Changing Basis:
Multistage Optimization for
Matroids and Matchings
Anupam Gupta
CMU
Kunal Talwar
Udi Wieder
MSR
MSR
Multi-stage Optimization
▪ Example: A chef needs to buy tomatoes each day, there are 𝑛 suppliers that post
prices every day. Goal: minimize the price paid.
▪ One instance optimization: trivial – buy from the cheapest supplier
▪ Twist: establishing (or breaking) a business relation comes with a fixed cost
▪ Still doable in this case (offline)
▪ More generally, in practice:
▪ As time goes by the optimization problem changes
▪ Change comes with a cost. Either movement cost, setup cost etc.
Multi-stage Optimization
▪ Question: If a single instance could be solved efficiently, can we
solve/approximate well the multi-stage version?
▪ Answer: NO
▪ 3-dimensional matching could be reduced to three instances of a partition matroid
▪ NP-Hard to approximate
▪ Focus of this work: The underlying optimization problem is same matroid
optimization, weight of elements vary in time.
▪ Covers a large class of problem.
▪ Well structured and heavily studied.
Matroid Multi-stage Maintanence (MMM)
Input:
▪ A fixed Matroid (𝐸, 𝑀) of rank 𝑟.
▪ Fixed acquisition cost 𝑎(𝑒) for acquiring a new item.
▪ Each time step 𝑡, holding costs 𝑐𝑡 (𝑒) for holding an item.
Constraint: ▪ Each time step 𝑡, hold a base 𝐵𝑡 ⊆ 𝐸.
Goal:
▪ Minimize total cost:
𝑡𝑐
𝐵𝑡 + 𝑎 𝐵𝑡 ∖ 𝐵𝑡−1
▪ Offline version: Both 𝑐 and 𝑎 are given in advance
▪ Online version: 𝑎 is given in advance. 𝑐𝑡 is given at the beginning of time 𝑡.
Matroid Multi-stage Maintanence (MMM)
▪ 𝑘-Uniform Matroid: Base is any set of cardinality 𝑘.
▪ Also known as weighted caching
▪ 𝑘 pages in the cache, each page has a different fetching cost
▪ Graphical Matroids: Independent sets are forests
▪ Each time step hold a spanning tree.
▪ Greedy algorithm solves any single static instance. How to cope with the
acquisition cost?
Related but Different Models
▪ Regret minimization: The cost is given at the end of the time step. In our case, at
the beginning.
▪ Two step optimization: Takes into account two time steps only.
▪ Dynamic Algorithms: Requires optimal solution each time step. Often limits the
amount of change between time steps and aims at minimizing running time.
▪ Example: maintaining a spanning tree with edge inserts and deletions
Related Work
▪ [Buchbinder Chen Naor Shamir 2012]: Same problem, uniform acquisition cost
▪ Provide fractional solutions only.
▪ log 𝑛 approximations
▪ Essentially, an online solution for the Base Polytope LP.
▪ [Bansal Buchbinder Naor 2008] show an integer solution for weighted caching.
▪ Metrical Task Systems – very generic, degenerates to trivial solution in our case –
O(n) approximation.
Main Result
▪ Online algorithm with competitive ratio 𝑂 log 𝐸 log 𝑟𝑇
▪ Matching lower bound
Time
Universe
size
Matroid
Rank
▪ Hardness of approximation if the underlying problem is maximum matching
The Matroid Polytope and the Base Polytope
▪ The Matroid Polytope is the convex hull of incidence vectors of independent sets
𝐸
▪ Matroid Polytope is 𝑥 ∈ ℝ≥0
| 𝑥 𝑆 ≤ 𝑟𝑎𝑛𝑘 𝑆 , ∀𝑆 ⊆ 𝐸
Sum of 𝑥 in the 𝑆
components
▪ The Base Polytope is the convex hull of all base vectors.
𝐸
▪ Base Polytope = Matroid Polytope ∩ 𝑥 ∈ ℝ≥0
| 𝑥 𝐸 = 𝑟𝑎𝑛𝑘 𝐸
First Attempt – Using an LP
▪ Goal: Minimize total cost:
▪ LP: Minimize:
𝑡𝑐
𝑡𝑐
𝐵𝑡 + 𝑎 𝐵𝑡 ∖ 𝐵𝑡−1 such that 𝐵𝑡 is a base
𝑋𝑡 + 𝑋𝑡 − 𝑋𝑡−1
1
such that 𝑋𝑡 in Base Polytope
▪ The LP could be approximated online [Buchbinder Chen Naor Shamir 2012]
▪ But… how to round? Each fractional solution is a convex combination of bases,
would randomized rounding work?
▪ Say we have two vectors 𝑋𝑡 , 𝑋𝑡−1 and matching distributions over bases 𝜆𝑡 , 𝜆𝑡−1 .
▪ The fractional cost is 𝑋𝑡 – 𝑋𝑡−1
1,
while the integer cost is 𝐸𝑀(𝜆𝑡 , 𝜆𝑡−1 ).
▪ Examples where ℓ1 distance is small and EM distance is large suggest that it is hard to
make this approach work.
Key Idea – reduction to spanning sets
▪ A Spanning Set: A set that contains a base = set of full rank
▪ Theorem: Allowing for spanning sets does not reduce the overall cost.
▪ Proof sketch:
▪ We have spanning sets 𝑆1 , 𝑆2 , … , 𝑆𝑇 and want to find bases.
Matroid doesn’t
change!
▪ Set 𝐵1 ⊆ 𝑆1 .
▪ Given 𝐵𝑡−1 and 𝑆𝑡 start with 𝐵𝑡−1 ∩ 𝑆𝑡 and extend it to any base 𝐵𝑡 ⊆ 𝑆𝑡
▪ An item does not drop from 𝐵𝑡 unless it also drops from 𝑆𝑡
▪ Reduction is online
▪ Projection to bases is correlated across time, sidesteps previous issue
Warm-up: Offline Case
▪ Minimize:
𝑡𝑐
𝑋𝑡 + 𝑋𝑡 − 𝑋𝑡−1
1
such that 𝑋𝑡 in Base Polytope
▪ Pick a random threshold 𝜏 ∈ 0,1 . Round 𝑋𝑡 𝑒 to 1 iff 𝑋𝑡 𝑒 ≥ 𝜏 and to 0 o/w
▪ Same expected total cost.
▪ Expected rank of rounded solution is a constant fraction of the rank at each time step.
▪ If repeated 𝑂(log rT) times obtain spanning sets w.h.p. 𝑂(log rT) approximation.
▪ May need to augment the outcome of the rounding but this carries little cost.
▪ There are simpler algorithms (Greedy!) for the offline case.
Online Case. Step I
▪ W.l.o.g each item 𝑒 ∈ 𝐸 has a time interval 𝐼𝑒 such that the item is available at no
cost in 𝐼𝑒 and unavailable outside 𝐼𝑒
▪ Reduction is exact in the offline case and has a constant approximation in the
online case.
▪ Minimize:
Acquisition
Cost
𝑎𝑒 𝑥𝑒
s.t
∀𝑡, 𝑋𝐸𝑡 ∈ Spanning Polytope
All items available
at time 𝑡
Convex hull of all
spanning sets
The Spanning Polytope
▪ Spanning Polytope of a matroid ℳ is the convex hull of all spanning sets.
▪ Definition: ℳ ∗ is the dual matroid of ℳ. The bases of ℳ ∗ are the complements
of the bases of ℳ.
▪ 𝑟∗ 𝑆 = 𝑟 𝐸 ∖ 𝑆 + 𝑆 − 𝑟 𝐸
▪ Observation: X in Spanning Polytope of ℳ iff 1 − 𝑋 in Matroid Polytope of ℳ ∗
Online Case. Step II
▪ Minimize:
𝑎𝑒 𝑥𝑒
Acquisition
Cost
▪ Minimize: 𝑎𝑒 𝑥𝑒
s.t
∀𝑡, 𝑥𝐸𝑡 ∈ Spanning Polytope
All items available
at time 𝑡
Convex hull of all
spanning sets
s.t
▪ ∀𝑡, S ⊆ 𝐸𝑡 𝑥 𝑆 ≥ 𝑟 𝐸 − 𝑟(𝐸 ∖ 𝑆)
▪ 𝑥𝑒 ∈ [0,1]
▪ This is a Covering LP, that (should be) simpler to solve
Recap
▪ Reduce to ‘interval instance’.
▪ 𝑂(1) approximation
▪ Solve online: Minimize: 𝑎𝑒 𝑥𝑒
▪ 𝑂 log 𝐸 approximation
▪ ∀𝑡, S ⊆ 𝐸𝑡 𝑥 𝑆 ≥ 𝑟 𝐸 − 𝑟(𝐸 ∖ 𝑆)
▪ 𝑥𝑒 ∈ [0,1]
▪ Round using threshold rounding
▪ 𝑂 log 𝑟𝑇 approximation
▪ Turn a Spanning Set into a Base
▪ Exact
▪ 𝑂 log 𝐸 log 𝑟𝑇 approximation
▪ Hardness: Ω log 𝐸 log 𝑇
Solving the LP
▪ Minimize: 𝑎𝑒 𝑥𝑒
▪ ∀𝑡, S ⊆ 𝐸𝑡
𝑥 𝑆 ≥ 𝑟 𝐸 − 𝑟(𝐸 ∖ 𝑆)
▪ 𝑥𝑒 ∈ [0,1]
▪ [Buchbinder Naor] provide a method for solving these types of LP’s online
▪ Can we use it in a Black Box way?
▪ Yes but… we have an exponential number of constraints
▪ Overcome the problem by invoking [BN] on adaptively selected sets of violated
constraints.
▪ Show that running time stays polynomial
Hardness of Perfect Matching
▪ We are given a graph 𝐺 = (𝑉, 𝐸) and need to maintain a perfect matching each
time step.
▪ Theorem: NP-Hard to distinguish between instances with cost 𝑉
▪ Even when holding costs are in 0, ∞
▪ Acquisition costs are uniform for all edges.
▪ Number of time steps is a small constant.
𝜖
and 𝑉
1−𝜖 .
Concluding Remarks
▪ Better algorithms for the uniform cost case.
▪ Greedy?
▪ Hardness/upper-bound for intersections of matroids.
▪ Matchings in bipartite graphs.