W.-S. Luk of Fudan University gave a lecture on feasibility problems. He discussed the feasible flow problem and feasible potential problem. For feasible flow, the problem has a solution if and only if the demand is less than or equal to the capacity of every cut. For feasible potential, the problem has a solution if and only if the upper span is greater than or equal to zero for every cycle. Both problems can be solved efficiently using network flow algorithms or by finding a negative cycle. He also provided examples and remarks on reducing problems.

1. Lecture 2: Feasibility Problems
Wai-Shing Luk (陆伟成)
Fudan University
2012 年 8 月 11 日
W.-S. Luk (Fudan Univ.) Lecture 2: Feasibility Problems 2012 年 8 月 11 日 1/6

2. Feasibility Problems
Feasible Flow Problem: Feasible Potential Problem:
Find a flow x such that: Find a potential u such that:
c − ≤ x ≤ c + (element-wise) d − ≤ y ≤ d + (element-wise)
AT · x = b, b(V ) = 0 A·u =y
Can be solved efficiently using: Can be solved efficiently using:
Painted network algorithm Bellman-Ford algorithm
Minty’s algorithm If no feasible solution, return
If no feasible solution, return a “negative cycle”.
a “negative cut”.
W.-S. Luk (Fudan Univ.) Lecture 2: Feasibility Problems 2012 年 8 月 11 日 2/6

3. Examples
Genome-scale reaction network Timing constraints
[?] (primal) (co-domain)
A: Stoichiometric matrix AT : incident matrix of
S timing constraint graph
x: reactions between u: arrival time of clock
metabolites/proteins y : clock skew
c− ≤x ≤ c +: constraints d − ≤ y ≤ d + : setup- and
on reaction rates hold-time constraints
W.-S. Luk (Fudan Univ.) Lecture 2: Feasibility Problems 2012 年 8 月 11 日 3/6

4. Feasibility Flow Problem
Theorem
The problem has a feasible solution if and only if b(S) ≤ c + (Q) for all
cuts Q = [S, S ] where c + (Q) = upper capacity [?, p. 56].
Proof.
(if part) Let q = A · k be a cut vector (oriented) of Q. Then
c− ≤ x ≤ c+
=> q T x ≤ c + (Q)
=> (A · k)T x ≤ c + (Q)
=> k T AT x ≤ c + (Q)
=> k T b ≤ c + (Q)
=> b(S) ≤ c + (Q)
W.-S. Luk (Fudan Univ.) Lecture 2: Feasibility Problems 2012 年 8 月 11 日 4/6

5. Feasibility Flow Problem
Theorem
The problem has a feasible solution if and only if b(S) ≤ c + (Q) for all
cuts Q = [S, S ] where c + (Q) = upper capacity [?, p. 56].
Proof.
(if part) Let q = A · k be a cut vector (oriented) of Q. Then
c− ≤ x ≤ c+
=> q T x ≤ c + (Q)
=> (A · k)T x ≤ c + (Q)
=> k T AT x ≤ c + (Q)
=> k T b ≤ c + (Q)
=> b(S) ≤ c + (Q)
W.-S. Luk (Fudan Univ.) Lecture 2: Feasibility Problems 2012 年 8 月 11 日 4/6

6. Feasibility Flow Problem
Theorem
The problem has a feasible solution if and only if b(S) ≤ c + (Q) for all
cuts Q = [S, S ] where c + (Q) = upper capacity [?, p. 56].
Proof.
(if part) Let q = A · k be a cut vector (oriented) of Q. Then
c− ≤ x ≤ c+
=> q T x ≤ c + (Q)
=> (A · k)T x ≤ c + (Q)
=> k T AT x ≤ c + (Q)
=> k T b ≤ c + (Q)
=> b(S) ≤ c + (Q)
W.-S. Luk (Fudan Univ.) Lecture 2: Feasibility Problems 2012 年 8 月 11 日 4/6

7. Feasibility Flow Problem
Theorem
The problem has a feasible solution if and only if b(S) ≤ c + (Q) for all
cuts Q = [S, S ] where c + (Q) = upper capacity [?, p. 56].
Proof.
(if part) Let q = A · k be a cut vector (oriented) of Q. Then
c− ≤ x ≤ c+
=> q T x ≤ c + (Q)
=> (A · k)T x ≤ c + (Q)
=> k T AT x ≤ c + (Q)
=> k T b ≤ c + (Q)
=> b(S) ≤ c + (Q)
W.-S. Luk (Fudan Univ.) Lecture 2: Feasibility Problems 2012 年 8 月 11 日 4/6

8. Feasibility Flow Problem
Theorem
The problem has a feasible solution if and only if b(S) ≤ c + (Q) for all
cuts Q = [S, S ] where c + (Q) = upper capacity [?, p. 56].
Proof.
(if part) Let q = A · k be a cut vector (oriented) of Q. Then
c− ≤ x ≤ c+
=> q T x ≤ c + (Q)
=> (A · k)T x ≤ c + (Q)
=> k T AT x ≤ c + (Q)
=> k T b ≤ c + (Q)
=> b(S) ≤ c + (Q)
W.-S. Luk (Fudan Univ.) Lecture 2: Feasibility Problems 2012 年 8 月 11 日 4/6

9. Feasibility Flow Problem
Theorem
The problem has a feasible solution if and only if b(S) ≤ c + (Q) for all
cuts Q = [S, S ] where c + (Q) = upper capacity [?, p. 56].
Proof.
(if part) Let q = A · k be a cut vector (oriented) of Q. Then
c− ≤ x ≤ c+
=> q T x ≤ c + (Q)
=> (A · k)T x ≤ c + (Q)
=> k T AT x ≤ c + (Q)
=> k T b ≤ c + (Q)
=> b(S) ≤ c + (Q)
W.-S. Luk (Fudan Univ.) Lecture 2: Feasibility Problems 2012 年 8 月 11 日 4/6

10. Feasibility Potential Problem
Theorem
The problem has a feasible solution if and only if d + (P) ≥ 0 for all cycles
P where d + (P) = upper span [?, p. ??].
Proof.
(if part) Let τ be a path indicator vector (oriented) of P. Then
d− ≤ y ≤ d+
=> τ T y ≤ d + (P)
=> τ T (A · u) ≤ d + (P)
=> (AT τ )T u ≤ d + (P)
=> (∂P)T u ≤ d + (P)
=> 0 ≤ d + (P)
W.-S. Luk (Fudan Univ.) Lecture 2: Feasibility Problems 2012 年 8 月 11 日 5/6

11. Feasibility Potential Problem
Theorem
The problem has a feasible solution if and only if d + (P) ≥ 0 for all cycles
P where d + (P) = upper span [?, p. ??].
Proof.
(if part) Let τ be a path indicator vector (oriented) of P. Then
d− ≤ y ≤ d+
=> τ T y ≤ d + (P)
=> τ T (A · u) ≤ d + (P)
=> (AT τ )T u ≤ d + (P)
=> (∂P)T u ≤ d + (P)
=> 0 ≤ d + (P)
W.-S. Luk (Fudan Univ.) Lecture 2: Feasibility Problems 2012 年 8 月 11 日 5/6

12. Feasibility Potential Problem
Theorem
The problem has a feasible solution if and only if d + (P) ≥ 0 for all cycles
P where d + (P) = upper span [?, p. ??].
Proof.
(if part) Let τ be a path indicator vector (oriented) of P. Then
d− ≤ y ≤ d+
=> τ T y ≤ d + (P)
=> τ T (A · u) ≤ d + (P)
=> (AT τ )T u ≤ d + (P)
=> (∂P)T u ≤ d + (P)
=> 0 ≤ d + (P)
W.-S. Luk (Fudan Univ.) Lecture 2: Feasibility Problems 2012 年 8 月 11 日 5/6

13. Feasibility Potential Problem
Theorem
The problem has a feasible solution if and only if d + (P) ≥ 0 for all cycles
P where d + (P) = upper span [?, p. ??].
Proof.
(if part) Let τ be a path indicator vector (oriented) of P. Then
d− ≤ y ≤ d+
=> τ T y ≤ d + (P)
=> τ T (A · u) ≤ d + (P)
=> (AT τ )T u ≤ d + (P)
=> (∂P)T u ≤ d + (P)
=> 0 ≤ d + (P)
W.-S. Luk (Fudan Univ.) Lecture 2: Feasibility Problems 2012 年 8 月 11 日 5/6

14. Feasibility Potential Problem
Theorem
The problem has a feasible solution if and only if d + (P) ≥ 0 for all cycles
P where d + (P) = upper span [?, p. ??].
Proof.
(if part) Let τ be a path indicator vector (oriented) of P. Then
d− ≤ y ≤ d+
=> τ T y ≤ d + (P)
=> τ T (A · u) ≤ d + (P)
=> (AT τ )T u ≤ d + (P)
=> (∂P)T u ≤ d + (P)
=> 0 ≤ d + (P)
W.-S. Luk (Fudan Univ.) Lecture 2: Feasibility Problems 2012 年 8 月 11 日 5/6

15. Feasibility Potential Problem
Theorem
The problem has a feasible solution if and only if d + (P) ≥ 0 for all cycles
P where d + (P) = upper span [?, p. ??].
Proof.
(if part) Let τ be a path indicator vector (oriented) of P. Then
d− ≤ y ≤ d+
=> τ T y ≤ d + (P)
=> τ T (A · u) ≤ d + (P)
=> (AT τ )T u ≤ d + (P)
=> (∂P)T u ≤ d + (P)
=> 0 ≤ d + (P)
W.-S. Luk (Fudan Univ.) Lecture 2: Feasibility Problems 2012 年 8 月 11 日 5/6

16. Remarks
The only-if part of the proof is constructive. It can be done by
constructing an algorithm to obtain the feasible solution.
d + could be infinity or zero, etc.
By adding reverse edges for every edges, the feasible potential
problem can reduce to an elementary one:
Find a potential u such that
y ≤ d, (element-wise)
Au=y
where A is the incident matrix of the modified network.
W.-S. Luk (Fudan Univ.) Lecture 2: Feasibility Problems 2012 年 8 月 11 日 6/6