture of Goemans ’91 and improved on Goemans’06. Generalizes a result of Furer and Raghavachari ’92.

Outline Integrality of Spanning Tree B+1 result Extensions Conclusion and Open Problems Illustrate the iterative method on MBDST problem. 11/12/2018 9 9

Spanning Tree Polyhedron Integer program min e2 E ce xe s.t. e2 E(V) xe= |V|-1 xe 2 {0,1} Any tree has n-1 edges Seperation 11/12/2018

Spanning Tree Polyhedron Integer program min e2 E ce xe s.t. e2 E(V) xe= |V|-1 e2 E(S) xe· |S|-1 8 S½ V, |S|¸ 2 xe 2 {0,1} E(S): set of edges with both endpoints in S. Any tree has n-1 edges Subtour elimination constraints Seperation 11/12/2018

Spanning Tree Polyhedron Linear Integer program min e2 E ce xe s.t. e2 E(V) xe= |V|-1 e2 E(S) xe· |S|-1 8 S½ V, |S|¸ 2 xe 2 {0,1} 0· xe ·1 (Edmonds ‘71) Any tree has n-1 edges Subtour elimination constraints Equivalent compact formulations [Wong ’80] Polynomial time separation [Cunningham ’84] Seperation 11/12/2018

(Iterative) Rounding Spanning Trees Solve LP to obtain optimum extreme point x Remove all edges s.t. xe = 0 Return E (the non-zero edges) Claim: If |E|=|V|-1 then E is a MST. Proof: E feasible ) x(E)=|V|-1 ) xe=1 8 e2 E ) E is a tree. Edges included in F will form a spanning tree in the end. 11/12/2018

Extreme Points and Uncrossing Fact: x extreme ) d linearly independent tight constraints. Claim: x is extreme ) |E| · n-1 Put References, Put in 1-slide. (Looking Ahead) No 1-edge implies many fractional edges. Extreme point with few side constraints will imply few fractional edges. A contradiction. Put line later. 11/12/2018

Extreme Points and Uncrossing Fact: x extreme ) d linearly independent tight constraints. Claim: x is extreme ) |E| · n-1 e2 E(S) xe=|S|-1 Standard uncrossing ) linearly independent set of constraints defining x can be chosen to form a laminar family. Put References, Put in 1-slide. (Looking Ahead) No 1-edge implies many fractional edges. Extreme point with few side constraints will imply few fractional edges. A contradiction. Put line later. A[B B A AÅB 11/12/2018 11/12/2018

Extreme Points and Uncrossing Fact: x extreme ) d linearly independent tight constraints. Claim: x is extreme ) |E| · n-1 e2 E(S) xe=|S|-1 Standard uncrossing ) linearly independent set of constraints defining x can be chosen to form a laminar family. Put References, Put in 1-slide. (Looking Ahead) No 1-edge implies many fractional edges. Extreme point with few side constraints will imply few fractional edges. A contradiction. Put line later. 11/12/2018

Extreme Points and Uncrossing Claim: x is extreme ) |E| · n-1 Number of variables (dimension) = |E| Number of constraints = |L| ) |E|=|L| |L| is laminar ) |L|· n-1 ) |E|· n-1 Put References, Put in 1-slide. (Looking Ahead) No 1-edge implies many fractional edges. Extreme point with few side constraints will imply few fractional edges. A contradiction. Put line later. Theorem [Edmonds]: Spanning tree polyhedron is integral. 11/12/2018

Outline Integrality of Spanning Tree B+1 result Extensions Conclusion and Open Problems Illustrate the iterative method on MBDST problem. 11/12/2018 18 18

Base Problem and Constrained Problem MBDST problem Spanning Tree Problem min c(T) s.t. T is spanning tree degT(v) · Bv 8 v2W 11/12/2018

Bounded Degree Spanning Trees Extend spanning tree polyhedron min e2 E ce xe s.t. e2 E(V) xe= |V|-1 e2 E(S) xe· |S|-1 8 S ½ V e2 (v)xe· Bv 8 v 2 W xe¸ 0 Here W½V. Spanning tree Degree bounds Outline : 11/12/2018

Obtaining B+1 Algorithm Solve LP to obtain extreme point x. Remove all edges e s.t. xe=0. Return E. Pursued in LNSS. 11/12/2018

Obtaining B+1 Algorithm Relaxation Step While W  Solve LP to obtain extreme point x. Remove all edges e s.t. xe=0. If 9 v2 W such that degE(v)· Bv+1, then remove the degree constraint of v. Return E Lemma: Solution returned is a tree Optimal cost degE(v)· Bv+1 Pursued in LNSS.

Main Lemma Lemma: There exists v2 W such that degE(v)· Bv+1 where E is the support of x. Proof: Tight constraints = 1. Laminar Family: L 2 . Degree constraints: W Will show |E|> |L|+|W| ) contradiction Variables: Edges E Collect 1 token for Each member of L Each vertex in W Extra token Redistribute 1 token for each edge. |E| total token. See [Bansal, Khandekar, Nagarajan’08] 11/12/2018

Charging Argument Simpler argument due to Bansal, Khandekar and Nagarajan ‘08. Redistribution: Degree constraint for u and v get (1-xuv)/2 tokens each. u v $1 (1-xuv)/2 (1-xuv)/2 11/12/2018

Charging Argument Simpler argument due to Bansal, Khandekar and Nagarajan ‘08. Redistribution: Degree constraint for u and v get (1-xuv)/2 tokens each. u v xuv Smallest set containing u and v gets xuv tokens $1 Collection: w Tokens received : e2 (w) (1-xe)/2 ¸ (degE(w)-Bw)/2 ¸ 1 (since degree constraint present) 11/12/2018

Charging Argument Simpler argument due to Bansal, Khandekar and Nagarajan ‘08. Redistribution: Degree constraint for u and v get (1-xuv)/2 tokens each. u v xuv Smallest set containing u and v gets xuv tokens $1 Collection: S R2 R1 x(E(S))=|S|-1 - x(E(Ri))=|Ri|-1 Tokens to S= Integer 11/12/2018

General Methodology Constrained Problem Base Problem Side Constraints 11/12/2018

Minimum Bounded Spanning Tree Degree Constraints Thm[S, Lau ‘07]: (1,B+1)-approximation 11/12/2018

Degree Bounded Steiner Tree Iterative Rounding [Jain]: ½-edge ) 2-approximation Steiner Tree Degree Constraints Thm [LNSS ‘07]: Obtain (2,2B+3)-approximation 11/12/2018

Degree Bounded Steiner Tree Iterative Rounding [Jain]: ½-edge ) 2-approximation Steiner Tree Degree Constraints Thm [LS ‘08]: Obtain (2,B+3)-approximation 11/12/2018

Degree Bounded Arboresence Degree Constraints Thm [LNSS ‘07]: Obtain (2,2B+2)-approximation Thm [BKN ’08]: Obtain (B+4)-approximation. 11/12/2018

Bipartite Matching Bipartite Matching Bipartite Matching Multi-criteria Makespan Constraints Thm[S]: 2-approximation for scheduling unrelated parallel machines. Thm[FRS]: (1+²)-approximation for Multi-criteria bipartite matching. 11/12/2018

More General Structures Matroid Submodular Flow Degree Constraints Degree Constraints Thm[Kiraly, Lau, S ’08]: Additive approximation for degree constrained matroid basis and degree constrained submodular flow problem. See BKNP ’10 for more general structures. 11/12/2018

Multi-Criteria Spanning Tree lengthi (T) · Li Thm [Grandoni, Ravi, S. ’09]: Iterative Relaxation gives (1+²)-approximation for fixed ² 11/12/2018

General Methodology Base Problem Side Constraints “Structured” side constraints in “structured” problems are easy to handle Base Problem Side Constraints 11/12/2018

Applications* Topolgy Control for Future Airborne Networks, Krishnamurthi et al 2009. Deploying Mesh Nodes under non-uniform propogation, Robinson et al, 2009 11/12/2018

Conclusion Iterative Rounding and Relaxation Open Problems Obtain new iterative proofs of classical integrality results Extend these integrality results to multi-criteria optimization and obtain additive approximations Open Problems OPT+1 for Bin Packing? OPT + log2 n. Karp Karmarkar ’82. Discrepancy of Sets? Beck, Fiala ’81, Bansal ’10. Primal-Dual Algorithms? 11/12/2018

Thank You!